Hierarchical Levels of Abilities that Constitute Fraction Understanding at Elementary School

  • Aristoklis A. NicolaouEmail author
  • Demetra Pitta-Pantazi


This article examines whether the 7 abilities found in a previous study carried out by the authors to constitute fraction understanding of sixth grade elementary school students determine hierarchical levels of fraction understanding. The 7 abilities were as follows: (a) fraction recognition, (b) definitions and mathematical explanations for fractions, (c) argumentations and justifications about fractions, (d) relative magnitude of fractions, (e) representations of fractions, (f) connections of fractions with decimals, percentages, and division, and (g) reflection during the solution of fraction problems. The sample comprised of 182 sixth grade students that were clustered into 3 categories by means of latent class analysis: those of low fraction understanding, those of medium fraction understanding, and those of high fraction understanding. It was found that low fraction understanding students were sufficient in fraction recognition and relative magnitude of fractions, those belonging to the medium category in fraction recognition, relative magnitude of fractions, as well as in connections with decimals, percentages and division and representations of fractions, while high fraction understanding students were sufficient in all 7 abilities. It was also found that these levels were stable across time; the hierarchical levels were the same across three measurements that took place. Possible implications for fraction understanding are discussed, and directions for future research are drawn.


Elementary school Fraction understanding Hierarchical levels Sixth grade Students’ abilities 


  1. Behr, M. J., Lesh, R., Post, T. R. & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–126). New York: Academic. New York, NY.Google Scholar
  2. Charalambous, C. Y. & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64, 293–316. doi: 10.1007/s10649-006-9036-2.CrossRefGoogle Scholar
  3. Clarke, D. M. & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72(1), 127–138. doi: 10.1007/s10649-009-9198-9.CrossRefGoogle Scholar
  4. Cyprus National Mathematics Curriculum (2010). Mathematics curriculum in Cyprus for grades K-12. Nicosia, Cyprus: Lithostar. Retrieved 10 March, 2013, from
  5. Deliyianni, E. & Gagatsis, A. (2013). Tracing the development of representational flexibility and problem solving in fraction addition: A longitudinal study. Educational Psychology, 33(4), 427–442. doi: 10.1080/01443410.2013.765540.CrossRefGoogle Scholar
  6. Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131. doi: 10.1007/s10649-006-0400-z.CrossRefGoogle Scholar
  7. Gagatsis, A., Michaelidou, E. & Shiakalli, M. (2001). Representational theories and the learning of mathematics. Nicosia: ERASMUS IP 1. Nicosia, Cyprus.Google Scholar
  8. Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh (Ed.), Number and Measurement: Papers from a Research Workshop ERIC/SMEAC (pp. 101–144). Columbus, OH.Google Scholar
  9. 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). NJ: Erlbaum. Hillsdale, NJ.Google Scholar
  10. Lamon, S. J. (1999). Teaching fractions and ratios for understanding. Mahwah, New Jersey: Lawrence Erlbaum Associates. Mahwah, NJ.Google Scholar
  11. Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed). New York, NY:Routledge.Google Scholar
  12. Marcoulides, G. A. & Kyriakides, L. (2010). Structural equation modelling techniques. In B. Creemers, L. Kyriakides & P. Sammons (Eds.), Methodological advances in educational effectiveness research (pp. 277–303). London and New York: Routledge.Google Scholar
  13. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.Google Scholar
  14. Newstead, K. & Murray, H. (1998). Young students’ constructions of fraction. Proceedings of the 22nd Conference of the International Group for Psychology of Mathematics Education. Stellenbosch, South Africa: University of Stellenbosch.Google Scholar
  15. Nicolaou, A. & Pitta-Pantazi, D. (2011a). Factors that constitute understanding a mathematical concept at the elementary school: Fractions as the concept of reference. Article presented at the 4th Conference of the Union of Greek Researchers in Mathematics Education (pp. 351–361). University of Ioannina, Ioannina.Google Scholar
  16. Nicolaou, A. A. & Pitta-Pantazi, D. (2011b). A theoretical model for understanding fractions at elementary school. Proceedings of the 7th Conference of the European Society of Mathematics Education (pp. 366–375). Univeristy of Rzeszow, Poland.Google Scholar
  17. Niemi, D. (1996a). A fraction is not a piece of pie: Assessing exceptional performance and deep understanding in elementary school mathematics. Gifted Child Quarterly, 40(2), 70–80.CrossRefGoogle Scholar
  18. Niemi, D. (1996b). Assessing conceptual understanding in mathematics: Representations, problem solutions, justifications and explanations. Journal of Educational Research, 89(6), 351–363.CrossRefGoogle Scholar
  19. Niemi, D. (1996c). Instructional influences on content area explanations and representational knowledge: Evidence for the construct validity of measures of principled understanding. CSE Technical Report 403. CRESST/University of California, Los Angeles.Google Scholar
  20. Oppenheimer, L. & Hunting, R. P. (1999). Relating fractions & decimals: listening to students talk. Mathematics Teaching in the Middle School, 4(5), 318–321.Google Scholar
  21. Pantziara, M. & Philippou, G. (2012). Levels of students’ “conception” of fractions. Educational Studies in Mathematics, 79, 61–83. doi: 10.1007/s10649-011-9338-x.CrossRefGoogle Scholar
  22. Pirie, S. & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 165–190.CrossRefGoogle Scholar
  23. Scaptura, C., Suh, J. & Mahaffey, G. (2007). Masterpieces to mathematics: Using art to teach fraction, decimal, and percent equivalents. Mathematics Teaching in the Middle School, 13(1), 24–28.Google Scholar
  24. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.CrossRefGoogle Scholar
  25. Streefland, L. (1991). Fractions in realistic mathematics education: A paradigm of developmental research. Dordrecht, The Netherlands: Kluwer.Google Scholar
  26. Stylianides, A. J. (2007). The notion of proof in the context of elementary school mathematics. Educational Studies in Mathematics, 65, 1–20. doi: 10.1007/s10649-006-9038-0.CrossRefGoogle Scholar
  27. Vamvakoussi, X. & Vosniadou, S. (2010). How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181–209. doi: 10.1080/07370001003676603.CrossRefGoogle Scholar
  28. Whitin, D. J. & Whitin, P. (2012). Making sense of fractions and percentages. Teaching Children Mathematics, 18(8), 490–496.CrossRefGoogle Scholar

Copyright information

© Ministry of Science and Technology, Taiwan 2014

Authors and Affiliations

  • Aristoklis A. Nicolaou
    • 1
    Email author
  • Demetra Pitta-Pantazi
    • 2
  1. 1.Ministry of EducationLimassolCyprus
  2. 2.Department of EducationUniversity of CyprusLimassolCyprus

Personalised recommendations