Egyptian Fractions and Representation Registers in the Construction of the Fraction Concept

Francisco Clemente; Josep Maria Fortuny

doi:10.29333/ejmste/10992

Full Text (PDF)

Francisco Clemente ¹ ^* , Josep Maria Fortuny ¹

More Detail

¹ Departament de Didàctica de la Matemàtica i de les Ciències Experimentals, Universitat Autònoma de Barcelona, SPAIN^* Corresponding Author

Abstract

In this work we will analyze the relation between registers of representation and the construction of the fraction concept. Ninety-six students from first year of compulsory secondary education participated in the study, and performed equal share tasks in the context of Egyptian fractions (unit fractions with different denominators). The aim was to determine if, with these types of tasks, students could improve their learning of the different meanings of the fraction concept. Our results indicate that there seems to be a relationship between the meaning used and the representation chosen. Similarly, we found that—with these tasks—students significantly increased the number of registers of representation they used. Students who used distinct representations had to coordinate several registers, which might be interpreted as proof of the development of conceptual understanding.

Keywords

Egyptian fractions
fractions
rational numbers
registers of representation
share

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2021, Volume 17, Issue 7, Article No: em1984

https://doi.org/10.29333/ejmste/10992

Publication date: 19 Jun 2021

Article Views: 833

Article Downloads: 761

Open Access Disclosures References How to cite this article

Disclosure

Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

Ball D. L. (1993) Halves, pieces, and twoths: constructing representational contexts in teaching fractions. In T. Carpenter, E. Fennema, T. Romberg (Eds.), Rational numbers: An integration of research (pp 157-196). Erlbaum, Hillsdale.
Behr, M., Harel, G., Post, T. & Lesh, R. (1992). Rational number, ratio and proportion. In D. A. Grows (Ed.), Handbook on research on mathematics teaching and learning (pp. 296–333). Macmillan.
Bentley, K. (2004). Egyptian fractions in Modern Times. Mathematics in School, Jan., 9-10.
Berlinghoff, W. P., & Gouvea, F. Q. (2004). Math through the ages (Expanded edition). Washington, DC: Mathematical Association of America, Farmington, ME: Oxton House Publishers.
Charalambous, C., Delaney, S., Hsu, H.-Y. & Mesa, V. (2010). Comparative analysis of the addition and subtraction of fractions in textbooks from three countries. Mathematical Thinking and Learning, 12(2), 117-151. https://doi.org/10.1080/10986060903460070
Clarke, D. M., Roche, A. & Mitchell, A. (2011). One-to-one student interviews provide powerful insights and clear focus for the teaching of fractions in the middle years. In J. Way & J. Bobis (Eds.), Fractions: Teaching for understanding (pp. 23-32). The Australian Association of Mathematics Teachers.
Cramer, K. A., Post, T. R., & delMas, R. C. (2002). Initial fraction learning by fourth-and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111-144. https://doi.org/10.2307/749646
Cramer, K., Monson, D., Ahrendt, S., Wyberg, T., Pettis, C., & Fagerlund, C. (2019). Reconstructing the unit on the number line: Tasks to extend fourth graders’ fraction understandings. Investigations in Mathematics Learning, 11(3), 180-194. https://doi.org/10.1080/19477503.2018.1434594
D’Amore, B., Fandiño Pinilla M. I. & Iori M. (2013). Primi elementi di semiotica. La sua presenza e la sua importanza nel processo di insegnamento-apprendimento della matemática [First elements of semiotics. Its presence and its importance in the teaching-learning process of mathematics]. Pitagora.
Duval, R. (1993). Registres de représentation sémiotique et fonctionnement cognitif de la pensé [Semiotic representation registers and cognitive functioning of thought], Annales de Didactique et de Sciences Cognitives, 5, IREM de Strasbourg, pp. 37-65.
Duval, R. (2000). Basic issues for research in mathematics education. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 55-69), PME.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1-2), 103-131. https://doi.org/10.1007/s10649-006-0400-z
Empson, S. B., Junk, D., Dominguez, H. & Turner, E. (2006) Fractions as the coordination of multiplicatively related quantities: a cross-sectional study of children’s thinking. Educational Studies in Mathematics 63(1), 1-28. https://doi.org/10.1007/s10649-005-9000-6
Eves, H. (1969). An introduction to the history of mathematics (3d ed.). Holt, Rinehart and Winston.
Fauvel, J., & Gray, J. (Eds.). (1987). The history of mathematics: A reader. Mathematical Assn of Amer.
Fazio, L., & Siegler, R. S. (2011). Teaching fractions. International Academy of Education. Geneva.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures (Vol. 1). Springer Science & Business Media.
Furinghetti, F., & Radford, L. (2002). Historical conceptual developments and the teaching of mathematics: from philogenesis and ontogenesis theory to classroom practice. In Handbook of international research in mathematics education (pp. 643-666). Routledge. https://doi.org/10.4324/9781410602541-22
Halliday, M. A. K. (1975). Some aspects of sociolinguistics: Interactions between linguistics and mathematical educations, UNESCO, Copenhague, pp. 64-73.
Jaworski, B., & Goodchild, S. (2006) Inquiry community in an activity theory frame. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (vol. 3, pp. 353-360). PME.
Kaput, J. J. (1987). Representation systems and mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 19-26). Erbaum.
Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 49-84). Erlbaum.
Kosheleva, O., & Lyublinskaya, I. (2007). Using innovative fraction activities as a vehicle for examining conceptual understanding of fraction concepts in pre-service elementary teachers mathematical education. In Proceedings of the 29th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education PME-NA (pp. 25-28).
Lamon, S. J. (2007). Rational numbers and proportional reasoning: toward a theoretical framework for research. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp 629–667). IAP, Charlotte.
Lamon, S. J. (2020). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Routledge. https://doi.org/10.4324/9781003008057
Lingard, D. (1999). The history of mathematics: an essential component of the mathematics curriculum at all levels. In Proceedings of the International Conference on Mathematics Education into the 21st Century: Societal Challenges, Issues and Approaches (Vol. III, pp. 38-43), Cairo, Egypt.
Liu, C., Xin, Z., & Li, X. (2011). The development of Chinese students’ understanding of the fraction concepts from fifth to eighth grade, Journal of Mathematics Education, 4(2), 17-34.
MacGregor, N. (2011). A history of the world in 100 objects. Penguin.
NRICH (2020). Egyptian Fractions. https://nrich.maths.org/1173
O’Reilly, D. (1995). Creating Egyptian fractions. The Australian Mathematics Teacher, 51, 10-10.
Oliver, J. (2003). Fractions in ancient Egyptian times. Mathematics in School, 32(1), 14-16.
Pantziara, M., & Philippou, G. (2012). Levels of students’ “conception” of fractions. Educational Studies in mathematics, 79(1), 61-83. https://doi.org/10.1007/s10649-011-9338-x
Park, J., Güçler, B., & McCrory, R. (2013). Teaching prospective teachers about fractions: historical and pedagogical perspectives. Educational Studies in Mathematics, 82(3), 455-479. https://doi.org/10.1007/s10649-012-9440-8
Pimm, D. (2002). Symbols and meanings in school mathematics. Routledge. https://doi.org/10.4324/9780203428610
Pimm, D. (2014). Unthought knowns. For the Learning of Mathematics, 34(3), 15-16.
Pothier, Y. & Sawada, D. (1983). Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14, 307-317. https://doi.org/10.5951/jresematheduc.14.5.0307
Ribeiro, M., Mellone, M., & Jakobsen, A. (2016). Interpreting students’ non-standard reasoning: Insights for mathematics teacher education. For the Learning of Mathematics, 36(2), 8-13.
Simon, M. A., Placa, N., Avitzur, A., & Kara, M. (2018). Promoting a concept of fraction-as-measure: A study of the learning through activity research program. The Journal of Mathematical Behavior, 52, 122-133. https://doi.org/10.1016/j.jmathb.2018.03.004
Steffe, L. P., & Olive, J. (2009). Children’s fractional knowledge. Springer Science & Business Media. https://doi.org/10.1007/978-1-4419-0591-8
Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for research in mathematics education, 30(4), 390-416. https://doi.org/10.2307/749707
Wilkins, J. L., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International journal of STEM education, 5(1), 27. https://doi.org/10.1186/s40594-018-0119-2
Zhang, X., Clements, M. A., & Ellerton, N. F. (2014). Conceptual mis(understandings) of fractions: From area models to multiple embodiments. Mathematics Education Research Journal, 27, 233-261. https://doi.org/10.1007/s13394-014-0133-8

How to cite this article

APA

Clemente, F., & Fortuny, J. M. (2021). Egyptian Fractions and Representation Registers in the Construction of the Fraction Concept. Eurasia Journal of Mathematics, Science and Technology Education, 17(7), em1984. https://doi.org/10.29333/ejmste/10992

Vancouver

Clemente F, Fortuny JM. Egyptian Fractions and Representation Registers in the Construction of the Fraction Concept. EURASIA J Math Sci Tech Ed. 2021;17(7):em1984. https://doi.org/10.29333/ejmste/10992

AMA

Clemente F, Fortuny JM. Egyptian Fractions and Representation Registers in the Construction of the Fraction Concept. EURASIA J Math Sci Tech Ed. 2021;17(7), em1984. https://doi.org/10.29333/ejmste/10992

Chicago

Clemente, Francisco, and Josep Maria Fortuny. "Egyptian Fractions and Representation Registers in the Construction of the Fraction Concept". Eurasia Journal of Mathematics, Science and Technology Education 2021 17 no. 7 (2021): em1984. https://doi.org/10.29333/ejmste/10992

Harvard

Clemente, F., and Fortuny, J. M. (2021). Egyptian Fractions and Representation Registers in the Construction of the Fraction Concept. Eurasia Journal of Mathematics, Science and Technology Education, 17(7), em1984. https://doi.org/10.29333/ejmste/10992

MLA

Clemente, Francisco et al. "Egyptian Fractions and Representation Registers in the Construction of the Fraction Concept". Eurasia Journal of Mathematics, Science and Technology Education, vol. 17, no. 7, 2021, em1984. https://doi.org/10.29333/ejmste/10992