Feedback influences children's reasoning about math equivalence: A meta-analytic review

Thinking and Reasoning

Volumn 24, Issue 2, 2018, Pages 157-178

  Fyfe, Emily R ^a   Brown, Sarah A ^b  

^a INDIANA UNIVERSITY   (United States)

^b UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

conceptual knowledge; feedback; Math equivalence; meta analysis; procedural knowledge

Indexed keywords

EID: 85027063387     PISSN: 13546783     EISSN: 14640708     Source Type: Journal    
DOI: 10.1080/13546783.2017.1359208     Document Type: Article

References (74)

1. Alibali, M. W., (1999). How children change their minds: Strategy change can be gradual or abrupt. Developmental Psychology, 35, 127–145. doi:10.1037/0012-1649.35.1.127
2. Alibali, M. W., (2005). Mechanisms of change in the development of mathematical reasoning. Advances in Child Development and Behavior, 33, 79–123. doi:10.1016/S0065-2407(05)80005-9
3. Alibali, M. W., & Goldin-Meadow, S., (1993). Gesture-speech mismatch and mechanisms of learning: What the hands reveal about a child's state of mind. Cognitive Psychology, 25, 468–523.
4. Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C., (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9, 221–247. doi:10.1080/10986060701360902
5. Alibali, M. W., Phillips, K. M. O., & Fischer, A. D., (2009). Learning new problem-solving strategies leads to changes in problem representation. Cognitive Development, 24, 89–101. doi:10.1016/j.cogdev.2008.12.005
6. Baroody, A. J., & Ginsburg, H. P., (1983). The effects of instruction on children's understanding of the “equals” sign. Elementary School Journal, 84, 199–212. doi:10.1086/461356
7. Behr, M., Erlwanger, S., & Nichols, E., (1980). How children view the equal sign. Mathematics Teaching, 92, 13–15.
8. Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Lsler, I., & Kim, J. S., (2015). The development of children's early algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87. doi:10.5951/jresematheduc.46.1.0039
9. Booth, J. L., & Davenport, J. D., (2013). The role of problem representation and feature knowledge in algebraic equation-solving. Journal of Mathematical Behavior, 32, 415–423. doi:10.1016/j.jmathb.2013.04.003
10. Brown, S. A., & Alibali, M. W., (2015, March). Investigating the effectiveness of two commonclassroom events: Feedback and exposure to peers’ strategies. Poster presented at the Biennial Meeting of the Society for Research in Child Development (SRCD), Philadelphia, PA.
11. Brown, S. A., & Alibali, M. W., (2017,). Does peer collaboration lead to better learning? The case of mathematical equivalence. Poster to be presented at the Biennial Meeting of the Society for Research in Child Development (SRCD), Austin, TX.
12. Butler, R., (1987). Task-involving and ego-involving properties of evaluation: Effects of different feedback conditions on motivational perceptions, interest, and performance. Journal of Educational Psychology, 79, 474–482.
13. Byrd, C. E., McNeil, N. M., Chesney, D. L., & Matthews, P. G., (2015). A specific misconception of the equal sign acts as a barrier to children's learning of early algebra. Learning and Individual Differences, 38, 61–67. doi:10.1016/j.lindif.2015.01.001
14. Carpenter, T. P., Franke, M. L., & Levi, L., (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.
15. Clariana, R. B., Wagner, D., & Murphy, L. C., (2000). Applying a connectionist description of feedback timing. Educational Technology Research and Development, 48, 5–22. doi:10.1007/BF02319855
16. Conati, C., Jacques, N., & Muir, M., (2013). Understanding attention to adaptive hints in educational games: An eye-tracking study. International Journal of Artificial Intelligence in Education, 23, 136–161.
17. Corbalan, G., Paas, F., & Cuyper, H., (2010). Computer-based feedback in linear algebra: Effects on transfer performance and motivation. Computers & Education, 55, 692–703.
18. Crooks, N. M., & Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34, 344–377. doi:10.1016/j.dr.2014.10.001.
19. DeCaro, M. S., & Rittle-Johnson, B., (2012). Exploring mathematics problems prepares children to learn from instruction. Journal of Experimental Child Psychology, 113, 552–568.
20. Fyfe, E. R., (2016). Providing feedback on computer-based algebra homework in middle-school classrooms. Computers in Human Behavior, 63, 568–574. doi:10.1016/j.chb.2016.05.082
21. Fyfe, E. R., & McNeil, N. M., & Borjas, S., (2015). Benefits of “concreteness fading” for children's mathematics understanding. Learning and Instruction, 35, 104–120. doi:10.1016/j.learninstruc.2014.10.004
22. Fyfe, E. R., & Rittle-Johnson, B., (2016a). Feedback both helps and hinders learning: The causal role of prior knowledge. Journal of Educational Psychology, 108, 82–97. doi:10.1037/edu0000053
23. Fyfe, E. R., & Rittle-Johnson, B., (2016b). The benefits of computer-generated feedback for mathematics problem solving. Journal of Experimental Child Psychology, 147, 140–151. doi:10.1016/j.jecp.2016.03.009
24. Fyfe, E. R., & Rittle-Johnson, B., (2017). Mathematics practice without feedback: A desirable difficulty in a classroom setting. Instructional Science, 45, 177–194. doi:10.1007/s11251-016-9401-1
25. Fyfe, E. R., DeCaro, M. S., & Rittle-Johnson, B., (2015). When feedback is cognitively-demanding: The importance of working memory capacity. Instructional Science, 43, 73–91. doi:10.1007/s11251-014-9323-8
26. Fyfe, E. R., Matthews, P. G., Amsel, E., McEldoon, K. L., & McNeil, N. M., (2017). Assessing formal knowledge of math equivalence among algebra and pre-algebra students. Journal of Educational Psychology. doi:10.1037/edu00000208
27. Fyfe, E. R., Rittle-Johnson, B., & DeCaro, M. S., (2012). The effects of feedback during exploratory mathematics problem solving: Prior knowledge matters. Journal of Educational Psychology, 104, 1094–1108. doi:10.1037/a0028389
28. Hattie, J., & Gan, M., (2011). Instruction based on feedback. In R., Mayer & P., Alexander (Eds.), Handbook of research on learning and instruction (pp. 249–271). New York, NY: Routledge.
29. Hattie, J., & Timperley, H., (2007). The power of feedback. Review of Educational Research, 77(1), 81–112. doi:10.3102/003465430298487
30. Hattikudur, S., & Alibali, M. W., (2010). Learning about the equal sign: Does comparing with inequality symbols help? Journal of Experimental Child Psychology, 107, 15–30. doi:10.1016/j.jecp.2010.03.004
31. Higgins, J., & Green, S., (2011). Cochrane handbook for systematic reviews of interventions. The Cochrane Collaboration. John Wiley & Sons. Retrieved from http://handbook.cochrane.org.
32. Jacobs, P. I., & Kulkarni, S., (1966). A test of some assumptions underlying programed instruction. Psychological Reports, 18, 103–110.
33. Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D., (2007). Professional development focused on children's algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38, 258–288.
34. Kalyuga, S., (2007). Expertise reversal effect and its implications for learner-tailored instruction. Educational Psychology Review, 19, 509–539. doi:10.1007/s10648-007-9054-3
35. Karabenick, S., & Knapp, J., (1988). Effects of computer privacy on help seeking. Journal of Applied Social Psychology, 16, 461–472.
36. Kieran, C., (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317–326. doi:10.1007/BF00311062
37. Kluger, A. N., & DeNisi, A., (1996). Effects of feedback intervention on performance: A historical review, a meta-analysis, and a preliminary feedback intervention theory. Psychological Bulletin, 119, 254–284. doi:10.1037/0033-2909.119.2.254
38. Krause, U.-M., Stark, R., & Mandl, H., (2009). The effects of cooperative learning and feedback on e-learning in statistics. Learning and Instruction, 19, 158–170. doi:10.1016/j.learninstruc.2008.03.003
39. Kulhavy, R. W., (1977). Feedback in written instruction. Review of Educational Research, 47, 211–232. doi:10.2307/1170128
40. Li, X., Ding, M., Capraro, M. M., & Capraro, R. M., (2008). Sources of differences in children's understandings of mathematical equality: Comparative analysis of teacher guides and student texts in China and the United States. Cognition and Instruction, 26, 195–217. doi:10.1080/07370000801980845
41. Luchins, A. S., (1942). Mechanization in problem solving: The effect of Einstellung. Psychological Monographs, 54, 1–95.
42. MacGregor, M., & Stacey, K., (1997). Students' understanding of algebraic notation. Educational Studies in Mathematics, 33, 1–19.
43. Matthews, P., & Rittle-Johnson, B., (2009). In pursuit of knowledge: Comparing self-explanations, concepts, and procedures as pedagogical tools. Journal of Experimental Child Psychology, 104, 1–21. doi:10.1016/j.jecp.2008.08.004
44. McNeil, N. M., (2014). A change-resistance account of children's difficulties understanding mathematical equivalence. Child Development Perspectives, 8, 42–47. doi:10.1111/cdep.12062
45. McNeil, N. M., & Alibali, M., (2005). Why won't you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76, 883–899. doi:10.1111/j.1467-8624.2005.00884.x
46. McNeil, N. M., Chesney, D. L., Matthews, P. G., Fyfe, E. R., Petersen, L. A., & Dunwiddie, A. E., (2012). It pays to be organized: Organizing arithmetic practice around equivalent values facilitates understanding of math equivalence. Journal of Educational Psychology, 104, 1109–1121. doi:10.1037/a0028997
47. McNeil, N. M., Fyfe, E. R., & Dunwiddie, A. E., (2015). Arithmetic practice can be modified to promote understanding of mathematical equivalence. Journal of Educational Psychology, 107, 423–436. doi:10.1037/a0037687
48. McNeil, N. M., Fyfe, E. R., Petersen, L. A., Dunwiddie, A. E., & Brletic-Shipley, H., (2011). Benefits of practicing 4 = 2 + 2: Nontraditional problem formats facilitate children's understanding of mathematical equivalence. Child Development, 82, 1620–1633. doi:10.1111/j.1467-8624.2011.01622.x
49. Mory, E. H., (2004). Feedback research revisited. In D., Jonassen (Ed.), Handbook of research on educational communications and technology: A project for the Association for Educational Communications and Technology (2nd ed., pp. 745–783). Mahwah, NJ: Lawrence Erlbaum Associates.
50. Narciss, S., & Huth, K., (2006). Fostering achievement and motivation with bug-related tutoring feedback in a computer-based training for written subtraction. Learning and Instruction, 16, 310–322. doi:10.1016/j.learninstruc.2006.07.003
51. National Council of Teachers of Mathematics. (2014). Principles to action: Ensuring mathematical success for all. Reston, VA: Author.
52. National Governors Association Center for Best Practices and the Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Author. Retrieved from http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf
53. Nihalani, P. K., Mayrath, M., & Robinson, D. H., (2011). When feedback harms and collaboration helps in computer simulation environments: An expertise reversal effect. Journal of Educational Psychology, 103, 776–785. doi:10.1037/a0025276
54. Pashler, H., Cepeda, N. J., Wixted, J. T., & Rohrer, D., (2005). When does feedback facilitate learning of words? Journal of Experimental Psychology: Learning, Memory, and Cognition, 31, 3–8. doi:10.1037/0278-7393.31.1.3
55. Perry, M., (1991). Learning and transfer: Instructional conditions and conceptual change. Cognitive Development, 6, 449–468. doi:10.1016/0885-2014(91)90049-J
56. Phye, G. D., (1979). The processing of informative feedback about multiple-choice test performance. Contemporary Educational Psychology, 4, 381–394.
57. Powell, S. R., (2012). Equations and the equal sign in elementary mathematics textbooks. The Elementary School Journal, 112, 627–648.
58. Powell, S. R., & Fuchs, L. S., (2010). Contribution of equal-sign instruction beyond word-problem tutoring for third-grade students with mathematics difficulty. Journal of Educational Psychology, 102, 381–394. doi:10.1037/a0018447
59. Rakoczy, K., Harks, B., Klieme, E., Blum, W., & Hochweber, J., (2013). Written feedback in mathematics: Mediated by students’ perception, moderated by goal orientation. Learning and Instruction, 27, 63–73. doi:10.1016/j.learninstruc.2013.03.002
60. Renwick, E., (1932). Children's misconceptions concerning the symbols for mathematical equality. The British Journal of Educational Psychology, 2, 173–183.
61. Rey, G. D., & Buchwald, F., (2011). The expertise reversal effect: Cognitive load and motivational explanations. Journal of Experimental Psychology: Applied, 17, 33–48.
62. Rittle-Johnson, B., & Alibali, M., (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91, 175–189. doi:10.1037/0022-0663.91.1.175
63. Rittle-Johnson, B., Loehr, A. M., & Durkin, K., (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM Mathematics Education. doi:10.1007/s11858-017-0834-z
64. Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K., (2011). Assessing knowledge of mathematical equivalence: A construct modeling approach. Journal of Educational Psychology, 103, 85–104. doi:10.1037/a0021334
65. Rittle-Johnson, B., Schneider, M., & Star, J., (2015). Not a one-way street: Bi-directional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27. doi:10.1007/s10648-015-9302-x
66. Sherman, J., & Bisanz, J., (2009). Equivalence in symbolic and nonsymbolic contexts: Benefits of solving problems with manipulatives. Journal of Educational Psychology, 101, 88–100. doi:10.1037/a0013156
67. Shute, V. J., (2008). Focus on formative feedback. Review of Educational Research, 78, 153–189. doi:10.3102/0034654307313795
68. Snow, R. E., & Lohman, D. F., (1984). Toward a theory of cognitive aptitude for learning from instruction. Journal of Educational Psychology, 76, 347–376. doi:10.1037//0022-0663.76.3.347
69. Star, J. R., (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411.
70. Sweller, J., Ayres, P., & Kalyuga, S., (2011). The redundancy effect. In M. Spector & S. Lajoie (Eds.), Cognitive load theory (pp. 141–154). New York, NY: Springer.
71. Sweller, J., Merrienboer, J., & Paas, F., (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10, 251–296.
72. Timms, M., DeVelle, S., & Lay, D., (2016). Towards a model of how learners process feedback: A deeper look at learning. Australian Journal of Education, 60, 128–145.
73. Tudge, J. R. H., Winterhoff, P. A., & Hogan, D. M., (1996). The cognitive consequences of collaborative problem solving with and without feedback. Child Development, 67, 2892–2909.
74. Zelazo, P. D., Frye, D., & Rapus, T., (1996). An age-related dissociation between knowing rules and using them. Cognitive Development, 11, 37–63.