Journal of Mathematics Teacher Education

, Volume 19, Issue 2–3, pp 123–148 | Cite as

Reflective analysis as a tool for task redesign: The case of prospective elementary teachers solving and posing fraction comparison problems

  • Eva ThanheiserEmail author
  • Dana Olanoff
  • Amy Hillen
  • Ziv Feldman
  • Jennifer M. Tobias
  • Rachael M. Welder


Mathematical task design has been a central focus of the mathematics education research community over the last few years. In this study, six university teacher educators from six different US institutions formed a community of practice to explore key aspects of task design (planning, implementing, reflecting, and modifying) in the context of comparing fractions using reasoning and sense-making. By presenting results of their implementation of two tasks with 63 prospective elementary teachers across three institutions and their reflective analysis of the implementation, the authors highlight the importance of collecting and analyzing data and reflecting on this analysis to inform the redesign of tasks. The authors also found that considering different types of tasks (problem solving vs. problem posing) helps illuminate different aspects of prospective elementary teachers' understanding, which can inform task redesign. Finally the authors contribute to the knowledge base on reasoning strategies for comparing fractions and prospective elementary teachers’ knowledge of these strategies.


Task design Fraction concepts Mathematical content knowledge Prospective elementary teachers 


  1. Adler, J., & Ball, D. (2008). Mathematical knowledge for teaching. Retrieved October 9, 2011, from
  2. Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practitioners: Towards a practice-based theory of professional education. In G. Sykes & L. Darling-Hammond (Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 3–32). San Francisco: Jossey-Bass.Google Scholar
  3. Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15(5), 323–341.CrossRefGoogle Scholar
  4. Chinnappan, M., & Forrester, T. (2014). Generating procedural and conceptual knowledge of fractions by pre-service teachers. Mathematics Education Research Journal, 26(4), 871–896.CrossRefGoogle Scholar
  5. 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, 127–138.CrossRefGoogle Scholar
  6. Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in preservice teachers' practices. Educational Studies in Mathematics, 52, 243–270.CrossRefGoogle Scholar
  7. Hiebert, J., & Behr, M. J. (Eds.) (1988). Introduction: Capturing the major themes. In Number concepts and operations in the middle grades (pp. 1–18). Reston, VA: Lawrence Erlbaum Associates and National Council of Teachers of Mathematics.Google Scholar
  8. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Erlbaum.Google Scholar
  9. Hiebert, J., Gallimore, R., & Stigler, J. W. (2002). A knowledge base for the teaching profession: What would it look like and how can we get one? Educational Researcher, 31(5), 3–15.Google Scholar
  10. Hill, H. C., Ball, D., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.Google Scholar
  11. Hillen, A., Olanoff, D., Thanheiser, E., Feldman, Z., Tobias, J., & Welder, R. (in preparation). Supporting prospective teachers' fraction number sense through problem solving and problem posing. Mathematics Teacher EducatorGoogle Scholar
  12. Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Charlotte, NC: Information Age.Google Scholar
  13. Lamon, S. (2012). Teaching fractions and ratios for understanding: Essential knowledge and instructional strategies for teachers. New York and London: Routledge/Taylor & Francis Group.Google Scholar
  14. Liljedahl, P., Chernoff, E., & Zazkis, R. (2007). Interweaving mathematics and pedagogy in task design: A tale of one task. Journal of Mathematics Teacher Education, 10(4–6), 239–249.CrossRefGoogle Scholar
  15. Livy, S. (2011). We can order by rote but can’t partition: We didn’t learn a rule. Paper presented at the Mathematics: Traditions and [new] practices. In Proceedings of the 34th annual conference of the Mathematics Education Research Group of Australasia and the Australian Association of Mathematics Teachers, Adelaide.Google Scholar
  16. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in china and the United States. Mahwah, NJ: Erlbaum.Google Scholar
  17. Markovits, Z., & Sowder, J. (1994). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 25(1), 4–29.Google Scholar
  18. Masingila, J. O., Olanoff, D. E., & Kwaka, D. K. (2012). Who teaches mathematics content courses for prospective elementary teachers in the united states? Results of a national survey. Journal of Mathematics Teacher Education, 15(5), 347–358.Google Scholar
  19. McClain, K. (2003). Supporting preservice teachers’ understanding of place value and multidigit arithmetic. Mathematical Thinking and Learning, 5(4), 281–306.CrossRefGoogle Scholar
  20. National Governors Association Center for Best Practices and Council of Chief State School Officers. (2010). Common core state standards initiative for mathematics. Retrieved February, 10, 2012, from
  21. National Governors Association and Council of Chief State School Officers. (2010). Common core standards intitiative. Retrieved February, 10, 2012, from
  22. National Research Council. (2001). Knowing and learning mathematics for teaching: Proceedings of a workshop. Washington DC: National Academy Press.Google Scholar
  23. Newmann, F. M., King, M. B., & Carmichael, D. L. (2007). Authentic instruction and assessment: Common standards for rigor and relevance in teaching academic subjects. Des Moines, IA: Iowa: Department of Education.Google Scholar
  24. Olanoff, D., Lo, J. J., & Tobias, J. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on fractions. The Mathematics Enthusiast, 11(2), 267–310.Google Scholar
  25. Rowland, T. (2008). The purpose, design and use of examples in the teaching of elementary mathematics. Educational Studies in Mathematics, 69(2), 149–163.Google Scholar
  26. San Diego State Foundation, Philipp, R., Cabral, C., & Schappelle, B. (2012). Imap: Integrating mathematics and pedagogy: Searchable collection of children’s mathematical thinking video clips and facilitator’s guide. Boston: Allyn & Bacon.Google Scholar
  27. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.CrossRefGoogle Scholar
  28. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.Google Scholar
  29. Silverman, J., & Thompson, P. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education, 11(6), 499–511. doi: 10.1007/s10857-008-9089-5.CrossRefGoogle Scholar
  30. Skinner, E. A., & Pitzer, J. R. (2012). Developmental dynamics of student engagement, coping, and everyday resilience. In S. L. Christenson, A. L. Reschly, & C. Wylie (Eds.), Handbook of research on student engagement (pp. 21–44). New York, NY: Springer.CrossRefGoogle Scholar
  31. Smith, M. S. (2001). Practice-based professional development for teachers of mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  32. Sowder, J. T. (1988). Mental computation and number comparisons: The role in development of number sense and computational estimation. In J. Hiebert & M. J. Behr (Eds.), Number concepts and operations in the middle grades (pp. 182–197). Reston, VA: Lawrence Erlbaum and National Council of Teachers of Mathematics.Google Scholar
  33. Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2, 50–80.CrossRefGoogle Scholar
  34. Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: From research to practice. Mathematics Teaching in the Middle School, 3(4), 268–275.Google Scholar
  35. Stein, M. K., & Smith, M. (2011). 5 practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  36. Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.CrossRefGoogle Scholar
  37. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.Google Scholar
  38. Suzuka, K., Sleep, L., Ball, D., Bass, H., Lewis, J. M., & Thames, M. H. (2009). Designing and using tasks to teach mathematical knowledge for teaching. In D. S. Mewborn & H. S. Lee (Eds.), Scholarly practices and inquiry in the preparation of mathematics teachers (amte monograph series, volume 6) (pp. 7–23). San Diego, CA: Association of Mathematics Teacher Educators.Google Scholar
  39. Thanheiser, E., Browning, C., Edson, A. J., Lo, J., Whitacre, I., Olanoff, D., & Morton, C. (2014). Mathematical content knowledge for teaching elementary mathematics: What do we know, what do we not know, and where do we go? The Mathematics Enthusiast, 8.Google Scholar
  40. Tobias, J. (2009). Preservice teachers’ conceptualization of fraction multiplication. In S. L. Swars, D. W. Stinson & S. Lemons-Smith (Eds.), Proceedings of the 31st annual meeting of the north american chapter of the international group for the psychology of mathematics education (Vol. 5, pp. 1276–1283). Atlanta, GA: Georgia State University.Google Scholar
  41. Tobias, J. M., Olanoff, D., Hillen, A. F., Welder, R. M., Feldman, Z., & Thanheiser, E. (2014). Using research to modify elementary school tasks for use in teacher preparation. In K. Karp & A. R. McDuffie (Eds.), Annual perspectives in mathematics education 2014: Using research to improve instruction (pp. 181–192). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  42. Watson, A., & Mason, J. (2007). Taken-as-shared: A review of common assumptions about mathematical tasks in teacher education. Journal of Mathematics Teacher Education, 10(4–6), 205–215.CrossRefGoogle Scholar
  43. Yackel, E., Underwood, D., & Elias, N. (2007). Mathematical tasks designed to foster a reconceptualized view of early arithmetic. Journal of Mathematics Teacher Education, 10(4–6), 351–367.CrossRefGoogle Scholar
  44. 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, 383–403.CrossRefGoogle Scholar
  45. Zazkis, R., & Chernoff, E. (2008). What makes a counterexample exemplary? Educational Studies in Mathematics, 68(3), 195–208.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Eva Thanheiser
    • 1
    Email author
  • Dana Olanoff
    • 2
  • Amy Hillen
    • 3
  • Ziv Feldman
    • 4
  • Jennifer M. Tobias
    • 5
  • Rachael M. Welder
    • 6
  1. 1.Portland State UniversityPortlandUSA
  2. 2.Widener UniversityChesterUSA
  3. 3.Kennesaw State UniversityKennesawUSA
  4. 4.Boston UniversityBostonUSA
  5. 5.Illinois State UniversityNormalUSA
  6. 6.Western Washington UniversityBellinghamUSA

Personalised recommendations