203
Views
2
CrossRef citations to date
0
Altmetric
Article

When Constant in a Proportional Relationship Isn’t Constant—A Sign of Not-So-Shared Understandings

ORCID Icon, & ORCID Icon

References

  • Arican, M. (2018). Preservice middle and high school mathematics teachers’ strategies when solving proportion problems. International Journal of Science and Mathematics Education, 16(2), 315–335. https://doi.org/10.1007/s10763-016-9775-1
  • Arican, M. (2019). Preservice mathematics teachers’ understanding of and abilities to differentiate proportional relationships from nonproportional relationships. International Journal of Science and Mathematics Education, 17(7), 1423–1443. https://doi.org/10.1007/s10763-018-9931-x
  • Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(2), 132–144. https://doi.org/10.2307/749140
  • Beckmann, S., & Izsák, A. (2015). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. Journal for Research in Mathematics Education, 46(1), 17–38. https://doi.org/10.5951/jresematheduc.46.1.0017
  • Bédard, J., & Chi, M. T. H. (1992). Expertise. Current Directions in Psychological Science, 1(4), 135–139. https://doi.org/10.1111/1467-8721.ep10769799
  • Boulet, G. (2007). How does language impact the learning of mathematics? Let me count the ways. Journal of Teaching and Learning, 5(1), 1–12. https://doi.org/10.22329/jtl.v5i1.125
  • Bransford, J. D., Brown, A. L., & Cocking, R. R. (1999). How people learn. National Academy Press.
  • Brown, R. E., Weiland, T., & Orrill, C. H. (2019). Mathematics teachers’ use of knowledge resources when identifying proportional reasoning situations. International Journal of Science and Mathematics Education. https://doi.org/10.1007/s10763-019-10006-3
  • Burke, J. P. (2017). Case study of mathematical playfulness in an adult mathematics classroom setting [ Unpublished doctoral dissertation]. University of Massachusetts.
  • De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311–334. https://doi.org/10.1023/A:1021205413749
  • diSessa, A. A. (1988). Knowledge in pieces. In G. Forman & P. Pufall (Eds.), Constructivism in the computer age (pp. 49–70). Lawrence Erlbaum Associates, Inc.
  • diSessa, A. A. (2018). A friendly introduction to “Knowledge in Pieces”: Modeling types of knowledge and their roles in Learning. In G. Kaiser, H. Forgasz, M. Graven, A. Kuzniak, E. Simmt, & B. Xu (Eds.), Invited Lectures from the 13th international congress on mathematical education. ICME-13 monographs (pp. 65–84). Springer International Publishing. https://doi.org/10.1007/978-3-319-72170-5_5
  • diSessa, A. A., Sherin, B. L., & Levin, M. (2016). Knowledge analysis: An introduction. In A. A. diSessa, B. L. Sherin, & M. Levin (Eds.), Knowledge and interaction: A synthetic agenda for the learning sciences (pp. 30–71). Routledge.
  • Greer, B. (2010). Overview of papers: Why is linear thinking so dominant? Mathematical Thinking and Learning, 12(1), 109–115. https://doi.org/10.1080/10986060903465996
  • Hammer, D. (2000). Student resources for learning introductory physics. American Journal of Physics, Physics Education Research Supplement, 68(S1), S52–S59. https://doi.org/10.1119/1.19520
  • Harel, G., & Behr, M. (1989). Structure and hierarchy of missing value proportion problems and their representations. Journal of Mathematical Behavior, 8(1), 77–119. https://wayback.archive-it.org/6305/20150924115736/http://www.cehd.umn.edu/ci/rationalnumberproject/89_5.html
  • Izsák, A. (2008). Mathematical knowledge for teaching fraction multiplication. Cognition and Instruction, 26(1), 95–143. https://doi.org/10.1080/0737000070179852
  • Izsák, A., & Jacobson, E. (2017). Preservice teachers’ reasoning about relationships that are and are not proportional: A knowledge-in-pieces account. Journal for Research in Mathematics Education, 48(3), 300–339. https://doi.org/10.5951/jresematheduc.48.3.0300
  • Izsák, A., Orrill, C. H., Cohen, A. S., & Brown, R. E. (2010). Measuring middle grades teachers’ understanding of rational numbers with the mixture Rasch model. The Elementary School Journal, 110(3), 279–300. https://doi.org/10.1086/648979
  • Kaput, J. J. (1985). Multiplicative word problems and intensive quantities: An integrated software response (Report No. 85-19). Educational Technology Center. ( ERIC Document Reproduction Service No. ED 295 787).
  • 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). National Council of Teachers of Mathematics.
  • Levin, M. (2009). A conceptual change lens on the emergence of a novel strategy during mathematical problem solving. 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 (pp. 193–200). Georgia State University.
  • Levin, M., & Brar, R. (2010). Coordination and contextuality: Revealing the nature of emergent mathematical understanding by means of a clinical interview. In K. Gomez, L. Lyons, & J. Randinsky (Eds.), Learning in the disciplines proceedings of the 9th international conference of the learning sciences - volume, short papers, symposia, and selected abstracts (Vol. 2, pp. 478–479). International Society of the Learning Sciences.
  • Lobato, J., & Ellis, A. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. National Council of Teachers of Mathematics.
  • Modestou, M., & Gagatsis, A. (2007). Students’ improper proportional reasoning: A result of the epistemological obstacle of “linearity”. Educational Psychology, 27(1), 75–92. https://doi.org/10.1080/01443410601061462
  • Modestou, M., & Gagatsis, A. (2010). Cognitive and metacognitive aspects of proportional reasoning. Mathematical Thinking and Learning, 12(1), 36–53. https://doi.org/10.1080/10986060903465822
  • Moore, K. C., Silverman, J., Paoletti, T., & LaForest, K. (2014). Breaking conventions to support quantitative reasoning. Mathematics Teacher Educator, 2(2), 141–157. https://doi.org/10.5951/mathteaceduc.2.2.0141
  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.
  • NGSS Lead States. (2013) . Next generation science standards: For states, by states. The National Academies Press.
  • Orrill, C. H., & Brown, R. E. (2012). Making sense of double number lines in professional development: Exploring teachers’ understandings of proportional relationships. Journal of Mathematics Teacher Education, 15(5), 381–403. https://doi.org/10.1007/s10857-012-9218-z
  • Stake, R. E. (2000). Case studies. In N. Denzin & Y. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 455–486). Sage.
  • Stavy, R., & Tirosh, D. (2000). How students (mis)-understand science and mathematics: Intuitive rules. Teachers College Press.
  • Steen, L. (Ed.). (2001). Mathematics and democracy: The case for quantitative literacy. The National Council on Education and the Disciplines.
  • Thompson, P. W., Hatfield, N. J., Yoon, H., Joshua, S., & Byerley, C. (2017). Covariational reasoning among US and South Korean secondary mathematics teachers. The Journal of Mathematical Behavior, 48, 95–111. https://doi.org/10.1016/j.jmathb.2017.08.001
  • 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. https://doi.org/10.2307/749339
  • Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralization. Cognition and Instruction, 23(1), 57–86. https://doi.org/10.1207/s1532690xci2301_3
  • Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127–174). Academic Press.
  • Weiland, T., Orrill, C. H., Brown, R. E., & Nagar, G. G. (2019). Mathematics teachers’ ability to identify situations appropriate for proportional reasoning. Research in Mathematics Education, 21(3), 233–250. https://doi.org/10.1080/14794802.2019.1579668

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.