References
- Ainsworth, S., Bibby, P., & Wood, D. (2002). Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences, 11(1), 25-61. https://doi.org/10.1207/S15327809JLS1101_2
- Bottoms, G. (2003). Getting students ready for Algebra I: What middle grades students need to know and be able to do (Report No. 02V52). Atlanta, GA: Southern Regional Education Board. (ERIC Document Reproduction Service No. ED476617).
- Bruner, J. (1966). Toward a Theory of Instruction. Cambridge, MA: Harvard University Press.
- Creswell, J. W., & Plano Clark, V. L. (2011). Designing and conducting mixed methods research (2nded.). Thousand Oaks, CA: Sage.
- Donovan, M. S., & Bransford, J. D. (Eds.). (2005). How students learn: Mathematics in the classroom. Washington D. C.: The National Academies Press.
- Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt &M. Santos (Eds.), Proceedings of the 21st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3-26). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. (ERIC Document Reproduction Service No. ED433998)
- Fosnot, C. T., & Dolk, M. (2002). Young mathematicians at work: constructing fractions, decimals, and percents. Portsmouth, NH: Heinemann.
- Freudenthal, H. (1991). Revisiting mathematics education china lectures, Dordrecht: Kluwer Academic Publishers.
- Greeno, J. G. (1987). Instructional representations based on research about understanding. In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 61-88). New York: Academic Press.
- Greeno, J. G., & Hall, R. P. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78, 361-367.
- Heritage, M., & Niemi, D. (2006). Toward a framework for using student mathematical representations as formative assessments. Educational Assessment, 11(3-4), 265-282. https://doi.org/10.1080/10627197.2006.9652992
- Huck, S. W. (2008). Reading statistics and research. Boston, MA: Pearson Education.
- Jigyel, K., & Afamasaga-Fuata'i, K. (2007). Students' conceptions of models of fractions and equivalence. Australian Mathematics Teacher, 63(4), 17-25.
- Johnson, R. B., & Onwuegbuzie, A. J. (2004). Mixed methods research: A research paradigm whose time has come. Educational Researcher, 33(7), 14-26. doi:10.3102/0013189X033007014
- Lamon, S. J. (2001). Presenting and representing: From fractions to rational numbers. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics (pp. 146-165). Reston, Virginia: National Council of Teachers of Mathematics.
- Muzheve, M. T., & Capraro, R. M. (2006). An exploration of the role natural language and idiosyncratic representations in teaching how to convert among fractions, decimals, and percents. The Journal of Mathematics Behavior, 31, 1-14.
- Nastasi, B. K., Hitchcock, J. H., & Brown, L. M. (2010). An inclusive framework for conceptualizing mixed methods design typologies. In A. Tashakkori & C. Teddlie (Eds.), Handbook of mixed methods in social and behavioral research (2nd ed., pp. 305-338). Thousand Oaks, CA: Sage.
- National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
- National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
- Newton, K. J., & Sands, J. (2012). Why don't we just divide across? Mathematics Teaching in the Middle School, 17(6), 340-345. https://doi.org/10.5951/mathteacmiddscho.17.6.0340
- Ng, S. F., & Lee, K. (2009). The model method: Singapore children's tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40(3), 282-313
- Niemi, D. (1996). Assessing conceptual understanding in mathematics: Representations, problem solutions, justification and explanations. Journal of Educational Research, 89, 351-363. https://doi.org/10.1080/00220671.1996.9941339
- Onwuegbuzie, A. J., & Teddlie, C. (2003). A framework for analyzing data in mixed methods research. In A. Tashakkori & C. Teddlie (Eds.), Handbook of mixed methods in social and behavioral research (pp. 351-383). Thousand Oaks, CA: Sage.
- Piaget, J. (1957). Construction of reality in the child. London: Routledge & Kegan Paul.
- Radford, L. (2003). Gestures, speech, and sprouting of signs: A semiotic-cultural approach to students' types of generalization, Mathematical Thinking and Learning, 5(1), 37-70. https://doi.org/10.1207/S15327833MTL0501_02
- Rasmussen, C., Heck, D. J., Tarr, J. E., Knuth, E., White, D. Y., Lambdin, D. V., . . . Barnes, D. (2011). Trends and issues in high school mathematics: research insights and needs. Journal for Research in Mathematics Education, 42(3), 204-219. https://doi.org/10.5951/jresematheduc.42.3.0204
- Saxe, G. B., Taylor, E. V., McIntosh, C., & Gearhart, M. (2005). Representing fractions with standard notation: A developmental analysis. Journal for Research in Mathematics Education, 36(2), 137-157.
- Shaughnessy, M. M. (2011). Identify fractions and decimals on a number line. Teaching Children Mathematics, 17(7), 428-434.
- Schnotz, W., & Bannert, M. (2003). Construction and interference in learning from multiple representations. Learning and Instruction, 13(2), 141-156. https://doi.org/10.1016/S0959-4752(02)00017-8
- SPSS Inc. (2013). SPSS 22.0 for Windows. [Computer software]. Chicago: SPSS Inc.
- Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers' beliefs and practices related to mathematics instruction. Teaching and Teacher Education, 17(2), 213-226. https://doi.org/10.1016/S0742-051X(00)00052-4
- Teddlie, C., & Tashakkori, A. (2006). Foundations of mixed methods research. Thousand Oaks, CA: Sage.
- Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54, 9-35. https://doi.org/10.1023/B:EDUC.0000005212.03219.dc
- Van Merrienboer, J. J., & Kirschner, P. A. (2012). Ten steps to complex learning: A systematic approach to four-component instructional design. Routledge.