Mathematical learning models that depend on prior knowledge and instructional strategies

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2. MIT used the mechanics baseline test (MBT) (Ref. [33]), whereas UMn used the force concept inventory (FCI) (Ref. [32]), as did Hake (Ref. [1]).
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10. It is common to administer the same instrument for both, but the post-test for the previous year can serve as the pretest for the current year if suitably calibrated.
11. J. D. Bransford and D. L. Schwartz, Rethinking Transfer: A Simple Proposal with Multiple Implications, in Review of Research in Education, edited by Iran-Nejad and D. Pearson (American Educational Research Association, Washington, DC, 1999), p. 61.
12. This half-century-old pregain and/or postgain index was independently utilized by Hovland et al. who called it the “effectiveness index” (Ref. [13]), Gery who called it the “gap-closing parameter” (Ref. [14]), and Hake who called it the “normalized gain” (Ref. [1]) as we do.
13. C. I. Hovland, A. A. Lumsdaine, and F. D. Sheffield, A Baseline for Measurement of Percentage Change, in The Language of Social Research: A Reader in the Methodology of Social Research, edited by P. F. Lazarsfeld and M. Rosenberg (Free, New York, 1955), p. 77.
14. F. W. Gery, Does Mathematics Matter?, in Research Paper in Economic Education, edited by A. Welch (Joint Council on Economic Education, New York, 1972), p. 142.
15. We assume that any knowledge element tests either known or unknown (right or wrong) thus sidestepping complications due to well-established alternative conceptions (Refs. [16–18]) and mixed mental states (Refs. [19, 20]).
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19. Z. Hrepic, D. A. Zollmann, and S. Rebello, Identifying Students' Models of Sound Propagation, in Physics Education Research Conference, Boise, 2002, edited by S. Franklin, K. Cummings, and J. Marx (American Institute of Physics, New York, 2002), p. 59.
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22. R. A. Rescorla and A. R. Wagner, A Theory of Pavlovian Conditioning: Variations in the Effectiveness of Reinforcement and Nonreinforcement, in Classical Conditioning II: Current Research and Theory, edited by A. H. Black and W. F. Prokasy (Appleton-Century-Crofts, New York, 1972), p. 64.
23. R. C. Atkinson, G. H. Bower, and E. J. Crothers, An Introduction to Mathematical Learning Theory (Wiley, New York, 1965), Chaps. 1 and 2.
24. C. L. Hull, A Behavior System (Yale University Press, New Haven, 1952).
25. V. P. Coletta and J. A. Phillips, Interpreting FCI scores: Normalized gain, preinstruction scores, and scientific reasoning ability, Am. J. Phys. 73, 1172 (2005) [SPIN].
26. R. L. Hill, D. Grosnick, and D. Ober, Student preparation, aptitude, and performance in a first-semester algebra-based physics course, Bull. Am. Astron. Soc. 38, 1235 (2007).
27. masteringphysics, Addison-Wesley, (www.masteringphysics.com).
28. E.-S. Morote and D. E. Pritchard, Annual Meeting of the National Association for Research in Science Teaching, New Orleans, LA, 2002 (unpublished).
29. C. Henderson, P. Heller, and the UMN Physics Education Research Group, Contributed Talk, American Association of Physics Teachers Winter Meeting, Kissimmee, FL, 19 January 2000 (unpublished).
30. P. Heller, R. Keith, and S. Anderson, Teaching problem solving through cooperative grouping. Part 1: Group versus individual problem solving, Am. J. Phys. 60, 627 (1992) [SPIN][INSPEC].
31. P. Heller and M. Hollabaugh, Teaching problem solving through cooperative grouping. Part 2: Designing problems and structuring groups, Am. J. Phys. 60, 637 (1992) [SPIN][INSPEC][ADS].
32. D. Hestenes, M. Wells, and G. Swackhamer, Force concept inventory, Phys. Teach. 30, 141 (1992) [SPIN].
33. D. Hestenes and M. Wells, A mechanics baseline test, Phys. Teach. 30, 159 (1992) [SPIN].
34. E. Mazur, Peer Instruction: A User’s Manual (Prentice-Hall, Upper Saddle River, NJ, 1997).
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36. R. R. Hake, Physics Education Research Conference, Boise, Idaho, August 2002 (unpublished).
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38. R. K. Thorton, Uncommon Knowledge: Student Behavior Correlated to Conceptual Learning, in International School of Physics Enrico Fermi Course CLVI, Varenna, Italy, 2004, edited by E. F. Redish and M. Vicentini (IOS, Amsterdam, 2004).