Developing procedural flexibility: Are novices prepared to learn from comparing procedures?

British Journal of Educational Psychology

Volumn 82, Issue 3, 2012, Pages 436-455

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

^a VANDERBILT UNIVERSITY   (United States)

^b HARVARD GRADUATE SCHOOL OF EDUCATION   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ADOLESCENT; ANALYSIS OF VARIANCE; ARTICLE; CLINICAL TRIAL; CONCEPT FORMATION; CONTROLLED CLINICAL TRIAL; CONTROLLED STUDY; FEMALE; FOLLOW UP; HUMAN; LEARNING; LONG TERM MEMORY; MALE; MATHEMATICS; METHODOLOGY; PROBLEM SOLVING; PSYCHOLOGICAL ASPECT; RANDOMIZED CONTROLLED TRIAL; STUDENT; TEACHING; UNITED STATES;

ADOLESCENT; ANALYSIS OF VARIANCE; CONCEPT FORMATION; FEMALE; FOLLOW-UP STUDIES; HUMANS; LEARNING; MALE; MATHEMATICS; PROBLEM SOLVING; RETENTION (PSYCHOLOGY); STUDENTS; TEACHING; UNITED STATES;

EID: 84864615411     PISSN: 00070998     EISSN: 20448279     Source Type: Journal    
DOI: 10.1111/j.2044-8279.2011.02037.x     Document Type: Article

References (49)

1. Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning from examples: Instructional principles from the worked examples research. Review of Educational Research, 70, 181-214. doi:10.2307/1170661
2. Australian Education Ministers. (2006). Statements of learning for mathematics. Carlton South Vic, Australia: Curriculum Corporations.
3. Bailey, R., Day, R., Frey, P., Howard, A., Hutchens, D., McClain, K.,..Price, J. (2004). Mathematics: Applications and concepts, course 3. New York : Glencoe.
4. Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93, 373-397. Retrieved from: http://www.jstor.org/stable/1002018
5. Baroody, A. J., & Dowker, A. (2003). The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah, NJ : Erlbaum.
6. Becker, J. P., & Selter, C. (1996). Elementary school practices. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 511-564). Dordrecht, The Netherlands : Kluwer.
7. Beishuizen, M., van Putten, C. M., & van Mulken, F. (1997). Mental arithmetic and strategy use with indirect number problems up to one hundred. Learning and Instruction, 7, 87-106. doi:10.1016/S0959-4752(96)00012-6
8. Blöte, A. W., Klein, A. S., & Beishuizen, M. (2000). Mental computation and conceptual understanding. Learning and Instruction, 10, 221-247. doi:10.1016/S0959-4752(99)00028-6
9. Blöte, 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
10. Brophy, J. (1999). Teaching. Education Practices Series No. 1, International Bureau of Education. Retrieved from http://www.ibe.unesco.org
11. Carnine, D. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141. doi:10.1177/002221949703000201
12. Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998). A longitudinal study of invention and understanding in children's multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3-20. doi:10.2307/749715
13. Clarke, T., Ayres, P., & Sweller, J. (2005). The impact of sequencing and prior knowledge on learning mathematics through spreadsheet applications. Educational Technology Research and Development, 53, 15-24. doi:10.1007/BF02504794
14. Dowker, A. (1992). Computational estimation strategies of professional mathematicians. Journal for Research in Mathematics Education, 23, 45-55. doi:10.2307/749163
15. Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology, 95, 393-405. doi:10.1037/0022-0663.95.2.393
16. Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Human, P., Murray, H., ...Wearne, D. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25, 12-21. doi:10.2307/1176776
17. Hiebert, J., Gallimore, R., Garnier, H., Givvin, K. B., Hollingsworth, H., Jacobs, J., ...Kersting, N. (2003). Teaching mathematics in seven countries: Results from the timss 1999 video study (No. NCES 2003-013). Washington, DC : U.S. Department of Education, National Center for Education Statistics.
18. Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14, 251-283. doi:10.1207/s1532690xci1403_1
19. Johnson, D. W., & Johnson, R. T. (1994). Learning together and alone: Cooperative, competitive and individualistic learning (4th ed.). Boston, MA : Allyn and Bacon.
20. Kenny, D. A., Kashy, D. A., & Cook, W. L. (2006). Dyadic data analysis. New York, NY : Guilford Press.
21. Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). New York : Simon & Schuster.
22. Kilpatrick, J., Swafford, J. O., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC : National Academy Press.
23. Klein, A. S., Beishuizen, M., & Treffers, A. (1998). The empty number line in dutch second grades: Realistic versus gradual program design. Journal for Research in Mathematics Education, 29, 443-464. doi:10.2307/749861
24. Kultusministerkonferenz. (2004). Bildungsstandards im fach mathematik für den primarbereich [educational standards in mathematics for primary schools]. Luchterhand : München-Neuwied.
25. Kurtz, K., Miao, C.-H., & Gentner, D. (2001). Learning by analogical bootstrapping. The Journal of the Learning Sciences, 10, 417-446. doi:10.1207/S15327809JLS1004new_2
26. Lampert, M. (1990). When the problem is not the question and the solution is not the answer mathematical knowing and teaching. American Educational Research Journal, 27, 29-63. doi:10.3102/00028312027001029
27. Leikin, R. (2003). Problem-solving preferences of mathematics teachers: Focusing on symmetry. Journal of Mathematics Teacher Education, 6, 297-329. doi:10.1023/A:1026355525004
28. Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change: Contributions to children's learning of multiplication. Journal of Experimental Psychology: General, 124, 83-97. doi:10.1037//0096-3445.124.1.83
29. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA : NCTM.
30. National Mathematics Advisory Panel. (2008). Foundations of success: The final report of the national mathematics advisory panel. Washington, DC : U.S. Department of Education.
31. Peugh, J. L., & Enders, C. K. (2004). Missing data in educational research: A review of reporting practices and suggestions for improvement. Review of Educational Research, 74, 525-556. doi:10.3102/00346543074004525
32. Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316, 1128-1129. doi:10.1126/science.1142103
33. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346-362. doi:10.1037//0022-0663.93.2.346
34. 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, 561-574. doi:10.1037/0022-0663.99.3.561
35. 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
36. 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, 836-852. doi:10.1037/a0016026
37. Satterthwaite, F. E. (1946). An approximate distribution of estimation of variance components. Biometrics Bulletin, 2, 110-114. doi:10.2307/3002019
38. Schafer, J. L., & Graham, J. W. (2002). Missing data: Our view of the state of the art. Psychological Methods, 7, 147-177. doi:10.1037//1082-989X.7.2.147
39. Siegler, R. S. (1996). Emerging minds: The process of change in children's thinking. New York : Oxford University Press.
40. Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. (2005). 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, 287-301. doi:10.1016/j.jmathb.2005.09.009
41. Singapore Ministry of Education. (2006). Secondary mathematics syllabuses.
42. Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404-411. Retrieved from: http://www.jstor.org/stable/30034943
43. Star, J. R., & Newton, K. J. (2009). The nature and development of expert's strategy flexibility for solving equations. ZDM Mathematics Education, 41, 557-567. doi:10.1007/s11858-009-0185-5
44. 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
45. Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 101, 408-426. doi:10.1016/j.jecp.2008.11.004
46. Sweller, J., van Merrienboer, J. J. G., & Paas, F. G. W. C. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10, 251-296. doi:10.1023/B:TRUC.0000021808.72598.4d
47. Torbeyns, J., Ghesquière, P., & Verschaffel, L. (2009). Efficiency and flexibility of indirect addition in the domain of multi-digit subtraction. Learning and Instruction, 19, 1-12. doi:10.1016/j.learninstruc.2007.12.002
48. Treffers, A. (1991). Didactical background of a mathematics program for primary education. In L. Streefland (Ed.), Realistic mathematics education in primary school (pp. 21-56). Utrecht, The Netherlands : Freudenthal Institute.
49. 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