Compared With What? The Effects of Different Comparisons on Conceptual Knowledge and Procedural Flexibility for Equation Solving

Journal of Educational Psychology

Volumn 101, Issue 3, 2009, Pages 529-544

  Rittle Johnson, Bethany ^a   Star, Jon R ^b  

^a VANDERBILT UNIVERSITY   (United States)

^b HARVARD GRADUATE SCHOOL OF EDUCATION   (United States)

Author keywords

conceptual (declarative) knowledge; learning processes; mathematics concepts; mathematics education; procedural knowledge

Indexed keywords

EID: 67949124849     PISSN: 00220663     EISSN: None     Source Type: Journal    
DOI: 10.1037/a0014224     Document Type: Article

References (56)

1. Ball D.L. With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal 93 (1993) 373-397
2. Ballheim C. Readers respond to what's basic. Mathematics Education Dialogues 3 (1999) 11
3. Barzi F., and Woodward M. Imputations of missing values in practice: Results from imputations of serum cholesterol in 28 cohort studies. American Journal of Epidemiology 160 (2004) 34-45
4. Beishuizen M., van Putten C.M., and van Mulken F. Mental arithmetic and strategy use with indirect number problems up to one hundred. Learning and Instruction 7 (1997) 87-106
5. Bloom B.S., Engelhart M.D., Furst E.J., Hill W.H., and Krathwohl D.R. Taxonomy of educational objectives. In: Bloom B.S. (Ed). Taxonomy of educational objectives: The classification of educational goals by a committee of college and university examiners (1956), Longman, Greens, New York 186-193
6. Blöte A.W., Van der Burg E., and Klein A.S. Students' flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology 93 (2001) 627-638
7. Bransford J.D., Franks J.J., Vye N.J., and Sherwood R.D. New approaches to instruction: Because wisdom can't be told. In: Vosniadou S., and Ortony A. (Eds). Similarity and analogical reasoning (1989), Cambridge University Press, New York 470-497
8. Carpenter T.P., Franke M.L., Jacobs V.R., Fennema E., and Empson S.B. A longitudinal study of invention and understanding in children's multidigit addition and subtraction. Journal for Research in Mathematics Education 29 (1998) 3-20
9. Catrambone R., and Holyoak K.J. Overcoming contextual limitations on problem-solving transfer. Journal of Experimental Psychology: Learning, Memory, and Cognition 15 (1989) 1147-1156
10. Chazan D., and Ball D. Beyond being told not to tell. For the Learning of Mathematics 19 (1999) 2-10
11. Cummins D. Role of analogical reasoning in the induction of problem categories. Journal of Experimental Psychology: Learning, Memory, and Cognition 18 (1992) 1103-1124
12. Fraivillig J.L., Murphy L.A., and Fuson K. Advancing children's mathematical thinking in everyday mathematics classrooms. Journal for Research in Mathematics Education 30 (1999) 148-170
13. Gentner D. Structure-mapping: A theoretical framework for analogy. Cognitive Science 7 (1983) 155-170
14. Gentner D. The development of relational category knowledge. In: Rakison D.H., and Gershkoff-Stowe L. (Eds). Building object categories in developmental time (2005), Erlbaum, Mahwah, NH 245-275
15. Gentner D., Loewenstein J., and Thompson L. Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology 95 (2003) 393-405
16. Gentner D., and Namy L.L. The role of comparison in children's early word learning. In: Waxman S.R., and Hall D.G. (Eds). Weaving a lexicon (2004), MIT Press, Cambridge, MA 533-568
17. Gick M.L., and Holyoak K.J. Analogical Problem Solving. Cognitive Psychology 12 (1980) 306-355
18. Gick M.L., and Holyoak K.J. Schema induction and analogical transfer. Cognitive Psychology 15 (1983) 1-38
19. Gick M.L., and Paterson K. Do contrasting examples facilitate schema acquisition and analogical transfer?. Canadian Journal of Psychology 46 (1992) 539-550
20. Hiebert J. Conceptual and procedural knowledge: The case of mathematics (1986), Erlbaum, Hillsdale, NJ
21. Hiebert J., and Wearne D. Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction 14 (1996) 251-283
22. Hummel J.E., and Holyoak K.J. Distributed representations of structure: A theory of analogical access and mapping. Psychological Review 104 (1997) 427-466
23. Kenny D.A., Kashy D.A., and Cook W.L. Dyadic data analysis (2006), Guilford Press, New York
24. Kilpatrick J., Swafford J.O., and Findell B. Adding it up: Helping children learn mathematics (2001), National Academy Press, Washington, DC
25. Lampert M. When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal 27 (1990) 29-63
26. Little R.J.A. A test of missing completely at random for multivariate data with missing values. Journal of the American Statistical Association 83 (1988) 1198-1202
27. Namy L.L., and Gentner D. Making a silk purse out of two sow's ears: Young children's use of comparison in category learning. Journal of Experimental Psychology: General 131 (2002) 5-15
28. National Council of Teachers of Mathematics. Curriculum and evaluation standards for school mathematics (1989), Author, Reston, VA
29. National Council of Teachers of Mathematics. Principles and standards for school mathematics (2000), Author, Reston, VA
30. National Council of Teachers of Mathematics. Curriculum focal points for prekindergarten through Grade 8 mathematics (2006), Author, Reston, VA
31. National Mathematics Advisory Panel. Foundations of success: The final report of the National Mathematics Advisory Panel (2008), U.S. Department of Education, Washington, DC
32. Oakes L.M., and Ribar R.J. A comparison of infants' categorization in paired and successive presentation familiarization tasks. Infancy 7 (2005) 85-98
33. Paas F.G.W.C., and Van Merrienboer J.J.G. Variability of worked examples and transfer of geometrical problem-solving skills: A cognitive-load approach. Journal of Educational Psychology 86 (1994) 122-133
34. Peugh J.L., and Enders C.K. Missing data in educational research: A review of reporting practices and suggestions for improvement. Review of Educational Research 74 (2004) 525-556
35. Reed S.K. Constraints on the abstraction of solutions. Journal of Educational Psychology 81 (1989) 532-540
36. Reed S.K., Ackinclose C.C., and Voss A.A. Selecting analogous problems: Similarity versus inclusiveness. Memory and Cognition 18 (1990) 83-98
37. Richland L.E., Holyoak K.J., and Stigler J.W. Analogy use in eighth-grade mathematics classrooms. Cognition and Instruction 22 (2004) 37-60
38. Richland L.E., Zur O., and Holyoak K.J. Cognitive supports for analogies in the mathematics classroom. Science 316 (2007) 1128-1129
39. Rittle-Johnson B., and Alibali M.W. Conceptual and procedural knowledge of mathematics: Does one lead to the other?. Journal of Educational Psychology 91 (1999) 175-189
40. Rittle-Johnson B., Siegler R.S., and Alibali M.W. Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology 93 (2001) 346-362
41. Rittle-Johnson B., and Star J.R. Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology 99 (2007) 561-574
42. Ross B.H., and Kennedy P.K. Generalizing from the use of earlier examples in problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition 16 (1990) 42-55
43. Schafer J.L., and Graham J.W. Missing data: Our view of the state of the art. Psychological Methods 7 (2002) 147-177
44. Schwartz D.L., and Bransford J.D. A time for telling. Cognition and Instruction 16 (1998) 475-522
45. Siegler R.S. Emerging minds: The process of change in children's thinking (1996), Oxford University Press, New York
46. Silver E.A., Ghousseini H., Gosen D., Charalambous C., and Strawhun B. Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior 24 (2005) 287-301
47. Singley M.K., and Anderson J.R. The transfer of cognitive skill (1989), Harvard University Press, Cambridge, MA
48. Star J.R. Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education 36 (2005) 404-411
49. Star J.R., and Rittle-Johnson B. Flexibility in problem solving: The case of equation solving. Learning and Instruction 18 (2008) 565-579 doi:10.1016/j.learninstruc. 2007.09.018
50. Star, J. R., & Rittle-Johnson, B. (2008b, March). The role of comparison in the development of flexible knowledge of computational estimation. Paper presented at the Annual Meeting of the American Educational Research Association, New York.
51. Star J.R., and Seifert C. The development of flexibility in equation solving. Contemporary Educational Psychology 31 (2006) 280-300
52. Tennyson C.L., Tennyson R.D., and Rothen W. Content structure and instructional control strategies as design variables in concept acquisition. Journal of Educational Psychology 72 (1980) 499-505
53. Tennyson R.D., and Tennyson C.L. Rule acquisition design strategy variables: Degree of instance, divergence, sequence, and instance analysis. Journal of Educational Psychology 67 (1975) 852-859
54. VanderStoep S.W., and Seifert C.M. Learning "how" versus learning "when": Improving transfer of problem-solving principles. Journal of the Learning Sciences 3 (1993) 93-111
55. VanLehn K. Cognitive skill acquisition. Annual Review of Psychology 47 (1996) 513-539
56. Waxman S.R., and Klibanoff R.S. The role of comparison in the extension of novel adjectives. Developmental Psychology 36 (2000) 571-581