Skip to main navigation menu Skip to main content Skip to site footer
Logo Revista Latinoamericana de Investigación en Matemática Educativa

Artículos

Vol. 17 No. 1 (2014): Marzo

THE TEACHING AND LEARNING OF RATIO, PROPORTION AND PROPORTIONALITY: THE STATE OF THE ART

  • Gilberto Obando Z
  • Carlos Eduardo Vasco
  • Luis Carlos Arboleda
DOI
https://doi.org/10.12802/relime.13.1713
Submitted
July 12, 2023
Published
2023-07-13

Abstract

Ratio, proportion and proportionality are a field that has been the object of a wide research in the last fifty years. Recent evaluations show that, for most students, it is still difficult to learn these objects of knowledge, which is an actual indicator that more didactic research should be made in order to come up with new ways to understand these problem areas. This way, it could have a great impact on the educational system. In this article, we revise and we make comments on some of the recent research on ratio, proportion and proportionality. According to an analysis perspective, we classify them into three groups: the cognitive and the epistemic perspective, and the semiotic-anthropological one. We finally pose some problems of research for the conclusion.

 

References

  1. Adjiage, R. (1999). L’expression des nombres rationnels et leur enseignement initial (Thèse de Doctorat inédite). Université Louis Pasteur, Strasbourg, France.
  2. Adjiage, R. (2005). Diversité et invariants des problèmes mettant en jeu des rapports. Annales de Didactique et de Sciences Cognitives, 10, 95-129.
  3. Adjiage, R. (2007). Rationnels et proportionnalité: complexité et enseignement au début du collège. Pétit X, 74, 5-33.
  4. Adjiage, R. & Pluvinage, F. (2007). An experiment in teaching ratio and proportion. Educational Studies in Mathematics, 65(2), 149-175. doi: 10.1007/s10649-006-9049-x
  5. Behr, M., Harel, G., & Post, T. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning. New York, NY: Macmillan Publishing Company.
  6. Behr, M., Khoury, H., Harel, G., Post, T., & Lesh, R. (1997). Conceptual units analysis of preservice elementary school teachers’ strategies on a Rational-Number-as-Operator task. Journal for Research in Mathematics Education, 28(1), 48-69.
  7. Ben-Chaim, D., Keret, Y., & Ilany, B.-S. (2012). Ratio and Proportion: Research and teaching in mathematics teachers’ education. doi: 10.1007/978-94-6091-784-4.
  8. Bolea, P., Bosch, M., & Gascon, J. (2001). La transposición didáctica de organizaciones matemáticas en procesos de algebrización: el caso de la proporcionalidad. Recherches en Didactique des Mathématiques, 21(3), 247-304.
  9. Bosch, M. (1994). La dimensión ostensiva de la actividad matemática. El caso de la proporcionalidad (Tesis de Doctorado no publicada). Univesitat Autònoma de Barcelona, Barcelona, España.
  10. Bosch, M., García, F., Gascón, J., & Higueras, L. (2006). La modelización matemática y el problema de la articulación de la matemática escolar. Una propuesta desde la Teoría Antropológica de lo Didáctico. Educación Matemática, 18(2), 37-74.
  11. Botero, O. (2006). Conceptualizacion del pensamiento multiplicativo en niños de segundo y tercero de educación básica a partir del estudio de la variación (Tesis de Maestría no publicada). Universidad de Antioquia, Medellín, Colombia.
  12. Brousseau, G., Brousseau, N., & Warfield, V. (2004). Rationals and decimals as required in the school curriculum. Part 1: Rationals as measurement. The Journal of Mathematical Behavior, 23(1), 1-20. doi: 10.1016/j.jmathb.2003.12.001
  13. Brousseau, G., Brousseau, N., & Warfield, V. (2007). Rationals and decimals as required in the school curriculum. Part 2: From rationals to decimals. The Journal of Mathematical Behavior, 26(4), 281-300. doi: 10.1016/j.jmathb.2007.09.001
  14. Brousseau, G., Brousseau, N., & Warfield, V. (2008). Rationals and decimals as required in the school curriculum. Part 3: Rationals and decimals as linear functions. The Journal of Mathematical Behavior, 27(3), 153-176. doi: 10.1016/j.jmathb.2008.07.006
  15. Bryant, P. & Nunes, T. (2009). Multiplicative reasoning and mathematics achievement. In M. Tzekaki, M. Kaldrimidou &, H. Sakonidi (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 217-224). Thessaloniki, Greece: PME.
  16. Comin, E. (2002). L’enseignement de la proporcionalité à l’école et au Collège. Recherches en Didactique des Mathématiques, 22(2-3), 135-182.
  17. Confrey, J., & Carrejo, D. (2005). Ratio and fraction: The difference between epistemological complementarity and conflict. In D. Carraher & R. Nemirovsky (Eds.), Journal for Research in Mathematics Education. Monograph, (Vol. 13). Reston, VA: NCTM.
  18. Confrey, J., & Maloney, A. (2008). From fraction to rational number: Diagnostic e-learning trajectories approach (DELTA) to rational number reasoning. Recuperado desde: http://cse.edc.org/dr-k12/Docs/Confrey_Presentation.pdf
  19. Confrey, J., Maloney, A., Nguyen, K., Mojica, G., & Myers, M. (2009). Equipartitioning/splitting as a foundation of rational number reasoning using learning trajectories. In M. Tzekaki, M. Kaldrimidou &, H. Sakonidi (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 345-352). Thessaloniki, Greece: PME.
  20. Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2/3), 135-164. doi: 10.1007/BF01273661
  21. Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66-86.
  22. Cortina, J., & Zúñiga, C. (2008). Ratio-like comparisons as an alternative to equal-partitioning in supporting initial learning of fractions. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano & A. Sepúlveda (Eds.), Proceedings of the Joint Meeting of Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education and 30 th North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 385-392). Morelia, México: PME-NA.
  23. De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311-334. doi: 10.1023/A:1021205413749
  24. De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2007). The illusion of linearity: From analysis to improvement. doi: 10.1007/978-0-387-71164-5.
  25. De Bock, D., Verschaffel, L., & Janssens, D. (2002). The effects of different problem presentations and formulations on the illusion of linearity in secondary school students. Mathematical Thinking and Learning, 4(1), 65-89. doi: 10.1207/S15327833MTL0401_3
  26. Deliyiani, E., Panaoura, A., Elia, I., & Gagatsis, A. (2008). Structural model for fraction understanding related to representations and problem solving. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano & A. Sepúlveda (Eds.), Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education and 30th North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 399-406). Morelia, México: PME-NA.
  27. Fandiño, M. (2009). Las Fracciones: aspectos conceptuales y didácticos. Bogotá, Colombia: Editorial Magisterio.
  28. Fernández, A.; & Puig, L. (2002). Análisis fenomenológico de los conceptos de razón, proporción y proporcionalidad. Gaceta de la Real Sociedad Matemática Española, 5(2), 397-416.
  29. Fernández, C.; & Llinares, S. (2010). Relaciones entre el pensamiento aditivo y mutiplicativo en estudiantes de educación primaria. El caso de la construcción de la idea de razón. Horizontes Educacionales, 15(1), 11-22.
  30. Fernández, C.; & Llinares, S. (2012). Características del desarrollo del razonamiento proporcional en la educación primaria y secundaria. Enseñanza de las Ciencias, 30(1), 129-142.
  31. Fernández, C., Llinares, S., Dooren, W. V., De Bock, D., & Verschaffel, L. (2010). How do proportional and additive methods develop along primary and secondary school? In M. Pinto & T. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 353-360). Belo Horizonte, Brazil: PME.
  32. Fernández, C., Llinares, S., van Dooren, W., De Bock, D., & Verschaffel, L. (2011). Effect of number structure and nature of quantities on secondary school students proportional reasoning. Studia Psychologica, 53(1), 69-82.
  33. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Hingham, MA: Kluwer Academic Publishers.
  34. García, F. (2005). La modelización como herramienta de articulación de la matemática escolar. De la proporcionalidad a las relaciones funcionales (Tesis de Doctorado no publicada). Universidad de Jaén, Jaén, España.
  35. García, F., Gascón, J., Higueras, L., & Bosch, M. (2006). Mathematical modelling as a tool for the connection of school mathematics. ZDM Mathematics Education, 38(3), 226-246. doi: 10.1007/BF02652807
  36. García, G., & Serrano, C. (1999). La comprensión de la proporcionalidad: una perspectiva social y cultural. Bogotá, Colombia: Asociación Colombiana de Matemática Educativa.
  37. Gómez, B. (2007). La razón en semejanza: El caso del perrito. En E. Castro & J. L. Lupiáñez (Eds.), Investigaciones en educación matemática: Pensamiento numérico (pp. 237-257). Granada, España: Editorial universitaria de Granada.
  38. Gómez, B., & Contreras, M. (2009). Sobre el análisis de los problemas multiplicativos relacionados con la división de fracciones. PNA, 3(4), 169-183.
  39. Guacaneme, E. (2002). Una mirada al tratamiento de la proporcionalidad en textos escolares de matemáticas. Revista EMA, 7(1), 3-42.
  40. Harel, G., Behr, M., Lesh, R., & Post, T. (1994). Invariance of ratio: the case of children’s anticipatory scheme for constancy of taste. Journal for Research in Mathematics Education, 25(4), 324-345.
  41. Hart, K. (1988). Ratio and proportion. In J. Hiebert & M. Behr (Eds.), Number concepts and operationsinthemiddlegrades (Vol.2,pp.198-219).Reston,VA:LawrenceErlbaumAssociates.
  42. Hersant, M. (2001). Interactions didactiques et pratiques d’enseignement, le cas de la proportionnalité au collège (Thèse de Doctorat inédite). Université Paris 7 - Denis Diderot, Paris, France.
  43. Hersant, M. (2005). La proportionnalité dans l’enseignement obligatoire en France, d’hier à aujourd’hui. Revue Repères IREM, (59), 5-41.
  44. Hersant, M., & Perrin-Glorian, M.-J. (2005). Characterization of an ordinary teaching practice with the help of the Theory of Didactic Situations. Educational Studies in Mathematics, 59(1/3), 113-151. doi: 10.1007/0-387-30451-7_5
  45. Hodgen, J., Kuchemann, D., Brown, M., & Coe, R. (2010). Multiplicative reasoning, ratio and decimals: a 30-year comparison of lower secondary students’ understandings. In M. Pinto & T. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 89-96). Belo Horizonte, Brazil: PME.
  46. Howe, C., Nunes, T., & Bryant, P. (2010). Intensive quantities: Why they matter to developmental research. British Journal of Developmental Psychology, 28(2), 307-329. doi: 10.1348/ 026151009X410362
  47. Howe, C., Nunes, T., & Bryant, P. (2011). Rational number and proportional reasoning: Using intensive quantities to promote achievement in mathematics and science. International Journal of Science and Mathematics Education, 9(2), 391-417. doi: 10.1007/s10763-010-9249-9
  48. Iannece, D., Mellone, M., & Tortora, R. (2010). Early multiplicative thought: a kindergarten path. In M. Pinto & T. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 121-127). Belo Horizonte, Brazil: PME.
  49. Kaput, J., & West, M. (1994). Missing-value poportional reasoning problems: factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 235-290). Albany, NY: State University of New York Press.
  50. Karplus, R., Pulos, S. & Stage, E. (1983). Proportional reasoning of early adolecents. In R. Lesh & M. Landau (Eds.), Adquisition of mathematics concepts and processes (pp. 45-90). New York, NY: Academic Press.
  51. Kieren, T. (1980). The rational number construct: Its elements and mechanisms. In T. E. Kieren (Ed.), Recent research on number learning (pp. 32-55). Columbus, OH: ERIC Publications; Reports - Research.
  52. Kieren, T. (1988). Personal knowledge of rational numbers: Its intuitive and formal development. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (Vol. 2, pp. 162-181). Reston, VA: Lawrence Erlbaum Associates.
  53. Koellner-Clark, K., & Lesh, R. (2003). Whodunit? Exploring proportional reasoning through the footprint problem. School Science & Mathematics, 103(2), 92-98. doi: 10.1111/ j.1949-8594.2003.tb18224.x
  54. Lachance, A., & Confrey, J. (2002). Helping students build a path of understanding from ratio and proportion to decimal notation. Journal of Mathematical Behavior, 20, 503-526. doi: 10.1016/ S0732-3123(02)00087-1
  55. Lamon, S. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In H. Guershon & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 89-120). Albany, NY: State University of New York Press
  56. Lamon, S. (2007). Rational number and proportional reasoning. Toward a theoretical framework for research. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (Vol. 1, pp. 629-667). New York, NY: Information Age Pub Inc.
  57. Lamon, S. (2012). Teaching fractions and ratios for understanding: essential content knowledge and instructional strategies for teachers (3ra Ed.). New York, NY: Taylor & Francis.
  58. Lesh, R., Post, T., & Behr, M. (1988). Proportional Reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (Vol. 2, pp. 93-117). Reston, VA: Lawrence Erlbaum associates.
  59. Lundberg, A. (2011). Proportion in mathematics textbooks in upper secondary school. En M. Pytlak, T. Rowland, & E. Swoboda (eds), Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 336-345). Rzeszów, Poland: University of Rzeszów.
  60. Mamede, E. (2010). Early years mathematics – the case of fractions. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), Proceedings of the sixth Congress of European Research in Mathematics Education (pp. 2607-2616). Lyon, France: Institut National de Recherche Pédagogique.
  61. Mamede, E., & Nunes, T. (2008). Building on children’s informal knowledge in the teaching of fractions. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano & A. Sepúlveda (Eds.), Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education and 30th North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 345-352). Morelia, México: Cinvestav-UMSNH.
  62. Martin, M., Mullis, I., & Foy, P. (2008). TIMSS 2007 International mathematics report: Findings form IEA’s trends in international mathematics and science study et the fourth and eight grades. Boston, MA: TIMSS & PIRLS International Study Center.
  63. Modestou, M., Elia, I., Gagatsis, A., & Spanoudis, G. (2008). Behind the scenes of pseudo-proportionality. International Journal of Mathematical Education in Science and Technology, 39(3), 313-324. doi: 10.1080/00207390701691541
  64. Modestou, M., & Gagatsis, A. (2007). Students’ improper proportional reasoning: A result of the epistemological obstacle of “linearity”. Educational Psychology, 27(1), 75-92. doi: 10.1080/ 01443410601061462
  65. Modestou, M., & Gagatsis, A. (2009). Proportional reasoning: the strategies behind the percentages. Acta Didactica Universitatis Comenianae, 9, 25-40.
  66. Modestou, M., & Gagatsis, A. (2010). Cognitive and metacognitive aspects of proportional reasoning. Mathematical Thinking and Learning, 12, 36-53. doi: 10.1080/10986060903465822
  67. Mullis, I., Martin, M., Olson, J., Berger, D., Milne, D., & Stanco, G. (Eds.). (2008). TIMSS 2007 encyclopedia: A guide to mathematics and science education around the world (Vol. 1 A-L, Vol. 2 M-Z). Boston, MA: TIMSS & PIRLS International Study Center.
  68. Noelting, G. (1980). The development of proportional reasoning and the ratio concept part I -differentiation of stages. Educational Studies in Mathematics, 11(2), 217-253. doi: 10.1007/ BF00304357
  69. Nunes, T. (2010). Continuities and discontinuities between informal and scientific mathematical thinking: insights for education. In M. Pinto & T. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 328-332). Belo Horizonte, Brazil: PME.
  70. Nunes, T., & Bryant, P. (2008). Rational numbers and intensive quantities: Challenges and insights to pupils´ implicit knowledge. Anales de Psicología, 24(2), 262-270.
  71. Nunes, T., Desli, D., & Bell, D. (2003). The development of children’s understanding of intensive quantities. International Journal of Educational Research, 7, 651-657. doi: 10.1016/ j.ijer.2004.10.002
  72. Obando, G. (2003). La enseñanza de los números racionales a partir de la relación parte-todo. Revista EMA, 8(2), 157-182.
  73. Ohlsson, S. (1988). Mathematical meaning an applicational meaning in the semantic of fractions and related concepts. In J. Hiebert & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades (Vol. 2, pp. 53-92). Reston, VA: Lawrence Erlbaum Associates.
  74. Pantziarra, M., & Pitta-Pantazi, D. (2005). The development of informal proportional thinking in primary school. En Marianna Bosch (ed), Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (pp 363-372). Sant Feliu de Guíxols, Spain: Universitat Ramon Llull.
  75. Piaget, J., & Inhelder, B. (1958). The growth of logical thinking from childhood to adolescence . (A. Parson, Trad.). United Stated: Basic Book, Inc.
  76. Pitta-Pantazi, D., & Christou, C. (2009). The structure of prospective kindergarten teachers’ proportional reasoning. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), Proceedings of the sixth Congress of European Research in Mathematics Education (pp. 2637-2646). Lyon, France: Institut National de Recherche Pédagogique.
  77. Ponte, J. P., & Marques, S. (2007). Proportion in school mathematics textbooks: a comparative study. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), Proceedings of the fifth Congress of European Research in Mathematics Education (pp. 2443-2452). Larnaca, Cyprus: University of Cyprus.
  78. Pontón, T. (2012). La comprensión de enunciados de problemas en la enseñanza y el aprendizaje inicial de los números racionales (Tesis de doctorado no publicada). Universidad del Valle, Cali, Colombia.
  79. Pontón, T. (2008). Una propuesta multirregistro para la conceptualización inicial de las fracciones (Tesis de Maestría no publicada). Universidad del Valle, Cali, Colombia.
  80. Pulos, S., & Tourniaire, F. (1985). Proportional Reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204. doi: 10.1007/BF02400937
  81. Rojas, P., Romero, J., Mora, L. O., Bonilla, M., Rodríguez, J., & Castillo, E. (2011). La multiplicación como cambio de unidad: estrategias para promover su aprendizaje. Bogota, Colombia: Fondo de publicaciones Universidad Distrital Francisco Jose de Caldas.
  82. Roth, W.-M., & Radford, L. (2011). A Cultural-Historical perspective on mathematics teaching and learning. Rotterdam, The Netherlands: Sense Publishers.
  83. Schliemann, A., Carraher, D., & Brizuela, B. (2000, Junio de 2010). From quantities to ratio, functions, and algebraic relations. Recuperado desde: http://ase.tufts.edu/education/ earlyalgebra/publications/2000-earlier/quantitiesRatios.pdf
  84. Schwartz, J. (1988). Intensive quantity and referent transforming arithmetic. In J. Hierbert & M. Behr (Eds.), Number concepts and operations in the middle grades (Vol. 2, pp. 41-52). Reston, VA: Lawrence Erlbaum Associates.
  85. Spinillo, A., & Bryant, P. (1991). Children’s Proportional Judgments: The Importance of “Half”. Child Development, 62(3), 427-440. doi: 10.1111/j.1467-8624.1991.tb01542.x
  86. Spinillo, A., & Bryant, P. (1999). Proportional reasoning in young children: part–part comparisons about continuous and discontinuous quantity. Mathematical Cognition, 5(2), 181-197. doi: 10.1080/135467999387298
  87. Steffe, L. (1994). Children’s Multiplying Schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3-40). New York, NY: State University of New York Press.
  88. Steffe, L., & Olive, J. (Eds.) (2010). Children’s Fractional Knowledge. doi:10.1007/978-1-4419-0591-8
  89. Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. Recuperado desde: http://ed.sc.edu/ite/nctm2003/FracsMultRsng.pdf
  90. TIMSS. (2009). TIMSS 2007 user guide for the international database. Boston, MA: TIMSS & PIRLS International Study Center.
  91. Tourniaire, F. (1986). Proportions in Elementary School. Educational Studies in Mathematics, 17(4), 401-412. doi: 10.1007/BF00311327
  92. Van Dooren, W., & De Bock, D. (2008). Pupils’ reasoning on proportionality: solving versus classifying missing-value word problems. In O. Figueras, J. Cortina, S. Alatorre, T. Rojano & A. Sepúlveda (Eds.), Proceedings of the 32nd Conference of the International Group for the Psychology of Mathematics Education and 30th North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 369-376). Morelia, México: Cinvestav-UMSNH.
  93. Van Dooren, W., De Bock, D., Gillard, E., & Verschaffel, L. (2009). Add? Or multiply? A study on the development of primary school students’ proportional reasoning skills. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 281-288). Thessaloniki, Greece: PME.
  94. Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralization. Cognition and Instruction, 23(1), 57-86. doi: 10.1207/s1532690xci2301_3
  95. Van Dooren, W., De Bock, D., Hessels, A., & Verschaffel, L. (2004). Students’ overreliance on proportionality: evidence from primary school pupils solving arithmetic word problem. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 385-392). Bergen, Norway: PME.
  96. Van Galen, F., Feijs, E., Figueiredo, N., Gravemeijer, K., van Herpen, E., & Keijzer, R. (2008). Fractions, percentages, decimals and proportions: A learning-teaching trajectory for grade 4, 5 and 6 (C. Frink, Trad.). Rotterdam, The Netherlands: Sense Publishers.
  97. Vasco, C. (1989). Dos nuevos grupos piagetianos en la lógica elemental. Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales, 17(64), 29-39.
  98. Vergnaud, G. (1988). Multiplicative Structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (Vol. 2, pp. 141-161). Reston, VA: Lawrence Erlbaum associates.
  99. Vergnaud, G. (1991). El niño, las matematicas y la realidad: Problemas de la enseñanza de las matemáticas en la escuela primaria (L. O. Segura, Trad.). México, D.F.: Trillas.
  100. Vergnaud, G. (1994). Multiplicative conceptual field: what and why? In H. Guershon & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 41-60). New York, NY: State University of New York Press.

Downloads

Download data is not yet available.

Similar Articles

<< < 10 11 12 13 14 15 16 17 18 > >> 

You may also start an advanced similarity search for this article.