Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions
 Boris Koichu,
 Guershon Harel,
 Alfred Manaster
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Twentyfour 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.
 Ball, D., Thames, M., & Phelps, G. (2008). Content knowledge for teaching. Journal of Teacher Education, 59, 389–407. CrossRef
 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. CrossRef
 Brown, S. (2001). Reconstructing school mathematics: Problems with problems and the real world. New York: Peter Lang.
 Clark, D., & Sukenik, M. (2006). Assessing fraction understanding using taskbased 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.
 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.
 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. CrossRef
 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. CrossRef
 Dey, I. (1999). Grounding grounded theory: Guidelines for qualitative inquiry. San Diego, CA: Academic Press.
 Ericsson, K., & Simon, H. (1993). Protocol analysis: Verbal reports as data (revised version). Cambridge, MA: MIT Press.
 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. CrossRef
 Glaser, B., & Strauss, A. (1967). The discovery of grounded theory. Chicago: Aldine.
 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.
 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.
 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.
 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.
 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.
 Harel, G. (2008a). DNR perspective on mathematics curriculum and instruction: Focus on proving, Part I. Zentralblatt fuer Didaktik der Mathematik, 40, 487–500. CrossRef
 Harel, G. (2008b). DNR perspective on mathematics curriculum and instruction, Part II. Zentralblatt fuer Didaktik der Mathematik, 40, 893–907. CrossRef
 Harel, G., & Koichu, B. (2010). An operational definition of learning. Journal of Mathematical Behavior, 29, 115–124. CrossRef
 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.
 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.
 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.
 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.
 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.
 Koichu, B., & Harel, G. (2007). Triadic interaction in clinical taskbased interviews with mathematics teachers. Educational Studies in Mathematics, 65, 349–365. CrossRef
 Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Erlbaum.
 Mack, N. (1995). Confounding wholenumber and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26(5), 422–441. CrossRef
 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. CrossRef
 Piaget, J. (1985). The equilibration of cognitive structures: The central problem of intellectual development. Chicago: University of Chicago.
 Silver, E. A., MamonaDowns, J., Leung, S. S., & Penney, K. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27, 293–309. CrossRef
 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. CrossRef
 TolukUçar, Z. (2009). Developing preservice teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25, 166–175. CrossRef
 Title
 Ways of thinking associated with mathematics teachers’ problem posing in the context of division of fractions
 Journal

Instructional Science
Volume 41, Issue 4 , pp 681698
 Cover Date
 20130701
 DOI
 10.1007/s1125101292541
 Print ISSN
 00204277
 Online ISSN
 15731952
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

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

 Boris Koichu ^{(1)}
 Guershon Harel ^{(2)}
 Alfred Manaster ^{(2)}
 Author Affiliations

 1. Department of Education in Technology and Science, Technion – Israel Institute of Technology, 32000, Haifa, Israel
 2. Department of Mathematics, University of California, San Diego, La Jolla, CA, 920930112, USA