Instructional Science

, Volume 41, Issue 4, pp 681–698 | Cite as

Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions

  • Boris Koichu
  • Guershon Harel
  • Alfred Manaster


Twenty-four mathematics teachers were asked to think aloud when posing a word problem whose solution could be found by computing 4/5 divided by 2/3. The data consisted of verbal protocols along with the written notes made by the subjects. The qualitative analysis of the data was focused on identifying the structures of the problems produced and the associated ways of thinking involved in constructing the problems. The results suggest that success in doing the interview task was associated with perception the given fractions as operands for the division operation and, at the same time, the divisor 2/3 as an operator acting over 4/5. The lack of success was associated with perception of division of fractions as division of divisions of whole numbers and using the result of division of fractions as the only reference point. The study sheds new light on the teachers’ difficulties with conceptualization of fractions.


Mathematics teachers Problem posing Division of fractions Ways of thinking Protocol analysis Intuitive models of division 


  1. Ball, D., Thames, M., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education, 59, 389–407.CrossRefGoogle Scholar
  2. Borko, H., Eisenhart, M., Brown, C., Underhill, R., Jones, D., & Agard, P. C. (1992). Learning to teach hard mathematics: Do novices and their instructors give up too easily? Journal for Research in Mathematics Education, 23, 194–222.CrossRefGoogle Scholar
  3. Brown, S. (2001). Reconstructing school mathematics: Problems with problems and the real world. New York: Peter Lang.Google Scholar
  4. Clark, D., & Sukenik, M. (2006). Assessing fraction understanding using task-based interviews. In J. Novotna, H. Moraova, M. Kratka & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 337–344). Prague: Charles University.Google Scholar
  5. Clement, J. (2000). Analysis of clinical interviews: Foundation and model viability. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 547–589). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  6. Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11, 395–415.CrossRefGoogle Scholar
  7. De Corte, E., & Verschaffel, L. (1996). An empirical test of the impact of primitive intuitive models of operations on solving word problems with a multiplicative structure. Learning and Instruction, 6, 219–243.CrossRefGoogle Scholar
  8. Dey, I. (1999). Grounding grounded theory: Guidelines for qualitative inquiry. San Diego, CA: Academic Press.Google Scholar
  9. Ericsson, K., & Simon, H. (1993). Protocol analysis: Verbal reports as data (revised version). Cambridge, MA: MIT Press.Google Scholar
  10. Fischbein, E., Deri, M., Nello, M., & Marino, M. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16, 3–17.CrossRefGoogle Scholar
  11. Glaser, B., & Strauss, A. (1967). The discovery of grounded theory. Chicago: Aldine.Google Scholar
  12. Greer, B. (1987). Understanding of arithmetical operations as models of situations. In J. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics (pp. 60–80). Oxford: Clarendon.Google Scholar
  13. Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). New York: Macmillan.Google Scholar
  14. Greer, B., Verschaffel, L., & de Corte, E. (2002). The answer is really 4.5: Beliefs about word problems. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 271–292). Dordrecht: Kluwer.Google Scholar
  15. Harel, G. (1995). From naive interpretist to operation conserver. In J. Sowder & B. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 143–165). New York: SUNY Press.Google Scholar
  16. Harel, G. (2007). The DNR system as a conceptual framework for curriculum development and instruction. In R. Lesh, E. Hamilton, & J. Kaput (Eds.), Foundations for the future in mathematics education (pp. 263–280). Mahwah, NJ: Erlbaum.Google Scholar
  17. Harel, G. (2008a). DNR perspective on mathematics curriculum and instruction: Focus on proving, Part I. Zentralblatt fuer Didaktik der Mathematik, 40, 487–500.CrossRefGoogle Scholar
  18. Harel, G. (2008b). DNR perspective on mathematics curriculum and instruction, Part II. Zentralblatt fuer Didaktik der Mathematik, 40, 893–907.CrossRefGoogle Scholar
  19. Harel, G., & Koichu, B. (2010). An operational definition of learning. Journal of Mathematical Behavior, 29, 115–124.CrossRefGoogle Scholar
  20. Harel, G., & Sowder, L. (1998). Students’ proof schemes. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research on collegiate mathematics education (pp. 234–283). Providence, RI: AMS.Google Scholar
  21. Harel, G., Behr, M., Post, T., & Lesh, R. (1994). The impact of number type on the solution of multiplication and division problems: Further considerations. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 365–388). Albany, NY: State University of New York Press.Google Scholar
  22. Harel, G., Koichu, B., & Manaster, A. (2006). Algebra teachers’ ways of thinking characterizing the mental act of problem posing. In J. Novotna, H. Moraova, M. Kratka & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 241–248). Prague: Charles University.Google Scholar
  23. Joram, E., Gabriele, A. J., Bertheau, M., Gelman, R., & Subrahmanyam, K. (2005). Children’s use of the reference point strategy for measurement estimation. Journal for Research in Mathematics Education, 36, 4–23.Google Scholar
  24. Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390–419). New York: Macmillan.Google Scholar
  25. Koichu, B., & Harel, G. (2007). Triadic interaction in clinical task-based interviews with mathematics teachers. Educational Studies in Mathematics, 65, 349–365.CrossRefGoogle Scholar
  26. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Erlbaum.Google Scholar
  27. Mack, N. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26(5), 422–441.CrossRefGoogle Scholar
  28. Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.CrossRefGoogle Scholar
  29. Piaget, J. (1985). The equilibration of cognitive structures: The central problem of intellectual development. Chicago: University of Chicago.Google Scholar
  30. Silver, E. A., Mamona-Downs, J., Leung, S. S., & Penney, K. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27, 293–309.CrossRefGoogle Scholar
  31. Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31, 5–25.CrossRefGoogle Scholar
  32. Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25, 166–175.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Department of Education in Technology and ScienceTechnion – Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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