An Alternative Route to Teaching Fraction Division: Abstraction of Common Denominator Algorithm


İsmail Özgür ZEMBAT


Abstract

From a curricular stand point, the traditional invert and multiply algorithm for division of fractions provides few affordances for linking to a rich understanding of fractions. On the other hand, an alternative algorithm, called common denominator algorithm, has many such affordances. The current study serves as an argument for shifting curriculum for fraction division from use of invert and multiply algorithm as a basis to the use of common denominator algorithm as a basis. This was accomplished with the analysis of learning of two prospective elementary teachers being an illustration of how to realize those conceptual affordances. In doing so, the article proposes an instructional sequence and details it by referring to both the (mathematical and pedagogical) advantages and the disadvantages. As a result, this algorithm has a conceptual basis depending on basic operations of partitioning, unitizing, and counting, which make it accessible to learners. Also, when participants are encouraged to construct this algorithm based on their work with diagrams, common denominator algorithm formalizes the work that they do with diagrams.


Keywords

Teaching fraction division, abstracting common denominator algorithm, curriculum development

Paper Details

Paper Details
Topic Elementary Education
Pages 399 - 422
Issue IEJEE, Volume 7, Issue 3
Date of acceptance 01 May 2015
Read (times) 674
Downloaded (times) 327

Author(s) Details

İsmail Özgür ZEMBAT

Mevlana (Rumi) University, Turkey, Turkey


References

Armstrong, B. E., & Bezuk, N. (1995). Multiplication and division of fractions: The search for meaning. In J. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 85-119). Albany, NY: State University of New York Press.

Ball, D. L. (1990). Prospective elementary and secondary teachers' understanding of division. Journal for Research in Mathematics Education, 21(2), 132-144.

Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D., & Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 23(3), 194-222.

Contreras, J. N. (1997). Learning to teach algebraic division for understanding: A comparison and contrast between two experienced teachers. In J. A. Dossey, J. O. Swafford, M. Parmantie & A. E. Dossey (Eds.), Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 535-544). Bloomington-Normal, IL: Eric Clearinghouse for Science, Mathematics, and Environmental Education.

Gregg, J., & Gregg, D. U. (2007). Interpreting the standard division algorithm in a `Candy Factory` context. Teaching Children Mathematics, 14(1) 25-31.

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). Hillsdale, NJ: Lawrence Erlbaum Associates, Publishers.

Lamon, S. (1996). The development of unitizing: Its role in children’s partitioning strategies. Journal for Research in Mathematics Education, 27(2), 170-193.

Li, Y. & Kulm, G. (2008). Knowledge and confidence of pre-service mathematics teachers: The case of fraction division. ZDM – The International Journal on Mathematics Education, 40(5), 833-843.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Perlwitz, M. D. (2004). Two students’ constructed strategies to divide fractions. Mathematics Teaching in the Middle School, 10(3), 122-126.

Piaget, J. (1964). Cognitive development in children: Piaget - development and learning. Journal of Research in Science Teaching, 2, 176-186.

Piaget, J. (1971). Biology and knowledge. Chicago: The University of Chicago.

Piaget, J. (1983). Piaget's theory. In W. Kesson (Ed.), History, theory, and methods, Handbook of child psychology (Vol. 1, pp. 103-128). New York: John Wiley & Sons.

Piaget, J. (2001). Studies in reflecting abstraction (Trans. R. L. Campbell). Sussex, England: Psychology Press.

Post, T. R., Harel, G., Behr, M. J., & Lesh, R. A. (1991). Intermediate teachers' knowledge of rational number concepts. In E. Fennema, T. P. Carpenter & S. J. Lamon (Eds.), Integrating research on teaching and learning mathematics (pp. 177-198). Albany, NY: State University of New York Press.

Sharp, J., & Adams, B. (2002). Children’s constructions of knowledge for fraction division after solving realistic problems. The Journal of Educational Research, 95(6), 333-347.

Simon, M. A. (1993). Prospective elementary teachers' knowledge of division. Journal for Research in Mathematics Education, 24(3), 233-254.

Simon, M. A., Saldanha, L., McClintock, E., Akar, G. K., Watanabe, T., & Zembat, I. O. (2010). A developing approach to studying students’ learning through their mathematical activity. Cognition and Instruction, 28(1), 70-112.

Simon, M. A., Tzur, R., Heinz, K., & Kinzel, M. (2004). Explicating a mechanism for conceptual learning: Elaborating the construct of reflective abstraction. Journal for Research in Mathematics Education, 35(5), 305-329.

Sowder, J. T. (1995). Instructing for rational number sense. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 15-30). Albany, NY: State University of New York Press.

Sowder, J. T., Philipp, R. A., Armstrong, B. E., & Schappelle, B. P. (1998). Middle-grade teachers' mathematical knowledge and its relationship to instruction: A research monograph. Albany, NY: State University of New York Press.

Steffe, L. P. (1991). The constructivist teaching experiment: Illustrations and implications. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp. 177-194). Dordrecth, The Netherlands: Kluwer Academic Publishers.

Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267-306). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Streefland, L. (1991). Fractions in realistic mathematics education. Dordrecht / Boston / London: Kluwer.

Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165-208.

Tzur, R., & Timmerman, M. (1997). Why do we invert and multiply? Elementary teachers' struggle to conceptualize division of fractions. In J. A. Dossey, J. O. Swafford, M. Parmantie & A. E. Dossey (Eds.), Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 553-559). Bloomington-Normal, IL: Eric Clearinghouse for Science, Mathematics, and Environmental Education.

Usiskin, Z. (2007). Some thoughts about fractions. Mathematics Teaching in the Middle School, 12(7), 370-373.

 Zembat, I. O. (2004). Conceptual development of prospective elementary teachers: The case of division of fractions (Doctoral dissertation, The Pennsylvania State University). ProQuest Digital Dissertations Database. (Publication No. AAT 3148695)

Zembat, I. O. (2007). Sorun aynı – kavramlar; Kitle aynı - öğretmen adayları. İlköğretim Online, 6(2), 305-312.