Professional Development to Support Students’ Algebraic Reasoning: An Example from the Problem-Solving Cycle Model

  • Karen KoellnerEmail author
  • Jennifer Jacobs
  • Hilda Borko
  • Sarah Roberts
  • Craig Schneider
Part of the Advances in Mathematics Education book series (AME)


The Problem-Solving Cycle (PSC) model of mathematics Professional Development (PD) seeks to enhance teachers’ knowledge and skills in a variety of domains. In this paper we consider how participating in the PSC program, with a specific focus on algebra, impacted one teachers’ instructional practice. We explore the nature of change in the teacher’s classroom, as well as the connection between these changes, the focus of our PD, and the teacher’s intentions. Our analyses are based on a set of videotaped classroom lessons collected over two years, along with interviews and written reflections. We conclude that there was a close match between the teacher’s personal goals for improvement and our program goals, and notable shifts in his algebra instruction that was likely to have supported his students’ algebraic reasoning.


Pedagogical Content Knowledge Instructional Practice Support Student Algebraic Thinking Algebraic Reasoning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ball, D. L., & Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a practice-based theory of professional education. In L. Darling-Hammond & G. Sykes (Eds.), Teaching as the Learning Profession: Handbook of Policy and Practice (pp. 3–32). San Francisco: Jossey-Bass. Google Scholar
  2. Ball, D. L., Lubienski, S., & Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge. In V. Richardson (Ed.), Handbook of Research on Teaching (4th ed.). New York: Macmillan. Google Scholar
  3. Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412–446. Google Scholar
  4. Borko, H., Frykholm, J. A., Pittman, M. E., Eiteljorg, E., Nelson, M., Jacobs, J. K., Clark, K. K., & Schneider, C. (2005). Preparing teachers to foster algebraic thinking. Zentralblatt für Didaktik der Mathematik: International Reviews on Mathematical Education, 37(1), 43–52. CrossRefGoogle Scholar
  5. Borko, H., Jacobs, J., Eiteljorg, E., & Pittman, M. E. (2008). Video as a tool for fostering productive discourse in mathematics professional development. Teaching and Teacher Education, 24, 417–436. CrossRefGoogle Scholar
  6. Clark, K. K., Jacobs, J., Pittman, M., & Borko, H. (2005). Strategies for building mathematical communication in the middle school classroom: Modeled in professional development, implemented in the classroom. Current Issues in Middle Level Education, 11(2), 1–12. Google Scholar
  7. Cnop, I., & Grandsard, F. (1998). Teaching abstract algebra concepts using small group instruction. International Journal of Mathematical Education in Science & Technology, 29(6), 843–851. CrossRefGoogle Scholar
  8. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13. CrossRefGoogle Scholar
  9. Design-Based Research Collective (2003). Design-based research: An emerging paradigm for educational inquiry. Educational Researcher, 32(1), 5–8. CrossRefGoogle Scholar
  10. Driscoll, M. (1999). Fostering Algebraic Thinking: A Guide for Teachers, Grades 6–10. New York: Heinemann. Google Scholar
  11. Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24, 94–116. CrossRefGoogle Scholar
  12. Ferrinni-Mundy, J., & Schram, T. (1997). Recognizing and Recording Reform in Mathematics Education: Issues and Implications. Reston, VA: National Council of Teachers of Mathematics. Journal for Research in Mathematics Education Monograph, no 8. Google Scholar
  13. Greeno, J. G. (2003). Situative research relevant to standards for school mathematics. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 304–332). Reston, VA: National Council of Teachers of Mathematics. Google Scholar
  14. Grossman, P., Hammerness, K., & McDonald, M. (2009). Refining teaching, re-imagining teacher education. Teachers and Teaching: Theory and Practice, 15(2), 273–289. Google Scholar
  15. Hadjidemetriou, C., & Williams, J. (2002). Teachers’ pedagogical content knowledge: Graphs from a cognitivist to a situated perspective. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 57–64). Norwich, UK. Google Scholar
  16. Jacobs, J., Hiebert, J., Givvin, K. B., Hollingsworth, H., Garnier, H., & Wearne, D. (2006). Does eighth-grade teaching in the United States align with the NCTM Standards? Results from the TIMSS 1995 and 1999 Video Studies. Journal for Research in Mathematics Education, 37(1), 5–32. Google Scholar
  17. Jacobs, J., Borko, H., Koellner, K., Schneider, C., Eiteljorg, E., & Roberts, S. A. (2007). The problem-solving cycle: A model of mathematics professional development. Journal of Mathematics Education Leadership, 10(1), 42–57. Google Scholar
  18. Kaput, J. J. (2007). What is algebra? What is algebraic reasoning. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the Early Grades. London: Taylor & Francis. Google Scholar
  19. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics (Vol. 2, pp. 707–752). Charlotte, NC: Information Age Publishing. Google Scholar
  20. Koellner, K., Jacobs, J., Borko, H., Schneider, C., Pittman, M., Eiteljorg, E., Bunning, K., & Frykholm, J. (2007). The problem-solving cycle: A model to support the development of teachers’ professional knowledge. Mathematical Thinking and Learning, 9(3), 271–303. Google Scholar
  21. Lampert, M. (2001). Teaching Problems and the Problems of Teaching. New Haven: Yale University Press. Google Scholar
  22. Lave, J., & Wenger, E. (1991). Situated Learning: Legitimate Peripheral Participation. Cambridge, UK: Cambridge University Press. Google Scholar
  23. Learning Mathematics for Teaching (LMT). (2006). A coding rubric for measuring the quality of mathematics in instruction (Technical Report LMT1.06). Ann Arbor, MI: University of Michigan, School of Education. Google Scholar
  24. Miles, M. B., & Huberman, A. M. (1994). An Expanded Sourcebook: Qualitative Data Analysis. Thousand Oaks, CA: Sage. Google Scholar
  25. Moses, R. P., & Cobb, C. (2001). Organizing algebra: The need to voice a demand. Social Policy, 31(4), 4–12. Google Scholar
  26. Nathan, M. J., & Koedinger, K. R. (2000). Teachers’ and researchers’ beliefs about the development of algebraic reasoning. Journal for Research in Mathematics Education, 31, 168–190. CrossRefGoogle Scholar
  27. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: Author. Google Scholar
  28. National Mathematics Advisory Panel (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Google Scholar
  29. National Research Council (1998). The Nature and Role of Algebra in the K-14 Curriculum: Proceedings of a National Symposium. Washington, DC: National Academy Press. Google Scholar
  30. Putnam, R., & Borko, H. (2000). What do new views of knowledge and thinking have to say about research on teacher learning? Educational Researcher, 29(1), 4–15. Google Scholar
  31. Shulman, L. S. (1983). Autonomy and obligation: The remote control of teaching. In L. S. Shulman & G. Sykes (Eds.), Handbook of Teaching and Policy (pp. 484–504). New York: Longman. Google Scholar
  32. Winicki-Landman, G. (2001). Shhh… Let them think… Let them talk!. Australian Senior Mathematics Journal, 15(2), 30–38. Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Karen Koellner
    • 1
    Email author
  • Jennifer Jacobs
    • 2
  • Hilda Borko
    • 3
  • Sarah Roberts
    • 4
  • Craig Schneider
    • 2
  1. 1.School of EducationHunter CollegeNew YorkUSA
  2. 2.School of EducationUniversity of Colorado at BoulderBoulderUSA
  3. 3.School of EducationStanford UniversityStanfordUSA
  4. 4.Department of Curriculum and InstructionIowa State UniversityAmesUSA

Personalised recommendations