This study presents a qualitative analysis of the knowledge teachers in one professional development course used to reason about proportional relationships with double number lines. We work from the knowledge-in-pieces perspective to consider the existing knowledge the participants did or did not invoke when learning to reason with this new-to-them representation. We analyzed videotaped sessions of a group of urban middle-grade teachers across five class meetings. Our findings include discussion of the two pieces of knowledge that emerged as important for reasoning about proportions with the representation and three knowledge pieces that deterred meaning making. Implications for professional development are discussed as are the implications for conceptualizing teachers’ understanding through a knowledge-in-pieces lens.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
We use “coordinate” rather than composed unit throughout this paper to indicate instances in which it is clear that the teacher is reasoning about the quantities in a related way. We lack data necessary to determine whether these teachers were able to fully reason about the quantities as composed units (Lamon 1995).
Abels, M., Wijers, M., Pligge, M., & Hedges, T. (2006). Models you can count on. In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in context. Chicago: Encyclopædia Britannica, Inc.
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special. Journal of Teacher Education, 59(5), 389–407.
Confrey, J., & Maloney, A. (2010). The construction, refinements, and early validation of the equipartitioning learning trajectory. In K. Gomez, L. Lyons, & J. Radinsky (Eds.), Learning in the disciplines: ICLS 2010 conference proceedings (Vol. 1, pp. 968–975). Chicago: University of Illinois at Chicago.
Corina, J. L., Zhao, Q., Cobb, P., & McClain, K. (2004). Supporting students’ reasoning with inscriptions. In Y. B. Kafai, W. A. Sandoval, N. Enyedy, A. S. Nixon, & F. Herrera (Eds.), ICLS 2004: Embracing diversity in the learning sciences (pp. 142–149). Mahwah, NJ: Lawrence Erlbaum Associates.
Denzin, N. K. (1989). The research act: A theoretical introduction to sociological methods. Englewood Cliffs, NJ: Prentice Hall.
diSessa, A. (1988). Knowledge in pieces. In G. Forman & P. Putall (Eds.), Constructivism in the computer age (pp. 49–70). Hillsdale, NJ: Lawrence Erlbaum Associates.
diSessa, A. A. (2006). A history of conceptual change research: Threads and fault lines. In R. K. Sawyer (Ed.), The Cambridge handbook of the learning sciences (pp. 265–282). New York: Cambridge University Press.
Dufour-Janvier, B., Bednarz, N., & Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 109–222). Hillsdale, NJ: Lawrence Erlbaum Associates.
Harel, G., & Behr, M. (1995). Teachers’ solutions for multiplicative problems. Hiroshima Journal of Mathematics Education, 3, 31–51.
Hiebert, J., & Stigler, J. W. (2000). A proposal for improving classroom teaching: Lessons from the TIMSS Video Study. Elementary School Journal, 101(1), 3–20.
Hill, H. (2007). Mathematical knowledge of middle school teachers: Implications for the No Child Left Behind Act. Educational Evaluation and Policy Analysis, 29(2), 95–114.
Hill, H. C., Ball, D. L., & Shilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 373–400.
Izsák, A. (2008). Mathematical knowledge for teaching fraction multiplication. Cognition and Instruction, 26(1), 95–143.
Izsák, A., Orrill, C. H., Cohen, A., & Brown, R. E. (2010). Measuring middle grades teachers’ understanding of rational numbers with the mixture Rasch model. Elementary School Journal, 110(3), 279–300.
Küchemann, D., Hodgen, J., & Brown, M. (2011, February). Using the double number line to model multiplication. Paper presented at Seventh Annual Congress of the European Society for Research in Mathematics Education, Rzeszów, Poland.
Lamon, S. (1993). Ratio and proportion: Connecting content and children’s thinking. Journal for Research in Mathematics Education, 24(1), 41–61.
Lamon, S. (1995). Ratio and proportion: Elementary didactical phenomenology. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 167–198). Albany, NY: State University of New York Press.
Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Charlotte, NC: Information Age Publishing.
Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems on representation in the teaching and learning of mathematics (pp. 33–40). Hillsdale, NJ: Lawrence Erlbaum Associates.
Lobato, J., Ellis, A., & Muñoz, R. (2003). How “focusing phenomena” in the instructional environment support individual students’ generalizations. Mathematical Thinking and Learning, 5(1), 1–36.
Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. The Journal of Mathematical Behavior, 21, 87–116.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
Moss, J., & Case, R. (1999). Developing childrens’ understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122–147.
Post, T., Harel, G., Behr, M., & Lesh, R. (1988). Intermediate teachers knowledge of rational number concepts. In E. Fennema (Ed.), Papers from first Wisconsin symposium for research on teaching and learning mathematics (pp. 194–219). Madison, WI: Wisconsin Center for Education Research.
Riley, K. R. (2010). Teachers’ understanding of proportional reasoning. In P. Brosnan, D. B. Erchick, & L. Flevares (Eds.), Proceedings of the 32 nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 6, pp. 1055–1061). Columbus, OH: The Ohio State University.
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.
Simon, M. A., & Blume, G. W. (1994). Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers. Journal of Mathematical Behavior, 13(2), 183–197.
Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 43(3), 271–292.
Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A Constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3(2), 115–163.
Steffe, L. P., & Olive, J. (2009). Children’s fractional knowledge. New York: Springer.
Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Thousand Oaks, CA: Sage.
Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179–234). Albany, NY: SUNY Press.
Thompson, P. W., & Thompson, A. G. (1994). Talking about rates conceptually, part I: A teacher’s struggle. Journal for Research in Mathematics Education, 25(3), 279–303.
Thompson, A. G., & Thompson, P. W. (1996). Talking about rates conceptually, part II: Mathematical knowledge for teaching. Journal for Research in Mathematics Education, 27(1), 2–24.
Work on this project was supported by the National Science Foundation (DRL-0633975 and DRL-1036083). The opinions expressed here are those of the authors and do not necessarily reflect the views of NSF. The authors wish to thank Gunhan Caglayan for his work on the data analysis as well as the Does it Work team, the participating teachers, and the facilitator for their support with data collection.
Rights and permissions
About this article
Cite this article
Orrill, C.H., Brown, R.E. Making sense of double number lines in professional development: exploring teachers’ understandings of proportional relationships. J Math Teacher Educ 15, 381–403 (2012). https://doi.org/10.1007/s10857-012-9218-z
- Middle grades
- Professional development
- Proportional reasoning
- Drawn representations
- Teacher knowledge