Cognitive predictors of calculations and number line estimation with whole numbers and fractions among at-risk students

Journal of Educational Psychology

Volumn 108, Issue 2, 2016, Pages 214-228

  Namkung, Jessica M ^a   Fuchs, Lynn S ^b  

^a STATE UNIVERSITY OF NEW YORK   (United States)

^b VANDERBILT UNIVERSITY   (United States)

Author keywords

Calculations; Cognitive predictors; Fractions; Mathematics; Number line estimation; Whole numbers

Indexed keywords

EID: 84956592395     PISSN: 00220663     EISSN: 19392176     Source Type: Journal    
DOI: 10.1037/edu0000055     Document Type: Article

References (96)

1. Alloway, T. P. (2009). Working memory, but not IQ, predicts subsequent learning in children with learning difficulties. European Journal of Psychological Assessment, 25, 92-98. http://dx.doi.org/10.1027/1015-5759.25.2.92
2. Alloway, T. P., Gathercole, S. E., Kirkwood, H.,&Elliott, J. (2009). The cognitive and behavioral characteristics of children with low working memory. Child Development, 80, 606-621. http://dx.doi.org/10.1111/j.1467-8624.2009.01282.x
3. American Psychiatric Association. (1994). Diagnostic and statistical manual of mental disorders. (4th ed.). Washington, DC: Author.
4. Baddeley, A. D. (1986). Working memory. Oxford, UK: Oxford University Press.
5. Bailey, D. H., Siegler, R. S.,&Geary, D. C. (2014). Early predictors of middle school fraction knowledge. Developmental Science, 17, 775-785. http://dx.doi.org/10.1111/desc.12155
6. Baturo, A. R.,&Cooper, T. J. (1999). Fractions, reunitisation and the number-line representation. Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education.
7. Booth, J. L.,&Newton, K. J. (2012). Fractions: Could they really be the gatekeeper's doorman? Contemporary Educational Psychology, 37, 247-253. http://dx.doi.org/10.1016/j.cedpsych.2012.07.001
8. Booth, J. L.,&Siegler, R. S. (2006). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 42, 189-201. http://dx.doi.org/10.1037/0012-1649.41.6.189
9. Booth, J. L.,&Siegler, R. S. (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79, 1016-1031. http://dx.doi.org/10.1111/j.1467-8624.2008.01173.x
10. Bright, G. W., Behr, M. J., Post, T. R.,&Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19, 215-232. http://dx.doi.org/10.2307/749066
11. Bull, R.,&Johnston, R. S. (1997). Children's arithmetical difficulties: Contributions from processing speed, item identification, and short-term memory. Journal of Experimental Child Psychology, 65, 1-24. http://dx.doi.org/10.1006/jecp.1996.2358
12. Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46, 3-18. http://dx.doi.org/10.1111/j.1469-7610.2004.00374.x
13. Cramer, K. A., Post, T. R.,&delMas, R. C. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33, 111-144. http://dx.doi.org/10.2307/749646
14. Cramer, K. A.,&Wyberg, T. (2009). Efficacy of different concrete models for teaching the part-whole construct for fractions. Mathematical Thinking and Learning, 11, 226-257. http://dx.doi.org/10.1080/10986060903246479
15. Dark, V. J.,&Benbow, C. P. (1991). Differential enhancement of working memory with mathematical versus verbal precocity. Journal of Educational Psychology, 83, 48-60. http://dx.doi.org/10.1037/0022-0663.83.1.48
16. de Hevia, M. D., Vallar, G.,&Girelli, L. (2008). Visualizing numbers in the mind's eye: The role of visuo-spatial processes in numerical abilities. Neuroscience and Biobehavioral Reviews, 32, 1361-1372. http://dx.doi.org/10.1016/j.neubiorev.2008.05.015
17. Edwards, L., Salant, V., Howard, V. F., Brougher, J.,&McLaughlin, T. F. (1995). Effectiveness of self-management on attentional behavior and reading comprehension for children with attention deficit disorder. Child&Family Behavior Therapy, 17, 1-17. http://dx.doi.org/10.1300/J019v17n02_01
18. Feigenson, L., Dehaene, S.,&Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307-314. http://dx.doi.org/10.1016/j.tics.2004.05.002
19. Fuchs, L. S., Compton, D. L., Fuchs, D., Paulsen, K., Bryant, J. D., & Hamlett, C. L. (2005). The prevention, identification, and cognitive determinants of math difficulty. Journal of Educational Psychology, 97, 493-513. http://dx.doi.org/10.1037/0022-0663.97.3.493
20. Fuchs, L. S.,&Fuchs, D. (2005). Enhancing mathematical problem solving for students with disabilities. The Journal of Special Education, 39, 45-57. http://dx.doi.org/10.1177/00224669050390010501
21. Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R., Seethaler, P. M., Capizzi, A. M., Fletcher, J. M. (2006). The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Journal of Educational Psychology, 98, 29-43. http://dx.doi.org/10.1037/0022-0663.98.1.29
22. Fuchs, L. S., Fuchs, D., Stuebing, K., Fletcher, J. M., Hamlett, C. L., & Lambert, W. (2008). Problem solving and computational skill: Are they shared or distinct aspects of mathematical cognition? Journal of Educational Psychology, 100, 30-47. http://dx.doi.org/10.1037/0022-0663.100.1.30
23. Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D., Hamlett, C. L., & Bryant, J. D. (2010). The contributions of numerosity and domain-general abilities to school readiness. Child Development, 81, 1520-1533. http://dx.doi.org/10.1111/j.1467-8624.2010.01489.x
24. Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D., Hamlett, C. L., Seethaler, P. M., Schatschneider, C. (2010). Do different types of school mathematics development depend on different constellations of numerical versus general cognitive abilities? Developmental Psychology, 46, 1731-1746. http://dx.doi.org/10.1037/a0020662
25. Fuchs, L. S., Geary, D. C., Compton, D. L., Fuchs, D., Schatschneider, C., Hamlett, C. L., Changas, P. (2013). Effects of first-grade number knowledge tutoring with contrasting forms of practice. Journal of Educational Psychology, 105, 58-77. http://dx.doi.org/10.1037/a0030127
26. Fuchs, L. S., Hamlett, C. L.,&Powell, S. R. (2003). Math Fact Fluency and Double-Digit Additional and Subtraction Tests. Available from L.S. Fuchs, 228 Peabody, Vanderbilt University, Nashville, TN 37203.
27. Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J. M., Hamlett, C. L., Cirino, P. T., Changas, P. (2013). Improving at-risk learners' understanding of fractions. Journal of Educational Psychology, 105, 683-700. http://dx.doi.org/10.1037/a0032446
28. Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J. M., Malone, A., Hamlett, C. L., Changas, P. (in press). Effects of intervention to improve at-risk fourth graders' understanding, calculations, and word problems with fractions. The Elementary School Journal.
29. Fuchs, L. S., Schumacher, R. F., Sterba, S. K., Long, J., Namkung, J. M., Malone, A., Changas, P. (2014). Does working memory moderate the effects of fraction intervention? An aptitude-treatment interaction. Journal of Educational Psychology, 106, 499-514. http://dx.doi.org/10.1037/a0034341
30. Geary, D. C. (1990). A componential analysis of an early learning deficit in mathematics. Journal of Experimental Child Psychology, 49, 363-383.
31. Geary, D. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345-362. http://dx.doi.org/10.1037/0033-2909.114.2.345
32. Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4-15.
33. Geary, D. C., Bow-Thomas, C. C.,&Yao, Y. (1992). Counting knowledge and skill in cognitive addition: A comparison of normal and mathematically disabled children. Journal of Experimental Child Psychology, 54, 372-391. http://dx.doi.org/10.1016/0022-0965(92)90026-3
34. Geary, D. C., Brown, S. C.,&Samaranayake, V. A. (1991). Cognitive addition: A short longitudinal study of strategy choice and speed-of-processing differences in normal and mathematically disabled children. Developmental Psychology, 27, 787-797. http://dx.doi.org/10.1037/0012-1649.27.5.787
35. Geary, D. C., Hoard, M. K., Byrd-Craven, J.,&DeSoto, M. C. (2004). Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability. Journal of Experimental Child Psychology, 88, 121-151. http://dx.doi.org/10.1016/j.jecp.2004.03.002
36. Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L.,&Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78, 1343-1359. http://dx.doi.org/10.1111/j.1467-8624.2007.01069.x
37. Geary, D. C., Hoard, M. K.,&Hamson, C. O. (1999). Numerical and arithmetical cognition: Patterns of functions and deficits in children at risk for a mathematical disability. Journal of Experimental Child Psychology, 74, 213-239. http://dx.doi.org/10.1006/jecp.1999.2515
38. Geary, D. C., Hoard, M. K., Nugent, L.,&Byrd-Craven, J. (2008). Development of number line representations in children with mathematical learning disability. Developmental Neuropsychology, 33, 277-299. http://dx.doi.org/10.1080/87565640801982361
39. Gunderson, E. A., Ramirez, G., Beilock, S. L.,&Levine, S. C. (2012). The relation between spatial skill and early number knowledge: The role of the linear number line. Developmental Psychology, 48, 1229-1241. http://dx.doi.org/10.1037/a0027433
40. Harris, K. R., Friedlander, B. D., Saddler, B., Frizzelle, R.,&Graham, S. (2005). Self-monitoring of attention versus self-monitoring of academic performance effects among students with ADHD in the general education classroom. The Journal of Special Education, 39, 145-157. http://dx.doi.org/10.1177/00224669050390030201
41. Hecht, S. A. (1998). Toward an information-processing account of individual differences in fraction skills. Journal of Educational Psychology, 90, 545-559. http://dx.doi.org/10.1037/0022-0663.90.3.545
42. Hecht, S. A., Close, L.,&Santisi, M. (2003). Sources of individual differences in fraction skills. Journal of Experimental Child Psychology, 86, 277-302. http://dx.doi.org/10.1016/j.jecp.2003.08.003
43. Hecht, S. A., Torgesen, J. K., Wagner, R. K.,&Rashotte, C. A. (2001). The relations between phonological processing abilities and emerging individual differences in mathematical computation skills: A longitudinal study from second to fifth grades. Journal of Experimental Child Psychology, 79, 192-227. http://dx.doi.org/10.1006/jecp.2000.2586
44. Hecht, S. A.,&Vagi, K. J. (2010). Sources of group and individual differences in emerging fraction skills. Journal of Educational Psychology, 102, 843-859. http://dx.doi.org/10.1037/a0019824
45. Hecht, S. A.,&Vagi, K. J. (2012). Patterns of strengths and weaknesses in children's knowledge about fractions. Journal of Experimental Child Psychology, 111, 212-229.
46. Hiebert, J., Wearne, D.,&Taber, S. (1991). Fourth graders' gradual construction of decimal fractions during instruction using different physical representations. The Elementary School Journal, 91, 321-341. http://dx.doi.org/10.1086/461658
47. Hitch, J. G.,&McAuley, E. (1991). Working memory in children with specific arithmetical learning difficulties. British Journal of Psychology, 82, 375-386. http://dx.doi.org/10.1111/j.2044-8295.1991.tb02406.x
48. Hoffer, T. B., Venkataraman, L., Hedberg, E. C.,&Shagle, S. (2007). Final report on the National Survey of Algebra Teachers for the National Math Panel. Retrieved from http://www2.ed.gov/about/bdscomm/list/mathpanel/final-report-algebra-teachers.pdf
49. Howell, D. C. (2007). Statistical methods for psychology (6th ed.). Belmont, CA: Thompson Wadsworth.
50. Jordan, N. C., Hanich, L. B.,&Kaplan, D. (2003). Arithmetic fact mastery in young children: A longitudinal investigation. Journal of Experimental Child Psychology, 85, 103-119. http://dx.doi.org/10.1016/S0022-0965(03)00032-8
51. Jordan, N. C., Hansen, N., Fuchs, L. S., Siegler, R. S., Gersten, R., & Micklos, D. (2013). Developmental predictors of fraction concepts and procedures. Journal of Experimental Child Psychology, 116, 45-58. http://dx.doi.org/10.1016/j.jecp.2013.02.001
52. Jordan, N. C., Kaplan, D., Nabors Oláh, L.,&Locuniak, M. N. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematics difficulties. Child Development, 77, 153-175. http://dx.doi.org/10.1111/j.1467-8624.2006.00862.x
53. Jordan, N. C.,&Montani, T. O. (1997). Cognitive arithmetic and problem solving: A comparison of children with specific and general mathematics difficulties. Journal of Learning Disabilities, 30, 624-634. http://dx.doi.org/10.1177/002221949703000606
54. Kail, R. (1992). Processing speed, speech rate, and memory. Developmental Psychology, 28, 899-904. http://dx.doi.org/10.1037/0012-1649.28.5.899
55. Kail, R.,&Hall, L. K. (1994). Processing speed, naming speed, and reading. Developmental Psychology, 30, 949-954. http://dx.doi.org/10.1037/0012-1649.30.6.949
56. Lee, K., Bull, R.,&Ho, R. M. (2013). Developmental changes in executive functioning. Child Development, 84, 1933-1953. http://dx.doi.org/10.1111/cdev.12096
57. 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. http://dx.doi.org/10.1037/0096-3445.124.1.83
58. Lesh, R., Post, T.,&Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representations in the teaching and learning of mathematics (pp. 33-40). Hillsdale, NJ: Erlbaum.
59. Levine, S. C., Jordan, N. C.,&Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53, 72-103. http://dx.doi.org/10.1016/S0022-0965(05)80005-0
60. McAuley, T.,&White, D. A. (2011). A latent variables examination of processing speed, response inhibition, and working memory during typical development. Journal of Experimental Child Psychology, 108, 453-468. http://dx.doi.org/10.1016/j.jecp.2010.08.009
61. McLean, J. F.,&Hitch, G. J. (1999). Working memory impairments in children with specific arithmetic learning difficulties. Journal of Experimental Child Psychology, 74, 240-260. http://dx.doi.org/10.1006/jecp.1999.2516
62. Miura, I. T., Okamoto, Y., Vlahovic-Stetic, V., Kim, C. C.,&Han, J. H. (1999). Language supports for children's understanding of numerical fractions: Cross-national comparisons. Journal of Experimental Child Psychology, 74, 356-365. http://dx.doi.org/10.1006/jecp.1999.2519
63. Mix, K. S., Levine, S. C.,&Huttenlocher, J. (1999). Early fraction calculation ability. Developmental Psychology, 35, 164-174. http://dx.doi.org/10.1037/0012-1649.35.1.164
64. Muthén, L. K.,&Muthén, B. (1998-2010). Mplus user's guide: Version 6. Los Angeles, CA: Muthén&Muthén.
65. National Mathematics Advisory Panel. (2008). Foundations for Success: Final Report of the National Mathematics Advisory Panel. Washington, DC: United States Department of Education. Retrieved from http://www.ed.gov/about/bdscomm/list/mathpanel/reprot/final-report.pdf
66. Ni, Y.,&Zhou, Y. D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40, 27-52. http://dx.doi.org/10.1207/s15326985ep4001_3
67. Nutley, S. B., Söderqvist, S., Bryde, S., Thorell, L. B., Humphreys, K., & Klingberg, T. (2011). Gains in fluid intelligence after training non-verbal reasoning in 4-year-old children: A controlled, randomized study. Developmental Science, 14, 591-601. http://dx.doi.org/10.1111/j.1467-7687.2010.01022.x
68. Paik, J. H.,&Mix, K. S. (2003). U.S. and Korean children's comprehension of fraction names: A reexamination of cross-national differences. Child Development, 74, 144-154. http://dx.doi.org/10.1111/1467-8624.t01-1-00526
69. Peng, P., Congying, S., Beilei, L.,&Sha, T. (2012). Phonological storage and executive function deficits in children with mathematics difficulties. Journal of Experimental Child Psychology, 112, 452-466. http://dx.doi.org/10.1016/j.jecp.2012.04.004
70. Pickering, S.,&Gathercole, S. (2001). Working Memory Test Battery for Children. London, UK: The Psychological Corporation.
71. Primi, R., Ferrao, M. E.,&Almeida, L. S. (2010). Fluid intelligence as a predictor of learning: A longitudinal multilevel approach applied to math. Learning and Individual Differences, 20, 446-451. http://dx.doi.org/10.1016/j.lindif.2010.05.001
72. Raghubar, K., Cirino, P., Barnes, M., Ewing-Cobbs, L., Fletcher, J., & Fuchs, L. (2009). Errors in multi-digit arithmetic and behavioral inattention in children with math difficulties. Journal of Learning Disabilities, 42, 356-371. http://dx.doi.org/10.1177/0022219409335211
73. Rittle-Johnson, B.,&Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills: Studies in developmental psychology (pp. 75-110). Hove, England: Psychology Press.
74. 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. http://dx.doi.org/10.1037/0022-0663.93.2.346
75. Robinson, C. S., Menchetti, B. M.,&Torgesen, J. K. (2002). Towards a two-factor theory of one type of mathematics disabilities. Learning Disabilities Research&Practice, 17, 81-89. http://dx.doi.org/10.1111/1540-5826.00035
76. Schneider, M., Grabner, R. H.,&Paetsch, J. (2009). Mental number line, number line estimation, and mathematical achievement: Their interrelations in grades 5 and 6. Journal of Educational Psychology, 101, 359-372. http://dx.doi.org/10.1037/a0013840
77. Schumacher, R. F., Namkung, J. M.,&Fuchs, L. S. (2010). 2010 Fraction Battery. Available from L.S. Fuchs, 228 Peabody, Vanderbilt University, Nashville, TN 37203.
78. Seethaler, P. M., Fuchs, L. S., Star, J. R.,&Bryant, J. (2011). The cognitive predictors of computational skill with whole versus rational numbers: An exploratory study. Learning and Individual Differences, 21, 536-542. http://dx.doi.org/10.1016/j.lindif.2011.05.002
79. Shimabukuro, S. M., Prater, M. A., Jenkins, A.,&Edelen-Smith, P. (1999). The effects of self-monitoring of academic performance on students with learning disabilities and ADD/ADHD. Education&Treatment of Children, 22, 397-414.
80. Siegel, L. S.,&Ryan, E. B. (1989). The development of working memory in normally achieving and subtypes of learning disabled children. Child Development, 60, 973-980. http://dx.doi.org/10.2307/1131037
81. Siegler, R. S.,&Booth, J. L. (2004). Development of numerical estimation in young children. Child Development, 75, 428-444. http://dx.doi.org/10.1111/j.1467-8624.2004.00684.x
82. Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., & Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U. S. Department of Education. Retrieved from whatworks.ed.gov/publications/practiceguides
83. Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23, 691-697. http://dx.doi.org/10.1177/0956797612440101
84. Siegler, R. S.,&Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49, 1994-2004. http://dx.doi.org/10.1037/a0031200
85. Siegler, R. S., Thompson, C. A.,&Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62, 273-296. http://dx.doi.org/10.1016/j.cogpsych.2011.03.001
86. Stafylidou, S.,&Vosniadou, S. (2004). The development of students' understanding of the numerical value of fractions. Learning and Instruction, 14, 503-518. http://dx.doi.org/10.1016/j.learninstruc.2004.06.015
87. Swanson, H. L. (2006). Cross-sectional and incremental changes in working memory and mathematical problem solving. Journal of Educational Psychology, 98, 265-281. http://dx.doi.org/10.1037/0022-0663.98.2.265
88. Swanson, J. M., Schuck, S., Mann, M., Carlson, C., Hartman, K., Sergeant, J. A., McCleary R. (2004). Categorical and dimensional definitions and evaluations of symptoms of ADHD: The SNAP and SWAN Rating Scales. Retrieved from http://www.ADHD.net
89. Tabachnick, B. G.,&Fidell, L. S. (2007). Using multivariate statistics (5th ed.). Boston, MA: Allyn&Bacon.
90. Test, D. W.,&Ellis, M. F. (2005). The effects of LAP fractions on addition and subtraction of fractions with students with mild disabilities. Education & Treatment of Children, 28, 11-24.
91. Wechsler, D. (1999). Wechsler Abbreviated Scale of Intelligence. San Antonio, TX: Psychological Corporation.
92. Wilkinson, G. S. (2008). Wide Range Achievement Test (Rev. 4). Wilmington, DE: Wide Range.
93. Woodcock, R. W. (1997). Woodcock Diagnostic Reading Battery. Itasca, IL: Riverside.
94. Woodcock, R. W., McGrew, K. S.,&Mather, N. (2001). Woodcock-Johnson III. Itasca, IL: Riverside.
95. Zorzi, M., Priftis, K., Meneghello, F., Marenzi, R.,&Umiltà, C. (2006). The spatial representation of numerical and non-numerical sequences: Evidence from neglect. Neuropsychologia, 44, 1061-1067. http://dx.doi.org/10.1016/j.neuropsychologia.2005.10.025
96. Zorzi, M., Priftis, K.,&Umiltà, C. (2002). Brain damage: Neglect disrupts the mental number line. Nature, 417, 138-139. http://dx.doi.org/10.1038/417138a