Misconceptions on part-part-whole proportional relationships using proportional division problems

References

• Adjiage, R., & Pluvinage, F. (2007). An experiment in teaching ratio and proportion. Educational Studies in Mathematics, 65(2), 149–175. doi:10.1007/s10649-006-9049-x
   Google Scholar
• Alatorre, S., & Figueras, O. (2005). A developmental model for proportional reasoning in ratio comparison tasks. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the international group for the psychology of mathematics education (Vol. 2, pp. 25–32). Australia: Melbourne, Victoria.
   Google Scholar
• Beckmann, S., & Izsak, 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. doi:10.5951/jresematheduc.46.1.0017
   Web of Science ®Google Scholar
• The Centre for Innovation in Mathematics Teaching. (n.d.). Unit 7 section 3: Proportional division. Retrieved March 15, 2018, from http://www.cimt.org.uk/projects/mepres/book8/bk8i7/bk8_7i3.htm
   Google Scholar
• Charters, E. (2003). The use of think-aloud methods in qualitative research: An introduction to think-aloud methods. Brock Education, 12(2), 68–82.
   Google Scholar
• Clark, M. R., Berenson, S. B., & Cavey, L. O. (2003). A comparison of ratios and fractions and their roles as tools in proportional reasoning. The Journal of Mathematical Behavior, 22(3), 297–317. doi:10.1016/S0732-3123(03)00023-3
   Google Scholar
• The Common Core Standards Writing Team. (2012). Progressions for the common core state standards in mathematics: 6–7, ratios and proportional relationships. Retrieved from http://ime.math.arizona.edu/progressions/
   Google Scholar
• Creswell, J. W. (2007). Qualitative inquiry & research design: Choosing among five approaches (2nd ed.). Thousand Oaks, CA: Sage Publications.
   Google Scholar
• De Bock, D., Verschaffel, L., Janssens, D., Van Dooren, W., & Claes, K. (2003). Do realistic contexts and graphical representations always have a beneficial impact on students’ performance? Negative evidence from a study on modelling non-linear geometry problems. Learning and Instruction, 13(4), 441–463. doi:10.1016/S0959-4752(02)00040-3
   Web of Science ®Google Scholar
• de la Torre, J., Tjoe, H., Rhoads, K., & Lam, D. (2013). Conceptual and theoretical issues in proportional reasoning. International Journal for Studies in Mathematics Education, 6(1), 21–38.
   Google Scholar
• Fernandez, C., Llinares, S., Modestou, M., & Gagatsis, A. (2010). Proportional reasoning: How task variables influence the development of students’ strategies from primary to secondary school. Acta Didactica Universitatis Comenianae Mathematics (10), 1–18.
   Google Scholar
• Freudenthal, H. (1983). Ratio and proportion. In Didactical phenomenology of mathematical structures | Hans Freudenthal | Springer (pp. 178–209). Springer. Retrieved from www.springer.com/us/book/9789027715357
   Google Scholar
• Heinz, K., & Sterba-Boatwright, B. (2008). The when and why of using proportions. Mathematics Teachers, 101, 528–533.
   Google Scholar
• Jeong, Y., Levine, S. C., & Huttenlocher, J. (2007). The development of proportional reasoning: Effect of continuous versus discrete quantities. Journal of Cognition and Development, 8(2), 237–256. doi:10.1080/15248370701202471
   Web of Science ®Google Scholar
• Kaput, J. J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The Development of multiplicative reasoning in the learning of mathematics (pp. 179–289). Albany, NY: State University of New York Press.
   Google Scholar
• Karplus, R., Pulos, S., & Stage, E. K. (1983). Early adolescents’ proportional reasoning on ‘rate’ problems. Educational Studies in Mathematics, 14(3), 219–233. doi:10.1007/BF00410539
   Google Scholar
• Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, D.C.: National Academies Press. Retrieved from http://www.nap.edu/catalog/9822.html
   Google Scholar
• Lamon, S. J. (1993). Ratio and proportion: Connecting content and children’s thinking. Journal for Research in Mathematics Education, 24(1), 41–61. doi:10.2307/749385
   Web of Science ®Google Scholar
• Lamon, S. J. (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. 629–668). Charlotte, NC: Information Age Publishing.
   Google Scholar
• Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed.). New York, NY: Routledge.
   Google Scholar
• Lesh, R., Post, T., & Behr, M. J. (1988). Proportional reasoning. In J. Hiebert & M. J. Behr (Eds.), Number concepts and operations in the middle grades. [Hillsdale, N.J.]:Reston, VA: Lawrence Erlbaum Associates; National Council of Teachers of Mathematics.
   Google Scholar
• Lobato, J., & Ellis, A. B. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics in grades 6–8. Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Martinez, R., & I, J. (2018). Four ways to determine equivalent ratios. Mathematics Teaching in The Middle School, 24(1), 48-52. doi:10.5951/mathteacmiddscho.24.1.0048
   Google Scholar
• National Center for Education Statistics. (2013). The nation’s report card mathematics and reading 2013. Washington, D.C.: Institute of Education Science, U.S. Department of Education.
   Google Scholar
• National Council of Teachers of Mathematics. (2013). Teaching ratio and proportion in the middle grades (Research Brief). Reston, VA: NCTM.
   Google Scholar
• National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, D.C.: National Governors Association Center for Best Practices, Council of Chief State School Officers.
   Google Scholar
• National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, D.C.: Department of Education.
   Google Scholar
• Silver, E. A. (1981). Young adults’ thinking about rational numbers. In T. R. Post & M. P. Roberts (Eds.), Proceedings of the third annual meeting of the North American chapter of the international group for the psychology of mathematics education (pp. 149–159). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
   Google Scholar
• Van Dooren, W., De Bock, D., Evers, M., & Verschaffel, L. (2009). Students’ overuse of proportionality on missing-value problems: How numbers may change solutions. Journal for Research in Mathematics Education, 40(2), 187–211.
   Web of Science ®Google Scholar
• Van Dooren, W., De Bock, D., Janssens, D., & Verschaffel, L. (2007). Pupils’ over-reliance on linearity: A scholastic effect? British Journal of Educational Psychology, 77(2), 307–321. doi:10.1348/000709906X115967
   PubMedGoogle Scholar
• Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. J. Behr (Eds.), Number concepts and operations in the middle grades (pp. 127-174). [Hillsdale, N.J.]: Reston, VA: Lawrence Erlbaum Associates; National Council of Teachers of Mathematics.
   Google Scholar