It pays to be organized: Organizing arithmetic practice around equivalent values facilitates understanding of math equivalence

Journal of Educational Psychology

Volumn 104, Issue 4, 2012, Pages 1109-1121

  McNeil, Nicole M ^a   Chesney, Dana L ^a   Matthews, Percival G ^a   Fyfe, Emily R ^a   Petersen, Lori A ^a   Dunwiddie, April E ^a   Wheeler, Mary C ^a  

^a UNIVERSITY OF NOTRE DAME   (United States)

Author keywords

Arithmetic practice; Learning processes; Mathematics education; Organization of knowledge

Indexed keywords

EID: 84874447354     PISSN: 00220663     EISSN: None     Source Type: Journal    
DOI: 10.1037/a0028997     Document Type: Article

References (67)

1. Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school through college. Washington, DC: U.S. Department of Education.
2. Agresti, A. (1996). An introduction to categorical data analysis. New York, NY: Wiley.
3. 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
4. Alibali, M. W., & Meredith, L. (2009, October). Sources of change in children's encoding of mathematics problems. In B. Rittle-Johnson (Organizer), Understanding knowledge change: Investigations on how children learn mathematics and literacy skills. Symposium presented at the biennial meeting of the Cognitive Development Society, San Anto-nio, TX.
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 Jour-nal, 84, 198-212. 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. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36, 412-446.
9. Bradley, R. H., & Corwyn, R. F. (2002). Socioeconomic status and child development. Annual Review of Psychology, 53, 371-399. doi:10.1146/ annurev.psych.53.100901.135233
10. Bruner, J. S. (1957). On perceptual readiness. Psychological Review, 64, 123-152. doi:10.1037/h0043805
11. Canobi, K. (2009). Concept-procedure interactions in children' addition and subtraction. Journal of Experimental Child Psychology, 102, 131-149. doi:10.1016/j.jecp.2008.07.008
12. Case, R. (1978). Intellectual development from birth to adulthood: A neo-Piagetian approach. In R. S. Siegler (Ed.), Children's thinking: What develops? (pp. 37-71). Hillsdale, NJ: Erlbaum.
13. Chase, W. G., & Simon, H. A. (1973). Perception in chess. Cognitive Psychology, 4, 55-81. doi:10.1016/0010-0285(73)90004-2
14. Chi, M. T. H., & Ceci, S. J. (1987). Content knowledge: Its role, repre-sentation, and restructuring in memory development. In L. Lipsitt (Ed.), Advances in child development and behavior (Vol. 20, pp. 91-143). New York, NY: Academic Press. doi:10.1016/S0065-2407(08)60401-2
15. Cobb, P. (1987). An investigation of young children's academic arithmetic contexts. Educational Studies in Mathematics, 18, 109-124. doi: 10.1007/BF00314722
16. Denmark, T., Barco, E., & Voran, J. (1976). Final report: A teaching experiment on equality (PMDC Technical Report No. 6). Tallahassee, FL: Florida State University.
17. Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understand-ing of equality: A foundation for algebra. Teaching Children Mathemat-ics, 6, 232-236.
18. Fyfe, E. R., & McNeil, N. M. (2009). Benefits of "concreteness fading" for children with low knowledge of math equivalence. Poster presented at the biennial meeting of the Cognitive Development Society, San Anto-nio, TX.
19. Geary, D. C., Bow Thomas, C. C., Liu, F., & Siegler, R. S. (1996). Development of arithmetical competencies in Chinese and American children: Influence of age, language, and schooling. Child Development, 67, 2022-2044. doi:10.2307/1131607
20. Gelman, R., & Williams, E. (1998). Enabling constraints for cognitive development and learning: Domain specificity and epigenesis. In W. Damon, D. Kuhn, & R. Siegler, (Eds.). Handbook of child psychology: Vol. 2. Cognition, perception and language (5th ed., pp. 575-630). New York, NY: Wiley.
21. 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
22. Hoover, H. D., Dunbar, S. B., & Frisbie, D. A. (2003). Iowa Tests of Basic Skills (ITBS). Rolling Meadows, IL: Riverside.
23. Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on children's algebraic rea-soning in elementary school. Journal for Research in Mathematics Education, 38, 258-288.
24. Jordan, N. C., Huttenlocher, J., & Levine, S. C. (1992). Differential calculation abilities in young children from middle-and low-income families. Developmental Psychology, 28, 644-653. doi:10.1037/0012-1649.28.4.644
25. Kieran, C. (1980). The interpretation of the equal sign: Symbol for an equiv-alent relation vs. an operator symbol. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education (pp. 163-169). Berkeley: University of California.
26. Kieran, C. (1981). Concepts associated with the equality symbol. Educa-tional Studies in Mathematics, 12, 317-326. doi:10.1007/BF00311062
27. Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws, Handbook of research on mathematics teaching and learning (pp. 390-419). New York, NY: Macmillan.
28. Klibanoff, R. S., Levine, S. C., Huttenlocher, J., Vasilyeva, M., & Hedges, L. V. (2006). Preschool children's mathematical knowledge: The effect of teacher "math talk". Developmental Psychology, 42, 59-69. doi: 10.1037/0012-1649.42.1.59
29. Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equa-tions. Journal for Research in Mathematics Education, 37, 297-312.
30. LeFevre, J. A., Sadesky, G. S., & Bisanz, J. (1996). Selection of procedures in mental addition: Reassessing the problem size effect in adults. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, 216-230. doi:10.1037/0278-7393.22.1.216
31. Li, X., Ding, M., Capraro, M. M., & Capraro, R. M. (2008). Sources of differences in children's understandings of mathematical equality: Com-parative analysis of teacher guides and student texts in China and the United States. Cognition and Instruction, 26, 195-217. doi:10.1080/ 07370000801980845
32. Lindberg, M. A. (1980). Is knowledge base development a necessary and sufficient condition for memory development? Journal of Experimental Child Psychology, 30, 401-410. doi:10.1016/0022-0965(80)90046-6
33. MacGregor, M., & Stacey, K. (1999). A flying start to algebra. Teaching Children Mathematics, 6, 78-85.
34. 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
35. McGilly, K., & Siegler, R. S. (1990). The influence of encoding and strategic knowledge on children's choices among serial-recall strategies. Develop-mental Psychology, 26, 931-941. doi:10.1037/0012-1649.26.6.931
36. McNeil, N. M. (2005). When the basics backfire: The change-resistance account of equation-learning difficulties. Unpublished doctoral disser-tation, Department of Psychology, University of Wisconsin, Madison.
37. McNeil, N. M. (2007). U-shaped development in math: 7-year-olds out-perform 9-year-olds on equivalence problems. Developmental Psychol-ogy, 43, 687-695. doi:10.1037/0012-1649.43.3.687
38. McNeil, N. M. (2008). Limitations to teaching children 2 Universal-GreekwithMathPi.-1.H11001 2 = 4: Typical arithmetic problems can hinder learning of math equivalence. Child Development, 79, 1524-1537. doi:10.1111/j.1467-8624.2008.01203.x
39. McNeil, N. M. (2009, June). Modifying arithmetic practice to promote understanding of mathematical equivalence. Invited talk presented at the annual research conference of the Institute of Education Sciences, Wash-ington, DC. Retrievable from http://ies.ed.gov/director/conferences/ 09ies_conference/presentations.asp
40. McNeil, N. M., & Alibali, M. W. (2004). You'll see what you mean: Students encode equations based on their knowledge of arithmetic. Cognitive Science, 28, 451-466. doi:10.1167/j.cogsci2003.11.002
41. McNeil, N. M., & Alibali, M. W. (2005a). Knowledge change as a function of mathematics experience: All contexts are not created equal. Journal of Cognition and Development, 6, 285-306. doi: 10.1207/s15327647jcd0602_6
42. McNeil, N. M., & Alibali, M. W. (2005b). Why won't you change your mind? Knowledge of operational patterns hinders learning and perfor-mance on equations. Child Development, 76, 883-899. doi:10.1111/ j.1467-8624.2005.00884.x
43. McNeil, N. M., Fyfe, E. R., Petersen, L. A., Dunwiddie, A. E., & Brletic-Shipley, H. (2011). Benefits of practicing 4 = 2 Universal-GreekwithMathPi.-1.H11001 2: Nontraditional problem formats facilitate children's understanding of math equivalence. Child Development, 82, 1620-1633. doi:10.1111/j.1467-8624.2011 .01622.x
44. McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. S., Hattikudur, S., & Krill, D. E. (2006). Middle-school students' under-standing of the equal sign: The books they read can't help. Cognition and Instruction, 24, 367-385. doi:10.1207/s1532690xci2403_3
45. McNeil, N. M., Rittle-Johnson, B., Hattikudur, S., & Petersen, L. A. (2010). Continuity in representation between children and adults: Arith-metic knowledge hinders undergraduates' algebraic problem solving. Journal of Cognition and Development, 11, 437-457. doi:10.1080/ 15248372.2010.516421
46. Moses, R. P., & Cobb, C. E. (2001). Radical equations: Math literacy and civil rights. Boston, MA: Beacon Press.
47. National Center for Improving Student Learning and Achievement in Mathematics Education. (2000). Building a foundation for learning algebra in the elementary grades. Madison, WI: Author.
48. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
49. National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
50. Perry, M. (1991). Learning and transfer: Instructional conditions and conceptual change. Cognitive Development, 6, 449-468. doi:10.1016/ 0885-2014(91)90049-J
51. Perry, M., Church, R. B., & Goldin-Meadow, S. (1988). Transitional knowledge in the acquisition of concepts. Cognitive Development, 3, 359-400. doi:10.1016/0885-2014(88)90021-4
52. Piaget, J., & Szeminska, A. (1995). The child's conception of number (C. Gattegno & F. M. Hodgson, Trans.). In H. E. Gruber & J. J. Voneche (Eds.), The essential Piaget (pp. 298-341). Northvale, NJ: Jason Aron-son Inc. (Original work published 1941).
53. 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
54. Ramani, G. B., & Siegler, R. S. (2008). Promoting broad and stable improvements in low-income children's numerical knowledge through playing number board games. Child Development, 79, 375-394. doi: 10.1111/j.1467-8624.2007.01131.x
55. Renwick, E. (1932). Children's misconceptions concerning the symbols for mathematical equality. British Journal of Educational Psychology, 2, 173-183. doi:10.1111/j.2044-8279.1932.tb02743.x
56. Rittle-Johnson, B. (2006). Promoting transfer: Effects of self-explanation and direct instruction. Child Development, 77, 1-15. doi:10.1111/j.1467-8624.2006.00852.x
57. Rittle-Johnson, B., & Alibali, M. W. (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
58. Rumelhart, D. E. (1980). Schemata: The building block of cognition. In R. Spiro, B. Bruce, & W. F. Brewer (Eds.), Theoretical issues in reading comprehension (pp. 33-58). Hilsdale, NJ: Erlbaum.
59. Seo, K.-H., & Ginsburg, H. P. (2003). "You've got to carefully read the math sentence . . .": Classroom context and children's interpretations of the equals sign. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. 161-187). Mahwah, NJ: Erlbaum.
60. Sherman, J., & Bisanz, J. (2009). Equivalence in symbolic and non-symbolic contexts: Benefits of solving problems with manipulatives. Journal of Educational Psychology, 101, 88-100. doi:10.1037/a0013156
61. Siegler, R. S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8, 481-520. doi:10.1016/0010-0285(76)90016-5
62. Siegler, R. S. (1988). Individual differences in strategy choices: Good students, not-so-good students, and perfectionists. Child Development, 59, 833-851. doi:10.2307/1130252
63. Siegler, R. S. (2002). Microgenetic studies of self-explanations. In N. Granott & J. Parziale (Eds.), Microdevelopment: Transition processes in development and learning (pp. 31-58). New York, NY: Cambridge University Press.
64. Steinberg, R. (1985). Instruction on derived facts strategies in addition and subtraction. Journal for Research in Mathematics Education, 16, 337-355. doi:10.2307/749356
65. Steinberg, R. M., Sleeman, D. H., & Ktorza, D. (1990). Algebra students' knowledge of equivalence of equations. Journal for Research in Math-ematics Education, 22, 112-121. doi:10.2307/749588
66. Tran, H., & Weinraub, M. (2006). Child care effects in context: Quality, stability, and multiplicity in nonmaternal child care arrangements during the first 15 months of life. Developmental Psychology, 42, 566-582. doi:10.1037/0012-1649.42.3.566
67. Weaver, J. F. (1973). The symmetric property of the equality relation and young children's ability to solve open addition and subtraction sen-tences. Journal for Research in Mathematics Education, 4, 45-56. doi:10.2307/749023