The Quantitative Meaning of Fractions for Mexican Sixth-Grade Students
Keywords:
Mathematics education, elementary school, fractions.Abstract
This study consisted in the application of questionnaires to 297 sixth-grade students from 13 different elementary schools in Mexico. Pupils were asked to identify the quantity represented by common fractions (e.g., 1/2, 1/4, 1/3, 3/4). Findings suggest that many students are finishing elementary school in Mexico with a very limited understanding of fractions. Some seem not to have developed basic quantitative concepts that would allow them to readily and correctly recognize the quantitative meaning of the most common fractions, including “1/2”. We explain the implications of such findings for students’ future mathematical development.
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References
Backhoff, E., Andrade, E., Sánchez, A., Peón, M. y Bouzas, A. (2006). El aprendizaje del español y las matemáticas en la educación básica en México: Sexto de primaria y tercero de secundaria. México: Instituto Nacional para la Evaluación de la Educación.
Behr, M., Harel, G., Post, T. y Lesh, R. (1992). Rational number, ratio, and proportion. En D. Grows (Ed.), Handbook of research in mathematics teaching and learning (pp. 296-333). Nueva York: Macmillan.
Behr, M., Lesh, R., Post, T. y Silver, E. (1983). Rational number concepts. En R. Lesh y M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes (pp. 91-125). Nueva York: Academic Press.
Behr, M., Wachsmuth, I., Thomas, R. P. y Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323-341.
Clarke, D. M. y Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72, 127-138.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, Holanda: Kluwer.
Gee, J. P. (2008). A sociocultural perspective on opportunity to learn Assessment, equity, and opportunity to learn. Nueva York: Cambridge University Press.
Gould, P. (2005). Year 6 students’ methods of comparing the size of fractions. En P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce y A. Roche (Eds.), Proceedings of the Annual Conference of the Mathematics Education Research Group of Australasia. (pp. 393-400). Sydney, Australia: MERGA.
Hannula, M. S. (2003). Locating fraction on a number line. En N. Pateman, B. Dougherty y J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education, 3, pp. 17-24. Honolulu: PME.
Hart, K. (1989). Fractions: Equivalence and addition. In K. Hart, D. C. Johnson, M. Brown, L. Dickson y R. Clarkson (Eds.), Childrens’ mathematical frameworks 8-13: A study of classroom teaching (pp. 46-75). Windsor, Berkshire, Reino Unido: NFER-NELSON.
Kieren, T. E. (1980). The rational number construct–Its elements and mechanisms. En T. E. Kieren (Ed.), Recent research on number learning (pp. 125-149). Columbus, OH: ERIC/SMEAC.
Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. En T. P. Carpenter, E. Fennema y T. A. Romberg (Eds.), Rational numbers: An integration of research. (pp. 50-84). Hillsdale, NJ: Lawrence Erlabaum.
Kilpatrick, J., Swafford, J. y Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. En F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629-667). Charlotte, NC: Information Age Pub.
PISA 2006 en México. (2007). México, D.F.: Instituto Nacional para la Evaluación de la Educación.
Secretaría de Educación Pública (2009). Programas de estudio 2009. Quinto grado. Educación básica. Primaria. México: Autor.
Sfard, A. (2001). Equilibrar algo desequilibrado: Los estándares del NCTM a la luz de las teorías del aprendizaje de las matemáticas. Revista EMA, 6, 95-140.
Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8, 359-371.
Simon, M. A., Tzur, R., Heinz, K. y Kinzel, M. (2004). Explicating a mechanism for conceptual learning: Elaborating the construct of reflective abstraction. Journal for Research in Mathematics Education, 35, 305-329.
Steffe, L. P. y Gale, J. (1995). Constructivism in education. Hillsdale, NJ: Lawrence Erlbaum.
Strauss, A. y Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage Publications.
Thompson, P. W. y Saldanha, L. A. (2003). Fractions and multiplicative reasoning. En J. Kilpatrick, G. Martin y D. Schifter (Eds.), Research companion to the principles and standards for school mathematics (pp. 95-113). Reston, VA: National Council of Teachers of Mathematics.
van Galen, F., Feijs, E., Figueiredo, N., Gravemeijer, K., van Herpen, E. y Keijzer, R. (2008). Fractions, percentages, decimals and proportions. A learning-teaching trajectory for grade 4, 5 and 6. Rotterdam, Países Bajos: Sense Publishers.
Behr, M., Harel, G., Post, T. y Lesh, R. (1992). Rational number, ratio, and proportion. En D. Grows (Ed.), Handbook of research in mathematics teaching and learning (pp. 296-333). Nueva York: Macmillan.
Behr, M., Lesh, R., Post, T. y Silver, E. (1983). Rational number concepts. En R. Lesh y M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes (pp. 91-125). Nueva York: Academic Press.
Behr, M., Wachsmuth, I., Thomas, R. P. y Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323-341.
Clarke, D. M. y Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72, 127-138.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, Holanda: Kluwer.
Gee, J. P. (2008). A sociocultural perspective on opportunity to learn Assessment, equity, and opportunity to learn. Nueva York: Cambridge University Press.
Gould, P. (2005). Year 6 students’ methods of comparing the size of fractions. En P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce y A. Roche (Eds.), Proceedings of the Annual Conference of the Mathematics Education Research Group of Australasia. (pp. 393-400). Sydney, Australia: MERGA.
Hannula, M. S. (2003). Locating fraction on a number line. En N. Pateman, B. Dougherty y J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education, 3, pp. 17-24. Honolulu: PME.
Hart, K. (1989). Fractions: Equivalence and addition. In K. Hart, D. C. Johnson, M. Brown, L. Dickson y R. Clarkson (Eds.), Childrens’ mathematical frameworks 8-13: A study of classroom teaching (pp. 46-75). Windsor, Berkshire, Reino Unido: NFER-NELSON.
Kieren, T. E. (1980). The rational number construct–Its elements and mechanisms. En T. E. Kieren (Ed.), Recent research on number learning (pp. 125-149). Columbus, OH: ERIC/SMEAC.
Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. En T. P. Carpenter, E. Fennema y T. A. Romberg (Eds.), Rational numbers: An integration of research. (pp. 50-84). Hillsdale, NJ: Lawrence Erlabaum.
Kilpatrick, J., Swafford, J. y Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. En F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629-667). Charlotte, NC: Information Age Pub.
PISA 2006 en México. (2007). México, D.F.: Instituto Nacional para la Evaluación de la Educación.
Secretaría de Educación Pública (2009). Programas de estudio 2009. Quinto grado. Educación básica. Primaria. México: Autor.
Sfard, A. (2001). Equilibrar algo desequilibrado: Los estándares del NCTM a la luz de las teorías del aprendizaje de las matemáticas. Revista EMA, 6, 95-140.
Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8, 359-371.
Simon, M. A., Tzur, R., Heinz, K. y Kinzel, M. (2004). Explicating a mechanism for conceptual learning: Elaborating the construct of reflective abstraction. Journal for Research in Mathematics Education, 35, 305-329.
Steffe, L. P. y Gale, J. (1995). Constructivism in education. Hillsdale, NJ: Lawrence Erlbaum.
Strauss, A. y Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage Publications.
Thompson, P. W. y Saldanha, L. A. (2003). Fractions and multiplicative reasoning. En J. Kilpatrick, G. Martin y D. Schifter (Eds.), Research companion to the principles and standards for school mathematics (pp. 95-113). Reston, VA: National Council of Teachers of Mathematics.
van Galen, F., Feijs, E., Figueiredo, N., Gravemeijer, K., van Herpen, E. y Keijzer, R. (2008). Fractions, percentages, decimals and proportions. A learning-teaching trajectory for grade 4, 5 and 6. Rotterdam, Países Bajos: Sense Publishers.
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Published
2012-05-01
