Journal of Research in Science, Mathematics and Technology Education

Didactic proposal to overcome the difficulties in the learning of Area and Volume in Spanish Primary Education students

Raquel Fernández-Cézar 1 * ,
Melody García-Moya 2,
Paloma Ocaña Aranda 2,
Ignacio Rieiro-Marín 2
1 Castilla-La Mancha University2 Castilla-La Mancha University * Corresponding Author
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Journal of Research in Science, Mathematics and Technology Education, Volume 2, Issue 3, September 2019, pp. 151-178
OPEN ACCESS VIEWS: 639 DOWNLOADS: 475 Publication date: 15 Sep 2019
ABSTRACT
This work presents a didactic proposal for the learning and measure of surface area and body volume. This proposal is framed in the Anthropological Theory of the Didactic (ATD), based on the recognized errors in  the learning of these magnitudes and considering their connection with the typified learning difficulties or epistemological obstacles. The proposal was developed as a didactic sequence, including the tasks from didactic  situations (as considered by Brousseau) and with a cross-curricular perspective in relation to the social-systemic structure (ATD), without restricting them in any didactic unit. The praxeology was structured in accordance with  the approaches of the ATD, and the didactic methodology was based on the definition of the errors, which followed the phases of development of the usual models in the learning of Geometry.  These phases were defined under a  generic framework influenced by the developed Van Hiele model for the learning of Geometry. The tasks that composed the didactic sequences were created "ad-hoc” or extracted from adequate sources throughout the Spanish  curriculum of Primary Education. The proposal was designed to be applied in the 5th Primary Education grade. The collection of evidences on the students learning regarding the area and the volume after the implementation of the proposal constitutes the natural next step of this project.
KEYWORDS
Anth ropological Theory of the Didactic, Theory of Didactic Situations, Errors, Learning difficulties, Area, Volume, Primary Education.
CITATION (APA)
Fernández-Cézar, R., García-Moya, M., Ocaña Aranda, P., & Rieiro-Marín, I. (2019). Didactic proposal to overcome the difficulties in the learning of Area and Volume in Spanish Primary Education students. Journal of Research in Science, Mathematics and Technology Education, 2(3), 151-178. https://doi.org/10.31756/jrsmte.232
REFERENCES
  1. Artigue, M. (2014). Potentialities and limitations of the theory of didactic situations for addressing the teaching and learning of mathematics at university level. Research in Mathematics Education, 16(2), 135-138. https://doi.org/10.1080/14794802.2014.918348
  2. Bachellard, G. (2000). La formación del espíritu científico. Contribución a un psicoanálisis del conocimiento objetivo [The formation of the scientific spirit. Contribution to psychoanalysis of objective knowledge]. México, México: Siglo XXI editores, S.A.
  3. Bachellard, G. (2004). Estudios [Studies]. Bilbao, España: Amorrortu Editores.
  4. Brousseau, G. (July 1976). Les obstacles épistémologiques et les problems en mathématiques problems [Epistemological barriers and problems in mathematics]. En J. Vanhamme & W. Vanhamme (Eds.), La problématique l'enseignement des mathématiques [The problem and the teaching of mathematics]. Simposio llevado a cabo en el XXVIII Commission for the Study and Improvement of Mathematics Teaching, Louvain la Neuve, Bélgica.
  5. Brousseau, G. (1980). Problèmes de l’enseignement des décimaux [Problems of teaching decimals]. Recherches en Didactique des Mathématiques, 1(1), 11-59.
  6. Brousseau, G. (1981). Problèmes de didactique des décimaux [Problems of didactics of decimals]. Recherches en Didactique des Mathématiques, 2(1), 37-127.
  7. Brousseau, G. (1983). Les obstacles épistémologiques et les problemes en mathematiques [Epistemological obstacles and problems in mathematics]. Recherches en Didactique des Mathematiques, 4(2), 165-198.
  8. Brousseau, G. (1986). Fondements et méthodes de la didactique des mathematiques [Foundations and methods of mathematics didactics]. Recherches en Didactique des Mathématiques, 7(2), 33-115.
  9. Brousseau, G. (1989a). Les obstacles épistémologiques et la didactique des mathématiques [Epistemological obstacles and the didactics of mathematics]. En N, Bednarz & C, Garnier (Eds), Construction des saviors. Obstacles et conflits [Building Knowledge. Obstacles and Conflicts](pp.41-63). Montreal, Canada: CIRADE Les 'editions Arc Agency Inc.
  10. Brousseau, G. (1989b). Obstacles épistémologiques, conflits socio-cognitifs et ingenierie didactique [Epistemological obstacles, socio-cognitive con flicts and didacticengineering]. En N. Bednarz & C. Garnier (Eds.), Construction des savoirs. Obstacles et conflits [Building Knowledge. Obstacles and Conflicts] (pp. 277-285). Montreal, Canada: CIRADE Les 'editions Arc Agency Inc.
  11. Brousseau, G. (July 1996). L’Enseignantdans la Theorie des Situations Didactiques 1. Structure et fonctionnement du systèmedidactique [The Teacher in the Theory of Educational Situations 1. Structure and functioning of the didactic system]. In R. Noirfalise & M.J, Perrin-Gloriam (Eds.). Actes de la VIII° Ecoled’été dedidactique des mathématiques, IREM de Clermont-Ferrand, Clermont-Ferrand, France.
  12. Brousseau, G. (October - November1999). Educación y Didáctica de las matemáticas [Education and Mathematics Didactics]. V Congreso Nacional de Investigación Educativa. Aguas calientes, Mexico.
  13. Brousseau, G. (2000). Educación y didáctica de las matemáticas [Education and didactics of mathematics]. Educación Matemática, 12(1), 5-38.
  14. Brousseau, G. (2002). Theory of didactical situations in mathematics: didactique des mathematiques, 1970-1990 [Theory of didactical situations in mathematics: didactics of mathematics, 1970-1990]. Boston, United Kingdom: Kluwer academic publishers.
  15. Brousseau, G. (2007). Iniciación al estudio de la teoría de las situaciones didácticas [Introduction to the study of the theory of didactic situation]. Buenos Aires, Argentina: Editorial Zorzal.
  16. Cabello Pardos, A.B., Rodriguez Cartagena, Mª. I., Garbayo Moreno, M.M., & Hidalgo Herrero, M. (2014). Dificultades, conflictos, errores y obstáculos epistemológicos en la identificación visual del resto de la división con números decimals [Difficulties, conflicts, errors and obstacles epistemological in the visual identification of the rest of the division with decimal numbers]. Épsilon, 31(87), 55-70.
  17. Camacho Machín, M., García Déniz, M., Hernández Domínguez, J., Noda Herrera, Mª. A., & Socas Robaina, M. M.
  18. (2003). La Medida en la Educación Primaria [Measurement in Primary Education]. In M.M, Socas Robaina,
  19. (Coord). Materriales curriculares. Colección Cuadernos de Aula [Curricular materials Classroom Notebooks Collection].La Laguna, Canarias: Consejería de Educación Cultura y Deportes del Gobierno de Canarias.
  20. Chamorro, C., & Belmonte, J.M. (1988). El problema de la medida. Didáctica de las magnitudes lineales [The measurement problem. Didactics of linear magnitudes]. Madrid, Spain: Editorial Síntesis S.A.
  21. Chevallard, Y. (1997). La transposición didáctica, del saber sabio al saber enseñado [The didactic transposition, of knowledge wise to the knowledge taught]. Buenos Aires, Argentina: AIQUE Grupo Editor.
  22. Chevallard, Y., Bosch, M., & Gascón, J. (1997). Estudiar matemáticas. El eslabón perdido entre la Enseñanza y el Aprendizaje [To study math. The missing link between Teaching and Learning].Barcelona, Spain: Horsori.
  23. Chevallard, Y., Bosch, M., & Kim, S. (February 2015). What is a theory according to the anthropological theory of the didactic? CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Czech Republic, Prague.
  24. Corberán Salvador, S., Gutiérrez Rodríguez, Á., Huerta Palau, M.P., Jaime Pastor, A., Bautista Margarit Garrigues, J., Peñas Pascual, J. A., & Ruiz Pérez, E. (1994). Diseño y evaluación de una propuesta curricular de aprendizaje de la geometría en enseñanza secundaria basada en el modelo de razonamiento de Van-Hiele [Design and evaluation of a curricular proposal for learning geometry in secondar yeducation based on the Van-Hiele reasoning model]. Madrid, Spain: Centro de publicaciones del Ministerio de Educación y ciencia.
  25. Crowley, M.L. (1987). The Van Hiele Model of the Development of Geometry Thought. Learning and Teaching Geometry K-12, 1, 1-16.
  26. D'Amore, B., & Fandiño Pinilla, M. I. (2007). Relaciones entre área y perímetro: convicciones de maestros y de estudiantes [Relationsbetweenarea and perimeter: convictions of teachers and students]. Revista latinoamericana de investigación en matemática educativa, 10(1), 39-68.
  27. Del Olmo, M. A., Moreno, M. F., & Gil, F. (1989). Superficie y volumen, ¿algo más que el trabajo con fórmulas? [Surface and volume, something more than working with formulas?]. Madrid, Spain: Editorial Síntesis S. A.
  28. Di Blasi Regner, M., Espro, F., Lois, A., & Milevicich, L., (December 2003). Dificultades y Errores: un estudio de caso [Difficulties and Errors: a case study]. Comunicación breve presentada en el II Congreso Internacional de Matemática Aplicada a la Ingeniería y Enseñanza de la Matemática en Ingeniería, Buenos Aires, Argentina.
  29. Díez, Á., Cañadas, M.C., Picado, M., Rico, L., & Castro, E. (2016). Magnitudes y su medida en el currículo de primaria en España (1945-2013) [Magnitudes and their measurement in the primary curriculum in Spain (1945-2013)]. Revista de Currículum y Formación de Profesorado, 20(1), 341-363.
  30. Engler, A., Gregorini, M.I., Müller, D., Vrancken, S., & Hecklein, M. (2004). Los errores en el aprendizaje de matemática [Mistakes in learning mathematics]. Premisa, 6(23), 23-32.
  31. Engler, A., Gregorini, M. I., Müller, D., & Vrancken, S. (2006). Los errores en el aprendizaje de la matemática [Errors in learning mathematics].Premisa, 23, 23-32.
  32. Fouz, F. (2005). Modelo de Van Hiele para la didáctica de la Geometría [Van Hiele model for the teaching of Geometry]. In R. Ibáñez & M. Macho. (Ed.), Un paseo por la Geometría (2004-2005) [A walk through Geometry (20042005)] (pp. 67-82). Bilbao, España: UPV-EHU.
  33. Font, V. (2003). Matemáticas y cosas. Una mirada desde la Educación Matemática [Mathematics and things. A look from Mathematics Education]. Boletín de la Asociación Matemática Venezolana, 10(2), 249-279.
  34. Freudenthal, H. (1983). Didactical Phenomenology of MathematicalStructures. Dordrecht, Netherlands: Reidel Publishing Company.
  35. Godino, J.D. (2004). Didáctica de la matemática para maestros [Didactics of mathematics for teachers]. Granada, Spain: Publicación realizada en el marco del Proyecto de Investigación y Desarrollo del Ministerio de Ciencia y Tecnología y Fondos FEDER, BSO2002-02452.
  36. Godino, J.D., Batanero, C., & Roa, R. (2002). Medida de magnitudes y su Didáctica para Maestros [Measurement of magnitudes and its Didactics for Teachers].Granada, Spain: Publicación realizada en el marco del Proyecto de Investigación y Desarrollo del Ministerio de Ciencia y Tecnología, BSO2002-02452.
  37. Heinze, A. (2005). Mistake-handling activities in the mathematics classroom. In, H. L, Chick & J. L, Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (pp. 105-112). Melbourne, Australia: PME.
  38. Jaime, A., & Gutiérrez, A. (July-August 1994). A model of test design to assess the Van Hiele levels. In J. da Ponte & J. Matos (Eds.), Proceedings of the International Conference for the Psychology of Mathematics Education. PMEXVIII, Lisboa, Portugal.
  39. Ley Orgánica 2/2006, de 3 de mayo, de Educación [Organic Law 2/2006, of May 3, on Education].
  40. Ley Orgánica 8/2013, de 9 de diciembre, para la mejora de la calidad educativa [Organic Law 8/2013, of December 9, for the improvement of the quality of education].
  41. Lovell, K. (1986). Desarrollo de los conceptos básicos matemáticos y científicos en los niños [Development of mathematical and scientific basic concepts in children]. Madrid, Spain: Ediciones Morata S.A.
  42. Martínez, A., & Rieiro Marín, I. (1993). La medida y la magnitude [The Measure and the Magnitude].Madrid, Spain: Granada, D.L.
  43. Özerem, A. (2012). Misconceptions in geometry and suggested Solutions for seventh grade students. Procedia - Social and Behavioral Sciences, 55, 720-729. https://doi.org/10.1016/j.sbspro.2012.09.557
  44. Piaget, J., Inhelder, B., & Szeminska, A. (1960). The Child's Conception of Geometry. New York, USA: Basic Books, Inc. Editores.
  45. Radatz, H. (1980). Students’ errors in the Mathematical Learning Process: A Survey. For the Learning of Mathematics, 1(1), 16-20.
  46. Rico, L. (1995). Errores y dificultadesen el aprendizaje de las Matemáticas [Errors and difficulties in learning Mathematics]. In J. Kilpatrick., J. Gómez & L. Rico. (Eds), Errores en el aprendizaje de la Matemática [Errors in the learning of Mathematics] (pp. 69 – 108). Mexico, Mexico: Grupo Editorial Iberoamérica.
  47. Ruston, N. (2014). Common errors in Mathematics. Research Matters, 1, 8-18.
  48. Tobón, T. S., Prieto, P. H. J., & Fraile, G. J. A. (2010). Secuencias didácticas. Aprendizaje y Evaluación de Competencias [Didactic Sequences. Learning and Evaluation]. Mexico: Pearson- Prentice Hall.
  49. Trigueros, M. (November 2016). Tendiendo Puentes Entre Teorias: ¿Qué Puede Lograrse? [Tending Bridges Between Theories: What Can Be Achieved? = Building Bridges between Theories: What Can Be Achieved?]. In M.B. Wood., E. E. Turner., M. Civil & J.A. Eli (Eds). Proceedings of the 38th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Tucson, AZ: The University of Arizona.
  50. Van Hiele, P.M. (1986). Structure and insign. A theory of Mathematics Education. London, UK: Academic Press.
  51. Vojkuvkova, I., (May-June 2012). The Van Hiele Model of Geometric Thinking. In J. Safrankova & J. Pavlu. (Eds),
  52. WDS'12 Proceedings of Contributed Papers: Part I- Mathematics and Computer Sciences, Matfyzpress, Prague.
  53. Wagman, H. G. (1982). The Child´s Conception of Area Measure. In M.K. Rosskopt. (Ed), Children´s Mathematical Concepts. Six Piagetian studies in mathematics education (pp.71-110). New York, USA: Teacher Collage Press.
  54. Zabala, A. (1998). La Práctica Educativa. Cómo enseñar [The Educational Practice How to teach]. Barcelona: Editorial Graó.
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