Mathematical Flexibility: Aspects of a Continuum and the Role of Prior Knowledge

References

• Abdi, H. (2010). Guttman scaling. In N.J. Salkind, D.M., Dougherty, & B. Frey (Eds.): Encyclopedia of Research Design. Thousand Oaks (CA): Sage. pp. 558–560.
   Google Scholar
• Blӧte, A. W., Van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: instruction effects. Journal of Educational Psychology, 93, 627–638. doi: 10.1037//0022-0663.93.3.627
   Web of Science ®Google Scholar
• Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34.
   Web of Science ®Google Scholar
• Common Core State Standards Initiative (CCSSI) (2012). The Standards: Mathematics. http://www.corestandards.org/the-standards/mathematics. Accessed July 14, 2013.
   Google Scholar
• Fazio, L. K., DeWolf, M., & Siegler, R. S. (2016). Strategy use and strategy choice in fraction magnitude comparison. Journal of Experimental Psychology: Learning, Memory, and Cognition, 42, 1–16. doi: 10.1037/xlm0000153
   PubMed Web of Science ®Google Scholar
• Koedinger, K. R., Anderson, J. R., Hadley, W. H., & Mark, M. A. (1997). Intelligent tutoring goes to school in the big city. International Journal of Artificial Intelligence in Education, 8, 30–43.
   Google Scholar
• Mercier, E. M., & Higgins, S. E. (2013). Collaborative learning with multi-touch technology: Developing adaptive expertise. Learning and Instruction, 25, 13–23. doi: 10.1016/j.learninstruc.2012.10.004
   Web of Science ®Google Scholar
• 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 Science and Education. Washington, DC: National Academy Press.
   Google Scholar
• Newton, K. J., Willard, C., & Teufel, C. (2014). Struggling sixth grade students' ways of solving fraction computation problems. Elementary School Journal, 115(1), 1–21.
   Web of Science ®Google Scholar
• Newton, K., Star, J. R., & Lynch, K. (2010). Understanding the development of flexibility in struggling algebra students. Mathematical Thinking and Learning, 12(4), 282–305.
   Web of Science ®Google Scholar
• Opfer, J. E., & Siegler, R. S. (2007). Representational change and children’s numerical estimation. Cognitive Psychology, 55(3), 169–195. doi: 10.1016/j.cogpsych.2006.09.002
   PubMed Web of Science ®Google Scholar
• Peters, G., De Smedt, B., Torbeyns, J., Verschaffel, L., & Ghesquière, P. (2014). Subtraction by addition in children with mathematical learning disabilities. Learning and Instruction, 30, 1–8. doi: 10.1016/j.learninstruc.2013.11.001
   Web of Science ®Google Scholar
• Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574. doi: 10.1037/0022-0663.99.3.561
   Web of Science ®Google Scholar
• Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836–852. doi: 10.1037/a0016026
   Web of Science ®Google Scholar
• Rittle-Johnson, B., Star, J. R., & Durkin, K. (2012). Developing procedural flexibility: Are novices prepared to learn from comparing procedures?. British Journal of Educational Psychology, 82(3), 436–455. doi: 10.1111/j.2044-8279.2011.02037.x
   PubMed Web of Science ®Google Scholar
• Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47(6), 1525–1538. doi: 10.1037/a0024997
   PubMed Web of Science ®Google Scholar
• Schraw, G., & Moshman, D. (1995). Metacognitive theories. Educational Psychology Review, 7(4), 351–371. doi: 10.1007/BF02212307
   Web of Science ®Google Scholar
• Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.
   Web of Science ®Google Scholar
• Star, J. R., & Newton, K. J. (2009). The nature and development of experts' strategy flexibility for solving equations. ZDM - The International Journal on Mathematics Education, 41, 557–567.
   Google Scholar
• Star, J. R., Newton, K., Pollack, C., Kokka, K., Rittle-Johnson, B., & Durkin, K. (2015). Student, teacher, and instructional characteristics related to students' gains in flexibility. Contemporary Educational Psychology, 41, 198–208.
   Web of Science ®Google Scholar
• Star, J. R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lynch, K., Newton, K., & Gogolen, C. (2015). Learning from comparison in algebra. Contemporary Educational Psychology, 40, 41–54. doi: 10.1016/j.cedpsych.2014.05.005
   Web of Science ®Google Scholar
• Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565–579. doi: 10.1016/j.learninstruc.2007.09.018
   Web of Science ®Google Scholar
• Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31(3), 280–300. doi: 10.1016/j.cedpsych.2005.08.001
   Web of Science ®Google Scholar
• Torbeyns, J., Ghesquière, P., & Verschaffel, L. (2009). Efficiency and flexibility of indirect addition in the domain of multi-digit subtraction. Learning and Instruction, 19(1), 1–12. doi: 10.1016/j.learninstruc.2007.12.002
   Web of Science ®Google Scholar
• Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359. doi: 10.1007/BF03174765
   Web of Science ®Google Scholar
• Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2011). Analyzing and developing strategy flexibility in mathematics education. In: Elen J., Stahl E., Bromme R., Clarebout G. (Eds.) Links Between Beliefs and Cognitive Flexibility. Dordrecht: Springer.
   Google Scholar
• Verschaffel, L., Torbeyns, J., DeSmedt, B., Luwel, K., & Van Doreen, W. (2007). Strategy flexibility in children with low achievement in mathematics. Educational and Child Psychology, 24(2), 16–27.
   Google Scholar