Educational Studies in Mathematics

, Volume 84, Issue 1, pp 93–113 | Cite as

An exploratory study of pre-service middle school teachers’ knowledge of algebraic thinking

  • Marta T. Magiera
  • Leigh A. van den Kieboom
  • John C. Moyer


Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers’ ability to use the features of the algebraic thinking (AT) habit of mind “Building Rules to Represent Functions” and their ability to recognize and interpret the features of the same AT habit of mind in middle school students. We assessed the pre-service teachers’ own ability to use the AT habit of mind Building Rules to Represent Functions by examining their solutions to algebra-based tasks in a semester-long mathematics content course. We assessed the pre-service teachers’ ability to recognize and interpret students’ facility with the AT habit of mind Building Rules to Represent Functions by analyzing their interpretations of students’ written solutions to algebra-based tasks and their ability to plan and analyze AT interviews of middle school students during a concurrent field-based education course. The data revealed that the pre-service teachers had a limited ability to recognize the full richness of algebra-based tasks’ potential to elicit the features of Building Rules to Represent Functions in students. The pre-service teachers’ own overall AT ability to Build Rules to Represent Functions was related to their ability to recognize the overall ability of students to Build Rules to Represent Functions, as exhibited during one-on-one interviews, but not to their ability to recognize the overall ability to Build Rules to Represent Functions exhibited exclusively in students’ written work. Implications for mathematics teacher education are discussed.


Algebraic thinking Algebra instruction Teacher knowledge Teacher education 


  1. Algebra Working Group to the National Council of Teachers of Mathematics. (1997). A framework for constructing a vision of algebra: A discussion document. Reston, VA: NCTM.Google Scholar
  2. Borko, H., & Putnam, R. T. (1996). Learning to teach. In R. Calfee & D. Berliner (Eds.), Handbook of educational psychology (pp. 673–725). New York: Macmillan.Google Scholar
  3. Carpenter, T. P., & Fennema, E. (1992). Cognitively guided instruction: Building on the knowledge of students and teachers. In W. Secada (Ed.), Curriculum reform: The case of mathematics education in the United States. Special issue of International Journal of Educational Research (pp. 457–470). Elmsford, NY: Pergamon.Google Scholar
  4. Carpenter, T., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.Google Scholar
  5. Clement, L. (2001). What do students really know about functions? Mathematics Teacher, 94(9), 745–748.Google Scholar
  6. Cuoco, A., Goldenberg, P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curriculum. Journal of Mathematical Behavior, 15, 375–402.CrossRefGoogle Scholar
  7. Derry, S., Wilsman, M., & Hackbarth, A. (2007). Using contrasting cases to deepen teacher understanding of algebraic thinking. Mathematical Thinking and Learning, 9(3), 305–329.CrossRefGoogle Scholar
  8. Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6–10. Portsmouth, NH: Heinemann.Google Scholar
  9. Driscoll, M. (2001). The fostering of algebraic thinking toolkit: A guide for staff development (Introduction and analyzing written student work module). Portsmouth, NH: Heinemann.Google Scholar
  10. Driscoll, M., & Moyer, J. (2008). Using students’ work as a lens on algebraic thinking. In J. M. Bay-Williams & K. Karp (Eds.), Readings from NCTM publications for grades K-8 (pp. 26–32). Reston, VA: NCTM.Google Scholar
  11. Driscoll, M., Moyer, J., & Zawojewski, J. (1998). Helping teachers implement algebra for all in Milwaukee Public Schools. Journal of Mathematics Education Leadership, 2, 3–12.Google Scholar
  12. Hill, H. C. (2010). The nature and predictors of elementary teachers’ mathematical knowledge for teaching. Journal for Research in Mathematics Education, 41(5), 513–545.Google Scholar
  13. Hill, H. C., Rowan, B., & Ball, D. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Education Research Journal, 42(2), 371–406.CrossRefGoogle Scholar
  14. Kaput, J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. In S. Fennel (Ed.), The nature and role of algebra in the K-14 curriculum: Proceedings of a national symposium (pp. 25–26). Washington, DC: National Research Council, National Academy Press.Google Scholar
  15. Kieran, C. (1996). The changing face of school algebra. In C. Alsina, J. Alvarez, B. Hodgson, C. Laborde, & A. Pérez (Eds.), 8th International Congress on Mathematical Education: Selected lectures (pp. 271–290). Sevilla, Spain: S.A.E.M. Thales.Google Scholar
  16. Kieran, C. (2004). Algebraic thinking in the middle grades: What is it? The Mathematics Educator, 8(1), 139–151.Google Scholar
  17. Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Charlotte, NC: Information Age.Google Scholar
  18. Kieran, C., & Chalouh, L. (1993). Prealgebra: The transition from arithmetic to algebra. In D. T. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. 179–198). New York: Macmillan.Google Scholar
  19. Lesh, R., & Kelly, A. (2000). Multi-tiered teaching experiments. In A. Kelly & R. Lesh (Eds.), Research design in mathematics and science education (pp. 197–230). Amsterdam, Netherlands: Kluwer.Google Scholar
  20. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.Google Scholar
  21. Magiera, M. T., van den Kieboom, L., & Moyer, J. (2011). Relationships among features of pre-service teachers’ algebraic thinking. In B. Ubuz (Ed.), Proceedings of the 35th conference of the International Group for the Psychology in Mathematics Education, Vol.3 (pp. 169–176). Ankara, Turkey: PME.Google Scholar
  22. Mewborn, D. (2003). Teaching, teachers’ knowledge, and their professional development. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), Research companion to principles and standards for school mathematics (pp. 45–52). Reston, VA: NCTM.Google Scholar
  23. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.Google Scholar
  24. Philipp, R., Ambrose, R., Lamb, L., Sowder, J., Schappelle, B., Sowder, L., et al. (2007). Effects of early field experiences on the mathematical content knowledge and beliefs of prospective elementary school teachers: An experimental study. Journal for Research in Mathematics Education, 38(5), 438–476.Google Scholar
  25. Silver, A. E. (1997). Algebra for all: Increasing students’ access to algebraic ideas, not just algebra courses. Mathematics Teaching in the Middle School, 2(4), 204–207.Google Scholar
  26. Swafford, J. O., & Langrall, C. W. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31, 81–112.CrossRefGoogle Scholar
  27. Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5–25.CrossRefGoogle Scholar
  28. Vacc, N. N., & Bright, G. W. (1999). Elementary pre-service teachers’ changing beliefs and instructional use of children’s mathematical thinking. Journal for Research in Mathematics Education, 30(1), 89–110.CrossRefGoogle Scholar
  29. van den Kieboom, L. A., Magiera, M. T., & Moyer, J. C. (2010, April). Pre-service teachers’ knowledge of algebraic thinking and the characteristics of the questions posed for students. Paper presented at the 2010 annual meeting of the American Educational Research Association. Retrieved September 27, 2010 from the AERA Online Paper Repository Accessed 19 Feb 2011.
  30. van Dooren, W., Verschaffel, L., & Onghema, P. (2002). The impact of pre-service teachers’ content knowledge on their evaluation of students’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33(5), 319–351.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Marta T. Magiera
    • 1
  • Leigh A. van den Kieboom
    • 2
  • John C. Moyer
    • 3
  1. 1.Department Mathematics, Statistics, and Computer ScienceMarquette UniversityMilwaukeeUSA
  2. 2.College of EducationMarquette UniversityMilwaukeeUSA
  3. 3.Department of Mathematics, Statistics and Computer ScienceMarquette UniversityMilwaukeeUSA

Personalised recommendations