Learning from comparison in algebra

Contemporary Educational Psychology

Volumn 40, Issue , 2015, Pages 41-54

  Star, Jon R ^a   Pollack, Courtney ^a   Durkin, Kelley ^b   Rittle Johnson, Bethany ^c   Lynch, Kathleen ^a   Newton, Kristie ^d   Gogolen, Claire ^a  

^a HARVARD UNIVERSITY   (United States)

^b UNIVERSITY OF LOUISVILLE   (United States)

^c VANDERBILT UNIVERSITY   (United States)

^d TEMPLE UNIVERSITY   (United States)

Author keywords

Algebra; Comparison; Conceptual knowledge; Flexibility; Procedural knowledge; Supplemental curriculum

Indexed keywords

EID: 84920263141     PISSN: 0361476X     EISSN: 10902384     Source Type: Journal    
DOI: 10.1016/j.cedpsych.2014.05.005     Document Type: Article

References (60)

1. Adelman C. The toolbox revisited: Paths to degree completion from high school though college 2006, United States Department of Education, Washington, DC.
2. Atkinson R.K., Derry S.J., Renkl A., Wortham D. Learning from examples: Instructional principles from the worked examples research. Review of Educational Research 2000, 70(2):181-214.
3. Australian Education Ministers Statements of learning for mathematics 2006, Curriculum Corporations, Carlton South Vic, Australia.
4. Ball D.L. With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal 1993, 93(4):373-397.
5. Beaton A.E., Mullis I.V.S., Martin M.O., Gonzales E.J., Kelly D.L., Smith T.A. Mathematics achievement in the middle years: IEA's third international mathematics and science study 1996, Center for the Study of Testing, Evaluation, and Educational Policy, Boston College, Boston.
6. Blume G.W., Heckman D.S. What do students know about algebra and functions?. Results from the sixth mathematics assessment 1997, 225-277. National Council of Teachers of Mathematics, Reston, VA. P.A. Kenney, E.A. Silver (Eds.).
7. Brophy, J. (1999). Teaching. Education practices series no. 1, International Bureau of Education, http://www.ibe.unesco.org.
8. Brophy J.E., Good T.L. Teacher behavior and student achievement. Handbook of research on teaching 1986, 328-375. Macmillan, New York. 3rd ed. M.C. Wittrock (Ed.).
9. Cameron A.C., Trivedi P.K. Microeconometrics: Methods and applications 2005, Cambridge University Press, Cambridge.
10. Carroll W.M. Invented computational procedures of students in a standards-based curriculum. Journal of Mathematical Behavior 2000, 18(2):111-121.
11. CBT/McGraw Hill (2007). AcuityTM Algebra. http://www.acuityforschools.com.
12. Cohen D.K., Hill H. Learning policy: When state education reform works 2001, Yale University Press, New Haven, CT.
13. Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.
14. Davison, M., Fehr, C., & Seipel, B. (2011, September). From the lab to the classroom: Expanding and scaling up the curriculum domain. In Paper presented at the fall conference of the Society for Research on Educational Effectiveness, Washington, DC.
15. Durkin K., Rittle-Johnson B. The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction 2012, 22:206-214.
16. Education Commission of the States (1998). Equity 2000. Denver, CO. http://www.ecs.org/html/Document.asp?chouseid=1510.
17. Gentner D., Loewenstein J., Thompson L. Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology 2003, 95:393-405.
18. Guo J.P., Pang M.F. Learning a mathematical concept from comparing examples: The importance of variation and prior knowledge. European Journal of Psychology of Education 2011, 26:495-525.
19. Guskey T.R. Evaluating professional development 2000, Corwin Press, Thousand Oaks, CA.
20. Kieran C. The learning and teaching of school algebra. Handbook of research on mathematics teaching and learning 1992, 390-419. Simon & Schuster, New York. D. Grouws (Ed.).
21. Krutetskii V.A. The psychology of mathematical abilities in school children 1976, University of Chicago Press, Chicago.
22. Kultusministerkonferenz (2004). Bildungsstandards im fach mathematik für den primarbereich [educational standards in mathematics for primary schools]. Luchterhand: Munchen-Neuwied.
23. Kurtz K., Miao C.H., Gentner D. Learning by analogical bootstrapping. The Journal of the Learning Sciences 2001, 10:417-446.
24. Lampert M. When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal 1990, 27(1):29-63.
25. Results from the fourth mathematics assessment of the National Assessment of Educational Progress 1989, National Council of Teachers of Mathematics, Reston, VA. M.M. Lindquist (Ed.).
26. Lynch K., Star J.R. Teachers' views about multiple strategies in middle and high school mathematics. Mathematical Thinking and Learning 2014, 16(2):85-108. 10.1080/10986065.2014.889501.
27. Lynch K., Star J.R. Views of struggling students on instruction incorporating multiple strategies in Algebra I: An exploratory study. Journal for Research in Mathematics Education 2014, 45(1):6-18.
28. Mack N. Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education 1990, 21:16-32. .
29. Moses R.P., Cobb C.E. Radical equations: Civil rights from Mississippi to the Algebra Project 2001, Beacon Press, Boston.
30. Murnane R.J., Willett J.B. Methods matter: Improving causal inference in educational and social science research 2011, Oxford University Press, New York.
31. Namy L.L., 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 2002, 131:5-15.
32. National Association of Education Progress, Question Tool (2011). U.S. Department of Education http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx?subject=mathematics.
33. National Council of Teachers of Mathematics Principles and standards for school mathematics 2000, Author, Reston, VA.
34. National Mathematics Advisory Panel (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. [Retrieved 17.02.10]. http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf.
35. National Research Council Adding it up: Helping children learn mathematics 2001, National Academy Press, Washington, DC.
36. Newton K.J., Star J.R. Exploring the nature and impact of model teaching with worked example pairs. Mathematics Teacher Educator 2013, 2(1):86-102.
37. Newton K., Star J.R., Lynch K. Exploring the development of flexibility in struggling algebra students. Mathematical Thinking and Learning 2010, 12(4):282-305.
38. Oakes L.M., Ribar R.J. A comparison of infants' categorization in paired and successive presentation familiarization tasks. Infancy 2005, 7:85-98.
39. Pashler H., Bain P., Bottge B., Graesser A., Koedinger K., McDaniel M., et al. Organizing instruction and study to improve student learning (NCER 2007-2004) 2007, National Center for Education Research, Institute of Education Sciences, U.S. Department of Education, Washington, DC, .
40. Richland L.E., Zur O., Holyoak K.J. Cognitive supports for analogies in the mathematics classroom. Science 2007, 316:1128-1129.
41. Rittle-Johnson B., Alibali M.W. Conceptual and procedural knowledge of mathematics: Does one lead to the other?. Journal of Educational Psychology 1999, 91:1-16.
42. Rittle-Johnson B., Star J.R. Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology 2007, 99(3):561-574.
43. Rittle-Johnson B., Star J.R. Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology 2009, 101(3):529-544.
44. Rittle-Johnson B., Star J.R. The power of comparison in learning and instruction: Learning outcomes supported by different types of comparisons. Psychology of Learning and Motivation: Cognition in Education 2011, Vol. 55:199-226. Elsevier, San Diego.
45. Rittle-Johnson B., Star J.R., Durkin K. The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology 2009, 101(4):836-852.
46. Rittle-Johnson B., Star J.R., Durkin K. Developing procedural flexibility: When should multiple procedures be introduced?. British Journal of Educational Psychology 2012, 82:436-455.
47. Rubin D. Multiple imputation for nonresponse in surveys 1987, J. Wiley & Sons, New York.
48. Schmidt W.H., McKnight C.C., Cogan L.S., Jakwerth P.M., Houang R.T. Facing the consequences: Using TIMMS for a closer look at U.S. mathematics and science education 1999, Kluwer, Dordrecht.
49. Schneider M., Rittle-Johnson B., Star J.R. Relations between conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology 2011, 47(6):1525-1538.
50. Schwartz D.L., Bransford J.D. A time for telling. Cognition and Instruction 1998, 16(4):475-522.
51. Silver E.A., Ghousseini H., Gosen D., Charalambous C., Strawhum 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 2005, 24:287-301.
52. Singapore Ministry of Education (2006). Secondary mathematics syllabuses.
53. Star J.R. Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education 2005, 36(5):404-411.
54. Star J.R. Foregrounding procedural knowledge. Journal for Research in Mathematics Education 2007, 38(2):132-135.
55. Star J.R., Rittle-Johnson B. Flexibility in problem solving: The case of equation solving. Learning and Instruction 2008, 18:565-579.
56. Star J.R., Rittle-Johnson B. It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology 2009, 102:408-426.
57. Star J.R., Seifert C. The development of flexibility in equation solving. Contemporary Educational Psychology 2006, 31:280-300.
58. Treffers A. Didactical background of a mathematics program for primary education. Realistic mathematics education in primary school 1991, 21-56. Freudenthal Institute, Utrecht, The Netherlands. L. Streefland (Ed.).
59. Verschaffel L., Luwel K., Torbeyns J., Van Dooren W. Developing adaptive expertise: A feasible and valuable goal for (elementary) mathematics education?. Ciencias Psicologicas 2007, 2007(1):27-35.
60. Woodward J., Beckmann S., Driscoll M., Franke M., Herzig P., Jitendra A., et al. Improving mathematical problem solving in grades 4 through 8: A practice guide (NCEE 2012-4055) 2012, National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education, Washington, DC, .