Journal of Mathematics Teacher Education

, Volume 15, Issue 2, pp 109–130 | Cite as

Learning to pose cognitively demanding tasks through letter writing

  • Anderson Norton
  • Signe Kastberg


We have used letter writing as a means for preservice teachers (PSTs) to develop ability to design effective tasks, in terms of eliciting high levels of cognitive activity from students. Studies on student-dependent task analyses, by assessing the levels of cognitive demand indicated in students’ responses, have demonstrated significant growth among PSTs over the course of letter-writing exchanges. We examine growth with a qualitative analysis of two PSTs who became effective at designing tasks that elicited high levels of cognitive activity. In particular, we examine how those PSTs accounted for tasks that did not elicit the kinds of activity they expected and how they adjusted their tasks to elicit higher levels of activity. We found disparity between the two PSTs’ apparently successful approaches: one that fit the larger goals of the project and one that fit only the descriptions specified in the project rubric. The study affirms the potential value of letter-writing projects while introducing a concern that has implications for all professional development projects.


Mathematical tasks Preservice teachers Professional development Students’ mathematical thinking Task design Task posing 



We thank the Indiana Mathematics Initiative, Zachary Rutledge, and Kareston Hall for their support on the underlying project.


  1. Boston, M. D., & Smith, M. S. (2009). Transforming secondary mathematics teaching. Increasing the cognitive demands of instructional tasks used in teachers’ classrooms. Journal for Research in Mathematics Education, 40(20), 119–156.Google Scholar
  2. Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and construction of situations in teaching and learning mathematics. In H.-G. Steiner (Ed.), Theory of mathematics education: ICME 5 topic area and miniconference (pp. 110–119). Bielefeld, Germany: Institut fur Didaktik der Mathematik der Universitat Bielefeld.Google Scholar
  3. Brown, S. I., & Walter, M. I. (1990). The art of problem posing (2nd ed.). Hillsdale, NJ: Erlbaum.Google Scholar
  4. Chamberlin, M. T. (2005). Teachers’ discussions of students’ thinking: Meeting the challenge of attending to students’ thinking. Journal of Mathematics Teacher Education, 8(2), 141–170.CrossRefGoogle Scholar
  5. Cobb, P., Wood, T., Yackel, E., Nicholls, J., Wheatley, G., Trigatti, B., et al. (1991). Assessment of a problem-centered second-grade mathematics project. Journal for Research in Mathematics Education, 22(1), 3–29.CrossRefGoogle Scholar
  6. Crespo, S. (2000). Seeing more than right and wrong answers: Prospective teachers’ interpretations of students’ mathematical work. Journal of Mathematics Teacher Education, 3, 155–181.CrossRefGoogle Scholar
  7. Crespo, S. (2003). Learning to pose mathematical problems: Exploring changes in PSTs’ practices. Educational Studies in Mathematics, 52, 243–270.CrossRefGoogle Scholar
  8. D’Ambrosio, B. (2004). Preparing teachers to teach mathematics within a constructivist framework: The importance of listening to children. In T. Watanabe & D. Thompson (Eds.), The work of mathematics teacher educators: Exchanging ideas for effective practice (Vol. 1, pp. 135–150). San Diego, CA: Association of Mathematics Teacher Educators.Google Scholar
  9. Davis, B. (1997). Listening for differences: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education, 28, 355–376.CrossRefGoogle Scholar
  10. Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403–434.CrossRefGoogle Scholar
  11. Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support or inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education, 28(5), 524–549.CrossRefGoogle Scholar
  12. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.Google Scholar
  13. Horoks, J., & Robert, A. (2007). Tasks designed to highlight task-activity relationships. Journal of Mathematics Teacher Education, 10, 279–287.CrossRefGoogle Scholar
  14. Kagan, D. (1992). Professional growth among preservice and beginning teachers. Review of Educational Research, 62(2), 129–169.Google Scholar
  15. Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Hillsdale, NJ: Erlbaum.Google Scholar
  16. Kilpatrick, J. (2009, February). Going to war with the army you have. Paper presented at the 13th annual conference of the Association of Mathematics Teacher Educators, Orlando, FL.Google Scholar
  17. Kosko, K. W., Norton, A., Conn, A., & San Pedro, J. M. (2010). Letter writing: Providing preservice teachers with experience in posing appropriate mathematical tasks to high school students. In J. W. Lott & J. Luebeck (Eds.), Association of mathematics teacher educators monograph 7: Mathematics teaching: Putting research into practice at all levels (pp. 207–224). San Diego, CA: Association of Mathematics Teacher Educators.Google Scholar
  18. Krainer, K. (1993). Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction. Educational Studies in Mathematics, 24(1), 65–93.CrossRefGoogle Scholar
  19. Lester, F. K., & Kehle, P. E. (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism (pp. 501–518). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  20. Levin, D., Hammer, D., & Coffey, J. (2009). Novice teachers’ attention to student thinking. Journal of Teacher Education, 60, 142–154.CrossRefGoogle Scholar
  21. 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, 239–249.CrossRefGoogle Scholar
  22. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.Google Scholar
  23. Norton, A., & Rutledge, Z. (2010). Measuring responses to task posing cycles: Mathematical letter writing between algebra students and pre-service teachers. The Mathematics Educator, 19(2), 32–45.Google Scholar
  24. Norton, A., Rutledge, Z., Hall, K., & Norton, R. (2009). Mathematical letter writing: An opportunity for further partnership between high schools and universities. Mathematics Teacher, 103(5), 340–346.Google Scholar
  25. Polya, G. (1957). How to solve it (2nd ed.). New York: Doubleday.Google Scholar
  26. Prestage, S., & Perks, P. (2007). Developing teacher knowledge using a tool for creating tasks for the classroom. Journal of Mathematics Teacher Education, 10, 381–390.CrossRefGoogle Scholar
  27. Schifter, D. (1996). A constructivist perspective on teaching and learning mathematics. In C. T. Fosnot (Ed.), Constructivism: Theory, perspectives, and practice (pp. 73–80). New York, London: Teachers College Press.Google Scholar
  28. Schoen, H. L., & Charles, R. I. (Eds.). (2003). Teaching mathematics through problem solving: Grades 6–12. Reston, VA: NCTM.Google Scholar
  29. Silver, E. A., Mamona-Downs, J., Leung, S. S., & Kenney, P. A. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309.CrossRefGoogle Scholar
  30. Sim, J., & Wright, C. C. (2005). The Kappa Statistic in reliability studies: Use, interpretation, and sample size requirements. Physical Therapy, 85(3), 257–268.Google Scholar
  31. 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.Google Scholar
  32. Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College, Columbia University.Google Scholar
  33. Thompson, P. W., Carlson, M. P., & Silverman, J. (2007). The design of tasks in support of teachers’ development of coherent mathematical meanings. Journal of Mathematics Teacher Education, 10, 415–432.CrossRefGoogle Scholar
  34. von Glasersfeld, E., & Steffe, L. P. (1991). Conceptual models in educational research and practice. Journal of Educational Thought, 25(2), 91–103.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentVirginia TechBlacksburgUSA
  2. 2.Department of Curriculum and InstructionPurdue UniversityWest LafayetteUSA

Personalised recommendations