1932

Abstract

Children's failure to reason often leads to their mathematical performance being shaped by spurious associations from problem input and overgeneralization of inapplicable procedures rather than by whether answers and procedures make sense. In particular, imbalanced distributions of problems, particularly in textbooks, lead children to create spurious associations between arithmetic operations and the numbers they combine; when conceptual knowledge is absent, these spurious associations contribute to the implausible answers, flawed strategies, and violations of principles characteristic of children's mathematics in many areas. To illustrate mechanisms that create flawed strategies in some areas but not others, we contrast computer simulations of fraction and whole number arithmetic. Most of their mechanisms are similar, but the model of fraction arithmetic lacks conceptual knowledge that precludes strategies that violate basic mathematical principles. Presentingbalanced problem distributions and inculcating conceptual knowledge for distinguishing flawed from legitimate strategies are promising means for improving children's learning.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-devpsych-041620-031544
2020-12-15
2024-06-19
Loading full text...

Full text loading...

/deliver/fulltext/devpsych/2/1/annurev-devpsych-041620-031544.html?itemId=/content/journals/10.1146/annurev-devpsych-041620-031544&mimeType=html&fmt=ahah

Literature Cited

  1. Barbieri CA, Rodrigues J, Dyson N, Jordan NC 2020. Improving fraction understanding in sixth graders with mathematics difficulties: effects of a number line approach combined with cognitive learning strategies. J. Educ. Psychol. 112:628–48
    [Google Scholar]
  2. Blazar D, Heller B, Kane T, Polikoff M, Staiger D et al. 2019. Learning by the book: comparing math achievement growth by textbook in six common core states Res. Rep., Cent. for Educ. Policy Res., Harvard Univ Cambridge, MA: https://cepr.harvard.edu/files/cepr/files/cepr-curriculum-report_learning-by-the-book.pdf
    [Google Scholar]
  3. Braithwaite DW, Leib ER, Siegler RS, McMullen J 2019. Individual differences in fraction arithmetic learning. Cogn. Psychol. 112:81–98
    [Google Scholar]
  4. Braithwaite DW, Pyke AA, Siegler RS 2017. A computational model of fraction arithmetic. Psychol. Rev. 124:603–25
    [Google Scholar]
  5. Braithwaite DW, Siegler RS. 2018. Children learn spurious associations in their math textbooks: examples from fraction arithmetic. J. Exp. Psychol. Learn. Mem. Cogn. 44:1765–77
    [Google Scholar]
  6. Braithwaite DW, Siegler RS. 2020. Putting fractions together. J. Educ. Psychol. In press https://doi.org/10.1037/edu0000477
    [Crossref] [Google Scholar]
  7. Braithwaite DW, Tian J, Siegler RS 2018. Do children understand fraction addition. ? Dev. Sci. 21:e12601
    [Google Scholar]
  8. Brown G, Quinn RJ. 2006. Algebra students’ difficulty with fractions: an error analysis. Aust. Math. Teach. 62:28–40
    [Google Scholar]
  9. Byrnes JP, Wasik BA. 1991. Role of conceptual knowledge in mathematical procedural learning. Dev. Psychol. 27:777–86
    [Google Scholar]
  10. Cady JA, Hodges TE, Collins RL 2015. A comparison of textbooks’ presentation of fractions. Sch. Sci. Math. 115:105–16
    [Google Scholar]
  11. Cai J. 2014. Searching for evidence of curricular effect on the teaching and learning of mathematics: some insights from the LieCal project. Math. Educ. Res. J. 26:811–31
    [Google Scholar]
  12. Carpenter TP, Corbitt M, Kepner H, Lindquist M, Reys R 1980. Results of the second NAEP mathematics assessment: secondary school. Math. Teach. 73:329–38
    [Google Scholar]
  13. Charalambous CY, Delaney S, Hsu H-Y, Mesa V 2010. A comparative analysis of the addition and subtraction of fractions in textbooks from three countries. Math. Think. Learn. 12:117–51
    [Google Scholar]
  14. Charalambous CY, Pitta-Pantazi D. 2007. Drawing on a theoretical model to study students’ understandings of fractions. Educ. Stud. Math. 64:293–316
    [Google Scholar]
  15. Charles R, Caldwell J, Cavanagh M, Chancellor D, Copley J et al. 2012. EnVisionmath Glenview, IL: Pearson Educ. Common Core ed.
    [Google Scholar]
  16. Chingos MM, Whitehurst GJ. 2012. Choosing blindly: instructional materials, teacher effectiveness, and the common core Rep., Brookings Inst Washington, DC: https://www.brookings.edu/wp-content/uploads/2016/06/0410_curriculum_chingos_whitehurst.pdf
    [Google Scholar]
  17. Clements DH, Swaminathan S, Hannibal MAZ, Sarama J 1999. Young children's concepts of shape. J. Res. Math. Educ. 30:192–212
    [Google Scholar]
  18. Dixon JK, Adams TL, Larson M, Leiva M 2012a. GO MATH! Orlando, F:L: Houghton Mifflin Harcourt Publ. Co Common Core ed .
    [Google Scholar]
  19. Dixon JK, Adams TL, Larson M, Leiva M 2012b. GO MATH! Standards Practice Book Orlando, FL: Houghton Mifflin Harcourt Publ. Co Common Core ed .
    [Google Scholar]
  20. Falkner KP, Levi L, Carpenter TP 1999. Children's understanding of equality: a foundation for algebra. Teach. Child. Math. 6:232–36
    [Google Scholar]
  21. Fuchs LS, Schumacher RF, Long J, Namkung J, Hamlett CL et al. 2013. Improving at-risk learners’ understanding of fractions. J. Educ. Psychol. 105:683–700
    [Google Scholar]
  22. Gabriel F, Coché F, Szucs D, Carette V, Rey B, Content A 2012. Developing children's understanding of fractions: an intervention study. Mind Brain Educ 6:137–46
    [Google Scholar]
  23. Gabriel FC, Szucs D, Content A 2013. The development of the mental representations of the magnitude of fractions. PLOS ONE 8:e80016
    [Google Scholar]
  24. Gay AS, Aichele DB. 1997. Middle school students’ understanding of number sense related to percent. Sch. Sci. Math. 97:27–36
    [Google Scholar]
  25. Geary DC. 1996. The problem-size effect in mental addition: developmental and cross-national trends. Math. Cogn. 2:63–94
    [Google Scholar]
  26. Geary DC, vanMarle K. 2018. Growth of symbolic number knowledge accelerates after children understand cardinality. Cognition 177:69–78
    [Google Scholar]
  27. Gelman R. 1991. Epigenetic foundations of knowledge structures: initial and transcendent constructions. The Epigenesis of Mind: Essays on Biology and Cognition S Carey, R Gelman 293–322 London: Psychol. Press
    [Google Scholar]
  28. Gilmore C, Göbel SM, Inglis M 2018. An Introduction to Mathematical Cognition New York: Routledge
    [Google Scholar]
  29. Gunderson EA, Hamdan N, Hildebrand L, Bartek V 2019. Number line unidimensionality is a critical feature for promoting fraction magnitude concepts. J. Exp. Child Psychol. 187:104657
    [Google Scholar]
  30. Hamann MS, Ashcraft MH. 1986. Textbook presentations of the basic addition facts. Cogn. Instr. 3:173–202
    [Google Scholar]
  31. Hamdan N, Gunderson EA. 2017. The number line is a critical spatial-numerical representation: evidence from a fraction intervention. Dev. Psychol. 53:587–96
    [Google Scholar]
  32. Handel MJ. 2016. What do people do at work. ? J. Labour Mark. Res. 49:177–97
    [Google Scholar]
  33. Hannula MS. 2003. Locating fraction on a number line. Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education NA Pateman, BJ Dougherty, JT Zilliox 17–24 Berlin, Germany: Int. Group Psychol. Math. Educ.
    [Google Scholar]
  34. Hansen N, Jordan NC, Rodrigues J 2017. Identifying learning difficulties with fractions: a longitudinal study of student growth from third through sixth grade. Contemp. Educ. Psychol. 50:45–59
    [Google Scholar]
  35. Hansen N, Rodrigues J, Kane B 2019. Opportunities for students with learning disabilities to develop representational ability with fractions: a textbook analysis. N. J. Math. Teach. 77:24–37
    [Google Scholar]
  36. Hecht SA. 1998. Toward an information-processing account of individual differences in fraction skills. J. Educ. Psychol. 90:545–59
    [Google Scholar]
  37. Hecht SA, Vagi KJ. 2010. Sources of group and individual differences in emerging fraction skills. J. Educ. Psychol. 102:843–59
    [Google Scholar]
  38. Heffernan NT, Heffernan CL. 2014. The ASSISTments ecosystem: building a platform that brings scientists and teachers together for minimally invasive research on human learning and teaching. Int. J. Artif. Intell. Educ. 24:470–97
    [Google Scholar]
  39. Hiebert J, Wearne D. 1985. A model of students’ decimal computation procedures. Cogn. Instr. 2:175–205
    [Google Scholar]
  40. Horsley M, Sikorová Z. 2014. Classroom teaching and learning resources: international comparisons from TIMSS—a preliminary review. Orb. Sch. 8:43–60
    [Google Scholar]
  41. Huxley A. 1960. On Art and Artists New York: Harper & Bros
    [Google Scholar]
  42. Jordan NC, Hansen N, Fuchs LS, Siegler RS, Gersten R, Micklos D 2013. Developmental predictors of fraction concepts and procedures. J. Exp. Child Psychol. 116:45–58
    [Google Scholar]
  43. Kahneman D. 2011. Thinking, Fast and Slow New York: Farrar, Straus and Giroux
    [Google Scholar]
  44. Koedel C, Polikoff M. 2017. Big bang for just a few bucks: the impact of math textbooks in California. Evid. Speaks Rep. 2:1–7
    [Google Scholar]
  45. Landy D, Goldstone RL. 2007a. Formal notations are diagrams: evidence from a production task. Mem. Cogn. 35:2033–40
    [Google Scholar]
  46. Landy D, Goldstone RL. 2007b. How abstract is symbolic thought. ? J. Exp. Psychol. Learn. Mem. Cogn. 33:720–33
    [Google Scholar]
  47. Landy D, Goldstone RL. 2010. Proximity and precedence in arithmetic. Q. J. Exp. Psychol. 63:1953–68
    [Google Scholar]
  48. Lortie-Forgues H, Tian J, Siegler RS 2015. Why is learning fraction and decimal arithmetic so difficult. ? Dev. Rev. 38:201–21
    [Google Scholar]
  49. Mack NK. 1995. Confounding whole-number and fraction concepts when building on informal knowledge. J. Res. Math. Educ. 26:422–41
    [Google Scholar]
  50. Martin WG, Strutchens ME, Elliot PC 2007. The Learning of Mathematics, 69th Yearbook Reston, VA: Natl. Counc. Teach. Math.
    [Google Scholar]
  51. McCloskey M. 2007. Quantitative literacy and developmental dyscalculias. Why Is Math so Hard for Some Children? The Nature and Origins of Mathematical Learning Difficulties and Disabilities DB Berch, MMM Mazzocco 415–29 Baltimore, MD: Paul H. Brookes Publ.
    [Google Scholar]
  52. McNeil NM, Fyfe ER, Dunwiddie AE 2015. Arithmetic practice can be modified to promote understanding of mathematical equivalence. J. Educ. Psychol. 107:423–36
    [Google Scholar]
  53. McNeil NM, Grandau L, Knuth EJ, Alibali MW, Stephens AC et al. 2006. Middle-school students’ understanding of the equal sign: The books they read can't help. Cogn. Instr. 24:367–85
    [Google Scholar]
  54. Mix KS, Sandhofer CM, Moore JA, Russell C 2012. Acquisition of the cardinal word principle: the role of input. Early Child. Res. Q. 27:274–83
    [Google Scholar]
  55. Moseley BJ, Okamoto Y, Ishida J 2007. Comparing US and Japanese elementary school teachers’ facility for linking rational number representations. Int. J. Sci. Math. Educ. 5:165–85
    [Google Scholar]
  56. Moss J, Case R. 1999. Developing children's understanding of the rational numbers: a new model and an experimental curriculum. J. Res. Math. Educ. 30:122–47
    [Google Scholar]
  57. Newton KJ, Willard C, Teufel C 2014. An examination of the ways that students with learning disabilities solve fraction computation problems. Elem. Sch. J. 115:1–21
    [Google Scholar]
  58. Ni Y, Zhou Y-D. 2005. Teaching and learning fraction and rational numbers: the origins and implications of whole number bias. Educ. Psychol. 40:27–52
    [Google Scholar]
  59. Obersteiner A, Van Dooren W, Van Hoof J, Verschaffel L 2013. The natural number bias and magnitude representation in fraction comparison by expert mathematicians. Learn. Instr. 28:64–72
    [Google Scholar]
  60. Opfer VD, Kaufman JH, Pane JD, Thompson LE 2018. Aligned curricula and implementation of Common Core State Mathematics Standards: findings from the American Teacher Panel Rep. RR-2487-HCT, RAND Corp Santa Monica, CA: https://www.rand.org/pubs/research_reports/RR2487.html
    [Google Scholar]
  61. Powell SR. 2012. Equations and the equal sign in elementary mathematics textbooks. Elem. Sch. J. 112:627–48
    [Google Scholar]
  62. Powell SR, Nurnberger-Haag J. 2015. Everybody counts, but usually just to 10! A systematic analysis of number representations in children's books. Early Educ. Dev. 26:377–98
    [Google Scholar]
  63. Resnick I, Verdine BN, Golinkoff R, Hirsh-Pasek K 2016. Geometric toys in the attic? A corpus analysis of early exposure to geometric shapes. Early Child. Res. Q. 36:358–65
    [Google Scholar]
  64. Richland LE, Stigler JW, Holyoak KJ 2012. Teaching the conceptual structure of mathematics. Educ. Psychol. 47:189–203
    [Google Scholar]
  65. Rohrer D, Dedrick RF, Hartwig MK 2020. The scarcity of interleaved practice in mathematics textbooks. Educ. Psychol. Rev. 32:87383
    [Google Scholar]
  66. Satlow E, Newcombe N. 1998. When is a triangle not a triangle? Young children's developing concepts of geometric shape. Cogn. Dev. 13:547–59
    [Google Scholar]
  67. Saxe GB, Diakow R, Gearhart M 2013. Towards curricular coherence in integers and fractions: a study of the efficacy of a lesson sequence that uses the number line as the principal representational context. ZDM Math. Educ. 45:343–64
    [Google Scholar]
  68. Schmidt W. 2002. The benefit to subject-matter knowledge. Am. Educ. 26:18
    [Google Scholar]
  69. Shrager J, Siegler RS. 1998. SCADS: a model of children's strategy choices and strategy discoveries. Psychol. Sci. 9:405–10
    [Google Scholar]
  70. Sidney PG, Thompson CA, Fitzsimmons C, Taber JM 2019. Children's and adults’ math attitudes are differentiated by number type. J. Exp. Educ. https://doi.org/10.1080/00220973.2019.1653815
    [Crossref] [Google Scholar]
  71. Siegler RS. 2006. Microgenetic analyses of learning. Handbook of Child Psychology, Vol. 2. Cognition, Perception, and Language D Kuhn, RS Siegler, W Damon, RM Lerner 464–510 Hoboken, NJ: Wiley. , 6th ed..
    [Google Scholar]
  72. Siegler RS, Crowley K. 1994. Constraints on learning in nonprivileged domains. Cogn. Psychol. 27:194–226
    [Google Scholar]
  73. Siegler RS, Duncan GJ, Davis-Kean PE, Duckworth K, Claessens A et al. 2012. Early predictors of high school mathematics achievement. Psychol. Sci. 23:691–97
    [Google Scholar]
  74. Siegler RS, Jenkins EA. 1989. How Children Discover New Strategies Hillsdale, NJ: Erlbaum
    [Google Scholar]
  75. Siegler RS, Lortie-Forgues H. 2015. Conceptual knowledge of fraction arithmetic. J. Educ. Psychol. 107:909–18
    [Google Scholar]
  76. Siegler RS, Pyke AA. 2013. Developmental and individual differences in understanding of fractions. Dev. Psychol. 49:1994–2004
    [Google Scholar]
  77. Siegler RS, Shrager J. 1984. Strategy choices in addition and subtraction: How do children know what to do?. The Origins of Cognitive Skills C Sophian 229–93 Hillsdale, NJ: Erlbaum
    [Google Scholar]
  78. Siegler RS, Thompson CA, Schneider M 2011. An integrated theory of whole number and fractions development. Cogn. Psychol. 62:273–96
    [Google Scholar]
  79. Stanovich KE, West RF. 2000. Individual differences in reasoning: implications for the rationality debate. ? Behav. Brain Sci. 23:645–65
    [Google Scholar]
  80. Tian J, Braithwaite DW, Siegler RS 2020. Distributions of textbook problems predict student learning: data from decimal arithmetic. J. Educ. Psychol https://doi.org/10.1037/edu0000618
    [Crossref] [Google Scholar]
  81. Torbeyns J, Schneider M, Xin Z, Siegler RS 2015. Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learn. Instr. 37:5–13
    [Google Scholar]
  82. Tunç-Pekkan Z. 2015. An analysis of elementary school children's fractional knowledge depicted with circle, rectangle, and number line representations. Educ. Stud. Math. 89:419–41
    [Google Scholar]
  83. Univ. Chic. Sch. Math. Proj 2015a. Everyday Mathematics Assessment Handbook Columbus, OH: McGraw- Hill Educ. , 4th ed..
    [Google Scholar]
  84. Univ. Chic. Sch. Math. Proj 2015b. Everyday Mathematics Student Math Journal Vols. 1 and 2 Columbus, OH: McGraw-Hill Educ. , 4th ed..
    [Google Scholar]
  85. Univ. Chic. Sch. Math. Proj 2015c. Everyday Mathematics Student Reference Book Columbus, OH: McGraw-Hill Educ. , 4th ed..
    [Google Scholar]
  86. Valverde GA, Bianchi LJ, Wolfe RG, Schmidt WH, Houang RT 2002. According to the Book: Using TIMSS to Investigate the Translation of Policy into Practice Through the World of Textbooks New York: Springer Sci.+Bus. Media
    [Google Scholar]
  87. Vamvakoussi X, Vosniadou S. 2010. How many decimals are there between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cogn. Instr. 28:181–209
    [Google Scholar]
  88. Ward JM, Mazzocco MM, Bock AM, Prokes NA 2017. Are content and structural features of counting books aligned with research on numeracy development?. Early Child. Res. Q. 39:47–63
    [Google Scholar]
/content/journals/10.1146/annurev-devpsych-041620-031544
Loading
/content/journals/10.1146/annurev-devpsych-041620-031544
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error