Primary Teacher Students’ Understanding of Fraction Representational Knowledge in Slovenia and Kosovo
Keywords:
primary teacher student, fraction, representation, partwhole subconstruct, measure subconstruct
Abstract
The study of primary teacher students’ knowledge of fractions is very important because fractions present a principal and highly complex set of concepts and skills within mathematics. The present study examines primary teacher students’ knowledge of fraction representations in Slovenia and Kosovo. According to research, there are five subconstructs of fractions: the part-whole subconstruct, the measure subconstruct, the quotient subconstruct, the operator subconstruct and the ratio subconstruct. Our research focused on the part-whole and the measure subconstructs of fractions, creating nine tasks that were represented by different modes of representation: area/region, number line and sets of objects. The sample consisted of 76 primary teacher students in Slovenia and 93 primary teacher students in Kosovo. Both similarities and differences of the primary teacher students’ interpretations of the representations across the two countries were revealed and compared. The findings suggest that primary teacher students from both countries need to upgrade their understanding of fractions. The analysis confirms that the formal mathematical knowledge acquired by primary teacher students is not necessarily adequate for teaching elementary concepts in school.
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References
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doi:10.1007/s10649-015-9673-4
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Hamdan, N., & Gunderson, E. A. (2017). The number line is a critical spatial-numerical representation: Evidence from a fraction intervention. Developmental Psychology, 53(3), 587–596.
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Ni, Y., & Di-Zhou, Y. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.
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Olanoff, D., Lo, J.-J., & Tobias, J. M. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, 11(2), 267–310.
Pantziara, M., & Philippou, G. (2012). Levels of students’ ‘conception’ of fractions. Educational Studies in Mathematics, 79(1), 61–83.
Park, J., Güçler, B., & McCrory, R. (2013). Teaching prospective teachers about fractions: historical and
pedagogical perspectives. Educational Studies in Mathematics, 82(3), 455–479.
Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry. New York, NY: Basic Books.
Pothier, Y., & Sawada, D. (1983). Partitioning: The emergence of rational numbers ideas in young children. Journal for Research in Mathematics Education, 14(5), 307–317.
Saxe, G. B., Taylor, E. V., McIntosh, C., & Gearhart, M. (2005). Representing fractions with standard notation: A developmental analysis. Journal for Research in Mathematics Education, 36(2), 137–157.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.
Son, J.-W., & Lee, J.-E. (2016). Pre-service teachers’ understanding of fraction multiplication, representational knowledge, and computational skills. Mathematics Teacher Education and Development, 18(2), 5–28.
Steffe, P. L., & Olive, J. (2010). Children’s fractional knowledge. New York, NY: Springer.
Tsao, Y. L. (2005). The number sense of pre-servıce elementary. College Student Journal, 39(4), 647–679.
Tunç-Pekkan, Z. (2015). An analysis of elementary school children’s fractional knowledge depicted with circle, rectangle, and number line representations. Educational Studies in Mathematics, 89(3), 419–441.
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Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). Boston, MA: Pearson Allyn & Bacon.
Van Steenbrugge, H., Valcke, M., & Desoete, A. (2010). Mathematics learning difficulties: Teachers’ professional knowledge and the use of commercially available learning packages. Educational Studies, 36(1), 59–71.
Van Steenbrugge, H., Valcke, M., & Desoete, A. (2014). Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge? Curriculum Studies, 46(1), 138–161.
Vula, E., & Kastrati, J. K. (2018). Preservice teacher procedural and conceptual knowledge of fractions. In G. Stylianides & K. Hino, Research advances in the mathematical education of pre-service elementary teachers (pp. 111–123). Cham: Springer.
Wilson, S. M. (2010). Knowledge for teaching mathematics in a primary school: Perspectives of preservice teachers. Canterbury, UK: University of Canterbury.
Yang, D.-C., Reys, R. E., & Reys, B. J. (2009). Number sense strategies used by pre-service teachers in Taiwan. International Journal of Science and Mathematics Education, 7(2), 383–403.
Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education. The Elementary School Journal, 90(4), 449–466.
Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide. American Educator, 29(1), 14–17, 20–22, 43–46.
Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2009). The array representation and primary children’s understanding and reasoning in multiplication. Educational Studies in Mathematics, 70(3), 217–241.
Behr, M. J., Post, T. R., Harel, G., & Lesh, R. (1993). Rational numbers: Toward a semantic analysis - emphasis on the operator construct. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: An integration of research, (pp. 13–47). Hillsdale, N.J.: Lawrence Erlbaum Associates.
Bobos, G., & Sierpinska, A. (2017). Measurement approach to teaching fractions: A design experiment in a pre-service course for elementary teachers. International Journal for Mathematics Teaching and Learning, 18(2), 203–239.
Castro-Rodriguez, E., Pitta-Pantazi, D., Rico, L., & Gomez, P. (2016). Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics, 92(1), 129–146.
doi:10.1007/s10649-015-9673-4
Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293–316.
Chinnappan, M., & Forrester, P. (2014). Generating procedural and conceptual knowledge of fractions by pre-service teachers. Mathematics Education Research Journal, 26(4), 871–896.
Clarke, D., Roche, A., & Mitchell, A. (2007). Year six fraction understanding: A part of the whole story. Mathematics: Essential research, essential practice. Proceedings of the 30th annual conference of the mathematics education research group of Australia (pp. 207–216). Adelaide: MERGA.
de Castro, B. V. (2008). Cognitive models: The missing link to learning fraction multiplication and division. Asia Pacific Education Review, 9(2), 101–112.
Empson, S. B., & Levi, L. (2011). Extending children’s mathematics: fractions and decimals. Porstmouth,
NH: Heinemann.
Hackenberg, A. (2007). Units coordination and construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behaviour, 26(1), 27–47.
Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions. Journal of Educational Psychology, 102(2), 395–406.
Hamdan, N., & Gunderson, E. A. (2017). The number line is a critical spatial-numerical representation: Evidence from a fraction intervention. Developmental Psychology, 53(3), 587–596.
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406.
Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching. The Elementary School Journal, 105(1), 11–30.
Keijzer, R., & Terwel, J. (2001). Audrey’s acquisition of fractions: A case study into the learning of formal mathematics. Educational Studies in Mathematics, 47(1), 53–73.
Kieren, T. E. (1976). On the mathematical, cognitive and instructional foundations of rational numbers. In R. A. Lesh (Ed.), Number and Measurement (pp. 101–144). Fonte: ERIC Document Reproduction Service No. ED 120 027.
Kieren, T. E. (1993). Rational and fractional numbers: from quotient fields to recursive understanding. In P. T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers. An integration of research (pp. 49–84). New Jersey, NJ: Erlbaum.
Lamon, S. J. (2012). Teaching fractions and ratios for understanding (3rd ed). New York, NY and London, UK: Taylor & Francis.
Lin, C.-Y., Becker, J., Byun, M.-R., & Ko, Y.-Y. (2013). Enhancing pre-service teachers’ fraction knowledge through open approach instruction. Journal of Mathematical Behavior, 32(3), 309–330.
Ma, L. (1999). Knowing and teaching elementary school mathematics (Vol. 1). New Jersey, NJ: Routledge.
Manfreda Kolar, V., JanežiÄ, A., & Hodnik ÄŒadež, T. (2015). Diagnosing students’ difficulties in understanding the concept of fraction. In J. Novotna & Moraova, H (Eds.), Developing mathematical language and reasoning (pp. 232–240). Prague: Charles University, Faculty of Education.
Moss, J., & Case, R. (2011). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122–147.
National Mathematics Advisory Panel (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, D.C.: U.S. Department of Education.
Newton, K. J. (2008). An extensive analysis of preservice elementary teachers’ knowledge of fractions. American Educational Research Journal, 45(4), 1080–1110.
Ni, Y., & Di-Zhou, Y. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.
OECD (2016). PISA 2015 Results (Vol. 1): Excellence and equity in education. Paris: OECD Publishing.
Olanoff, D., Lo, J.-J., & Tobias, J. M. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, 11(2), 267–310.
Pantziara, M., & Philippou, G. (2012). Levels of students’ ‘conception’ of fractions. Educational Studies in Mathematics, 79(1), 61–83.
Park, J., Güçler, B., & McCrory, R. (2013). Teaching prospective teachers about fractions: historical and
pedagogical perspectives. Educational Studies in Mathematics, 82(3), 455–479.
Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry. New York, NY: Basic Books.
Pothier, Y., & Sawada, D. (1983). Partitioning: The emergence of rational numbers ideas in young children. Journal for Research in Mathematics Education, 14(5), 307–317.
Saxe, G. B., Taylor, E. V., McIntosh, C., & Gearhart, M. (2005). Representing fractions with standard notation: A developmental analysis. Journal for Research in Mathematics Education, 36(2), 137–157.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.
Son, J.-W., & Lee, J.-E. (2016). Pre-service teachers’ understanding of fraction multiplication, representational knowledge, and computational skills. Mathematics Teacher Education and Development, 18(2), 5–28.
Steffe, P. L., & Olive, J. (2010). Children’s fractional knowledge. New York, NY: Springer.
Tsao, Y. L. (2005). The number sense of pre-servıce elementary. College Student Journal, 39(4), 647–679.
Tunç-Pekkan, Z. (2015). An analysis of elementary school children’s fractional knowledge depicted with circle, rectangle, and number line representations. Educational Studies in Mathematics, 89(3), 419–441.
Understanding fractions: Interpretations and representations. An iTalk2Learn guide (2014). Retrieved from http://www.italk2learn.eu/wp-content/uploads/2014/11/Understanding-fractions-Interpretations-and-representations.pdf
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). Boston, MA: Pearson Allyn & Bacon.
Van Steenbrugge, H., Valcke, M., & Desoete, A. (2010). Mathematics learning difficulties: Teachers’ professional knowledge and the use of commercially available learning packages. Educational Studies, 36(1), 59–71.
Van Steenbrugge, H., Valcke, M., & Desoete, A. (2014). Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge? Curriculum Studies, 46(1), 138–161.
Vula, E., & Kastrati, J. K. (2018). Preservice teacher procedural and conceptual knowledge of fractions. In G. Stylianides & K. Hino, Research advances in the mathematical education of pre-service elementary teachers (pp. 111–123). Cham: Springer.
Wilson, S. M. (2010). Knowledge for teaching mathematics in a primary school: Perspectives of preservice teachers. Canterbury, UK: University of Canterbury.
Yang, D.-C., Reys, R. E., & Reys, B. J. (2009). Number sense strategies used by pre-service teachers in Taiwan. International Journal of Science and Mathematics Education, 7(2), 383–403.
Published
2018-06-27
How to Cite
Kolar, V. M., Hodnik ČadežT., & Vula, E. (2018). Primary Teacher Students’ Understanding of Fraction Representational Knowledge in Slovenia and Kosovo. Center for Educational Policy Studies Journal, 8(2), 71-96. https://doi.org/10.26529/cepsj.342
Section
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