Representational Flexibility and Problem-Solving Ability in Fraction and Decimal Number Addition: A Structural Model

  • Eleni Deliyianni
  • Athanasios Gagatsis
  • Iliada Elia
  • Areti Panaoura


The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1701 primary and secondary school students. Eight components, which all involve representational transformations, were encompassed under the construct of representational flexibility in fraction and decimal number addition. This structure reveals that, for both concepts, the representational transformation competences of recognition and conversion, and therefore representational flexibility as well, were affected by the complexity of the concepts involved and the direction of the conversion, respectively. Results also showed that two first-order factors were needed to explain the problem-solving ability in fraction and decimal number addition, indicating the differential effect of the modes of representation that is diagrammatic and verbal form on problem-solving ability irrespective of the concepts involved, as in the case of the conversions. Representational flexibility and problem-solving ability were found to be major components of students’ representational thinking of fraction and decimal number addition. The proposed framework was invariant across the primary and secondary school students. Theoretical and practical implications are discussed.


Decimal number addition Fraction addition Problem-solving ability Representational flexibility Representational transformations Structural equation model 



The study reported in this article was supported by a program grant from the University of Cyprus (MED19: “The role of multiple representations in mathematical learning during the transition within and between primary and secondary school”).

Supplementary material

10763_2015_9625_MOESM1_ESM.docx (224 kb)
ESM 1 (DOCX 224 kb)


  1. Acevedo Nistal, A., Van Dooren, W., Clarebout, G., Elen, J. & Verschaffel, L. (2009). Conceptualising, investigating and stimulating representational flexibility in mathematical problem solving and learning: A critical review. ZDM Mathematics Education, 41, 627–636.CrossRefGoogle Scholar
  2. Alajmi, A. H. (2012). How do elementary textbooks address fractions? A review of mathematics textbooks in the USA, Japan, and Kuwait. Educational Studies in Mathematics, 79(2), 239–261.CrossRefGoogle Scholar
  3. Barnett-Clarke, C., Fisher, W., Marks, R. & Ross, S. (2010). Developing essential understanding of rational number: Grades 3–5. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  4. Bentler, M. P. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 301–345.CrossRefGoogle Scholar
  5. Boulet, G. (1998). Didactical implications of children’s difficulties in learning the fraction concept. Focus on Learning Problems in Mathematics, 20(4), 19–34.Google Scholar
  6. Carlson, M. & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58, 45–75.CrossRefGoogle Scholar
  7. 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.CrossRefGoogle Scholar
  8. Cramer, K., Post, T. & delMas, R. (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.CrossRefGoogle Scholar
  9. Cramer, K. & Wyberg, T. (2009). Efficacy of different concrete models for teaching the part-whole construct for fractions. Mathematical Thinking and Learning, 11(4), 226–257.CrossRefGoogle Scholar
  10. Deliyianni, E., Elia, I., Panaoura, A. & Gagatsis, A. (2007). The functioning of representations in Cyprus mathematics textbooks. In E.P. Avgerinos & A. Gagatsis (Eds.). Current Trends in Mathematics Education (pp. 155–167). Rhodes, Greece: Cyprus Mathematics Society & University of Aegean.Google Scholar
  11. Desmet, L., Grégoire, J. & Mussolin, C. (2010). Developmental changes in the comparison of decimal fractions. Learning and Instruction, 20(6), 521–532.CrossRefGoogle Scholar
  12. Duval, R. (2006). A cognitive analysis of problems of comprehension in learning of mathematics. Educational Studies in Mathematics, 61, 103–131.CrossRefGoogle Scholar
  13. Elia, I., Gagatsis, A. & Demetriou, A. (2007). The effects of different modes of representation on the solution of one step additive problem. Learning and Instruction, 17, 658–672.Google Scholar
  14. Empson, S. B., Levi, L. & Carpenter, T. P. (2011). The algebraic nature of fractions: Developing relational thinking in elementary school. In J. Cai & E. J. Knuth (Eds.), Early algebraization (pp. 409–428). Berlin, Germany: Springer.Google Scholar
  15. Gagatsis, A. & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24, 645–657.Google Scholar
  16. Goldin, G. A. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 275–285). Reston, VA: The National Council of Teachers of Mathematics.Google Scholar
  17. Hiebert, J., Wearne, D. & Taber, S. (1991). Fourth graders’ gradual construction of decimal fractions during instruction using different physical representations. Elementary School Journal, 91, 321–341.CrossRefGoogle Scholar
  18. Hitt, F. (1998). Difficulties in the articulation of different representations linked to the concept of function. The Journal of Mathematical Behavior, 17(1), 123–134.CrossRefGoogle Scholar
  19. Irwin, K. C. (2001). Using everyday knowledge of decimals to enhance understanding. Journal for Research in Mathematics Education, 32(4), 399–420.CrossRefGoogle Scholar
  20. Iuculano, T. & Butterworth, B. (2011). Understanding the real value of fractions and decimals. The Quarterly Journal of Experimental Psychology, 64(11), 2088–2098.CrossRefGoogle Scholar
  21. Jordan, N. C., Hansen, N., Fuchs, L. S., Siegler, R. S., Gersten, R. & Micklos, D. (2013). Developmental predictors of fraction concepts and procedures. Journal of Experimental Child Psychology, 116, 45–58.CrossRefGoogle Scholar
  22. Keijzer, R. & Terwel, J. (2003). Learning for mathematical insight: A longitudinal comparative study on modeling. Learning and Instruction, 13, 285–304.CrossRefGoogle Scholar
  23. Kline, R. B. (1998). Principles and practice of structural equation modeling. New York, NY: Guilford.Google Scholar
  24. Kong, S. C. & Kwok, L. F. (2003). A graphical partitioning model for learning common fraction: Designing affordances on a web-supported learning environment. Computers and Education, 40, 137–155.CrossRefGoogle Scholar
  25. Lambdin, D. (2003). Benefits of teaching through problem solving. In F. Lester (Ed.), Teaching mathematics through problem solving: Prekindergarten-Grade 6 (pp. 3–13). Reston, VA: NCTM.Google Scholar
  26. Lamon, S. L. (2001). Presenting and representing: From fractions to rational numbers. In A. Cuoco & F. Curcio (Eds.), The role of representations in school mathematics—2001 yearbook (pp. 146–165). Reston, VA: NCTM.Google Scholar
  27. Lesh, R., Post, T. & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  28. Moss, J. & Case, R. (1999). 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.CrossRefGoogle Scholar
  29. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.Google Scholar
  30. Ni, Y. (2001). Semantic domains of rational numbers and the acquisition of fraction equivalence. Contemporary Educational Psychology, 26, 400–417.CrossRefGoogle Scholar
  31. Niemi, D. (1996). Assessing conceptual understanding in mathematics: Representations, problem solutions, justifications, and explanations. The Journal of Educational Research, 89(6), 351–363.CrossRefGoogle Scholar
  32. Schnotz, W. (2002). Towards an integrated view of learning from text and visual displays. Educational Psychology Review, 14(1), 101–120.CrossRefGoogle Scholar
  33. Schoenfeld, A. H., Smith, J. P. & Arcavi, A. (1993). Learning: The microgenetic analysis of one student’s evolving understanding of a complex subject matter domain. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 4, pp. 55–176). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  34. Sowder, J. (1997). Place value as the key to teaching decimal operations. Teaching Children Mathematics, 3(8), 448–453.Google Scholar
  35. Steffe, L. P. & Olive, J. (2010). Children’s fractional knowledge. New York, NY: Springer.Google Scholar
  36. Teppo, A. & van den Heuvel-Panhuizen, M. (2014). Visual representations as objects of analysis: The number line as an example. ZDM Mathematics Education, 46(1), 45–58. doi: 10.1007/s11858-013-0518-2.
  37. Thomas, M. O. J. (2008). Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 7(2), 71–91.Google Scholar
  38. Verschaffel, G., Greer, B. & Torbeyns, J. (2006). Numerical thinking. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present, and future (pp. 51–82). Rotterdam, Netherlands: Sense Publishers.Google Scholar
  39. Whitley, J., Lupart, J. & Beran, T. (2007). Differences in achievement between adolescents who remain in A K–8 school and those who transition to a junior high school. Canadian Journal of Education, 30(3), 649–669.CrossRefGoogle Scholar

Copyright information

© Ministry of Science and Technology, Taiwan 2015

Authors and Affiliations

  • Eleni Deliyianni
    • 1
  • Athanasios Gagatsis
    • 2
  • Iliada Elia
    • 2
  • Areti Panaoura
    • 3
  1. 1.Cyprus Pedagogical InstituteNicosiaCyprus
  2. 2.University of CyprusNicosiaCyprus
  3. 3.Frederick UniversityNicosiaCyprus

Personalised recommendations