Developing flexible procedural knowledge in undergraduate calculus

References

• Alfieri, L., Nokes-Malach, T. J., & Schunn, C. D. (2013). Learning through case comparisons: A meta-analytic review. Educational Psychologist, 48(2), 87–113. doi: 10.1080/00461520.2013.775712
   Web of Science ®Google Scholar
• Anzai, Y., & Simon, H. A. (1979). The theory of learning by doing. Psychological Review, 86(2), 124–140. doi: 10.1037/0033-295X.86.2.124
   PubMed Web of Science ®Google Scholar
• Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6, 1–32. doi: 10.1090/cbmath/006/01
   Google Scholar
• Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody, & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1–34). Mahwah: Erlbaum.
   Google Scholar
• Berthold, K., & Renkl, A. (2009). Instructional aids to support a conceptual understanding of multiple representations. Journal of Educational Psychology, 101, 70–87. doi: 10.1037/a0013247
   Web of Science ®Google Scholar
• Blöte, A. W., Klein, A. S., & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction, 10(3), 221–247. doi: 10.1016/S0959-4752(99)00028-6
   Web of Science ®Google Scholar
• Canobi, K. H. (2009). Concept–procedure interactions in children's addition and subtraction. Journal of Experimental Child Psychology, 102, 131–149. doi: 10.1016/j.jecp.2008.07.008
   PubMed Web of Science ®Google Scholar
• Catrambone, R., & Holyoak, K. J. (1989). Overcoming contextual limitations on problem solving transfer. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15(6), 1147–1156.
   Web of Science ®Google Scholar
• Chi, M., DeLeeuw, N., Chiu, M. H., & LaVancher, C. (1994). Eliciting self- explanations improves understanding. Cognitive Science, 18, 439–477.
   Web of Science ®Google Scholar
• Dowker, A. (1992). Computational estimation strategies of professional mathematicians. Journal for Research in Mathematics Education, 23, 45–55. doi: 10.2307/749163
   Web of Science ®Google Scholar
• Dubinsky, E., and McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton, et al. (Eds.), The teaching and learning of mathematics at university level: An ICMI study (pp. 273–280). Dordrecht, Netherlands: Kluwer Academic Publishers.
   Google Scholar
• Fraivillig, J. L., Murphy, L. A., & Fuson, K. (1999). Advancing children's mathematical thinking in everyday mathematics classrooms. Journal for Research in Mathematics Education, 30, 148–170. doi: 10.2307/749608
   Web of Science ®Google Scholar
• Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology, 95(2), 393–408. doi: 10.1037/0022-0663.95.2.393
   Web of Science ®Google Scholar
• Gray, E., & Tall, D. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141.
   Google Scholar
• Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Eds.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum Associates.
   Google Scholar
• Hughes-Hallet, D. (2006). What have we learned from calculus reform? The road to conceptual understanding. In N. Hastings (Eds.), A fresh start for collegiate mathematics (pp. 43–45). Washington, DC: MAA Press.
   Google Scholar
• Kilpatrick, J., Swafford, J. O., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
   Google Scholar
• Maciejewski, W., & Merchant, S. (2015). Mathematical tasks, study approaches, and course grades in undergraduate mathematics: A year-by-year analysis. International Journal of Mathematical Education in Science and Technology, doi:10.1080/0020739X.2015.1072881.
   Web of Science ®Google Scholar
• Mason, J. (2003). On the structure of attention in the learning of mathematics. Australian Mathematics Teacher, 59(4), 17–25.
   Google Scholar
• McNeil, N. M., Chesney, D. L., Matthews, P. G., Fyfe, E. R., Petersen, L. A., Dunwiddie, A. E., & Wheeler, M. C. (2012). It pays to be organized: Organizing arithmetic practice around equivalent values facilitates understanding of mathematical equivalence. Journal of Educational Psychology, 104(4), 1109–1121. doi: 10.1037/a0028997
   Web of Science ®Google Scholar
• National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
   Google Scholar
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
   Google Scholar
• National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston: Author.
   Google Scholar
• Piaget, J. (1970). Genetic epistemology. New York: W.W. Norton.
   Google Scholar
• Reed, S. K. (1989). Constraints on the abstraction of solutions. Journal of Educational Psychology, 81, 532–540. doi: 10.1037/0022-0663.81.4.532
   Web of Science ®Google Scholar
• Resnick, L. B., & Ford, W. W. (1981). The psychology of mathematics for instruction. Hillsdale: Lawrence Erlbaum Associates.
   Google Scholar
• Rittle-Johnson, B. (2006). Promoting transfer: Effects of self-explanation and direct instruction. Child Development, 77, 1–15. doi: 10.1111/j.1467-8624.2006.00852.x
   PubMed Web of Science ®Google Scholar
• Rittle-Johnson, B., & Koedinger, K. (2009). Iterating between lessons on concepts and procedures can improve mathematics knowledge. British Journal of Educational Psychology, 79(3), 483–500. doi: 10.1348/000709908X398106
   PubMed Web of Science ®Google Scholar
• Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. Cohen Kadosh and A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 1118–1134). Oxford: Oxford University Press.
   Google Scholar
• Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, doi:10.1007/s10648–015–9302-x.
   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. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101, 529–544. doi: 10.1037/a0014224
   Web of Science ®Google Scholar
• Rittle-Johnson, B., & Star, J. R. (2011). The power of comparison in learning and instruction: Learning outcomes supported by different types of comparisons. In J. P. Mestre, & B. H. Ross (Eds.), Cognition in education ( Vol. 55, pp. 199–226). Oxford, UK: Academic.
   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, 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 between 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
• Schoenfeld, A. (1995). A brief biography of calculus reform. UME Trends: News and Reports on Undergraduate Mathematics Education, 6, 3–5.
   Google Scholar
• Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as two sides of the same coin. Educational Studies in Mathematics, 22, 1–36. doi: 10.1007/BF00302715
   Google Scholar
• Simon, H. A., & Reed, S. K. (1976). Modeling strategy shifts in a problem-solving task. Cognitive Psychology, 8(1), 86–97. doi: 10.1016/0010-0285(76)90005-0
   Web of Science ®Google Scholar
• Sinclair, N., Pimm, D., & Higginson, W. (2006). Mathematics and the aesthetic: Modern approaches to an ancient affinity. New York: Springer-Verlag.
   Google Scholar
• Skemp, R. (1976). Instrumental and relational understanding. Mathematics Teaching, 77, 20–26.
   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. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38(2), 132–135.
   Web of Science ®Google Scholar
• Star, J. R., & Newton, K. J. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM Mathematics Education, 41, 557–567. doi: 10.1007/s11858-009-0185-5
   Google Scholar
• Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18, 565–579. doi: 10.1016/j.learninstruc.2007.09.018
   Web of Science ®Google Scholar
• Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 102, 408–426. doi: 10.1016/j.jecp.2008.11.004
   PubMed Web of Science ®Google Scholar
• Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280–300. doi: 10.1016/j.cedpsych.2005.08.001
   Web of Science ®Google Scholar
• Star, J. R., & Stylianides, G. J. (2013). Procedural and conceptual knowledge: Exploring the gap between knowledge type and knowledge quality. Canadian Journal of Science, Mathematics and Technology Education, 13, 169–181. doi: 10.1080/14926156.2013.784828
   Google Scholar
• Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world's teachers for improving education in the classroom. New York: Free Press.
   Google Scholar
• Tallman, M., & Carlson, M. P. (2012). A characterization of calculus I final exams in U.S. colleges and universities. Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education (pp. 217–226). Portland, OR: Portland State University.
   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, 335–359. doi: 10.1007/BF03174765
   Web of Science ®Google Scholar
• Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477. doi: 10.2307/749877
   Web of Science ®Google Scholar
• Yackel, E., Rasmussen, C., & King, K. (2000). Social and sociomathematical norms in an advanced undergraduate mathematics course. Journal of Mathematical Behaviour, 19, 275–287. doi: 10.1016/S0732-3123(00)00051-1
   Google Scholar