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The Role of Conceptual Subitising in the Development of Foundational Number Sense

  • Judy SayersEmail author
  • Paul Andrews
  • Lisa Björklund Boistrup
Chapter

Abstract

Evidence indicates that children with a well-developed number sense are more likely to experience long-term mathematical success than children without. However, number sense has remained an elusive construct. In this chapter, we summarise the development of an eight-dimensional framework categorising what we have come to call foundational number sense or those non-innate number-related competences typically taught during the first years of schooling. We also show, drawing on grade one lessons from Hungary and Sweden, how focused instruction on conceptual subitising, the teaching of children to identify and use easily recognisable groups of objects to structure children’s understanding of number, facilitates children’s acquisition of a range of foundational number sense-related competences.

Keywords

Number Sense Number Pattern Instructional Task Constant Comparison Analysis Missing Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We are grateful to our colleague Jenni Back for sharing her Hungarian data.

References

  1. Andrews, P., & Sayers, J. (2013). Comparative studies of mathematics teaching: Does the means of analysis determine the outcome? ZDM: The International Journal on Mathematics Education, 45(1), 133–144.CrossRefGoogle Scholar
  2. Andrews, P., & Sayers, J. (2015). Identifying opportunities for grade one children to acquire foundational number sense: Developing a framework for cross cultural classroom analyses. Early Childhood Education Journal. doi: 10.1007/s10643-014-0653-6.Google Scholar
  3. Andrews, P., Sayers, J., & Marschall, G. (2015). Developing foundational number sense: Number line examples from Poland and Russia. Paper presented to the ninth congress of European research in mathematics education (CERME9), Prague.Google Scholar
  4. Aubrey, C., Dahl, S., & Godfrey, R. (2006). Early mathematics development and later achievement: Further evidence. Mathematics Education Research Journal, 18(1), 27–46.CrossRefGoogle Scholar
  5. Aubrey, C., & Godfrey, R. (2003). The development of children’s early numeracy through key stage 1. British Educational Research Journal, 29(6), 821–840.CrossRefGoogle Scholar
  6. Aunio, P., & Niemivirta, M. (2010). Predicting children’s mathematical performance in grade one by early numeracy. Learning and Individual Differences, 20(5), 427–435.CrossRefGoogle Scholar
  7. Aunola, K., Leskinen, E., Lerkkanen, M.-K., & Nurmi, J.-E. (2004). Developmental dynamics of math performance from preschool to grade 2. Journal of Educational Psychology, 96(4), 699–713.CrossRefGoogle Scholar
  8. Back, J., Sayers, J., & Andrews, P. (2014). The development of foundational number sense in England and Hungary: A case study comparison. In B. Ubuz, Ç. Haser, & M. Mariotti (Eds.), Proceedings of the eighth congress of the European Society for Research in Mathematics Education (pp. 1835–1844). Ankara: Middle East Technical University on behalf of ERME.Google Scholar
  9. Baroody, A. (2004). The developmental bases for early childhood number and operations standards. In D. Clements & J. Sarama (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 173–219). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  10. Baroody, A., & Wilkins, J. (1999). The development of informal counting, number, and arithmetic skills and concepts. In J. Copley (Ed.), Mathematics in the early years (pp. 48–65). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  11. Battista, M., Clements, D., Arnoff, J., Battista, K., & Van Auken Borrow, C. (1998). Students’ spatial structuring of two-dimensional arrays of squares. Journal for Research in Mathematics Education, 29, 503–532.CrossRefGoogle Scholar
  12. Benoit, L., Lehalle, H., & Jouen, F. (2004). Do young children acquire number words through subitizing or counting? Cognitive Development, 19(3), 291–307.CrossRefGoogle Scholar
  13. Berch, D. B. (2005). Making sense of number sense. Journal of Learning Disabilities, 38(4), 333–339.CrossRefGoogle Scholar
  14. Booth, J., & Siegler, R. (2006). Developmental and individual differences in pure numerical estimation. Developmental Psychology, 42(1), 189–201.CrossRefGoogle Scholar
  15. Booth, J., & Siegler, R. (2008). Numerical magnitude representations influence arithmetic learning. Child Development, 79(4), 1016–1031.CrossRefGoogle Scholar
  16. Brewer, E. W. (2012). Secondary data analysis. In J. Goodwin (Ed.), Sage secondary data analysis (pp. 165–176). London: Sage.Google Scholar
  17. Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46(1), 3–18.CrossRefGoogle Scholar
  18. Casey, B., Kersh, J., & Young, J. (2004). Storytelling sagas: An effective medium for teaching early childhood mathematics. Early Childhood Research Quarterly, 19(1), 167–172.CrossRefGoogle Scholar
  19. Chard, D., Clarke, B., Baker, S., Otterstedt, J., Braun, D., & Katz, R. (2005). Using measures of number sense to screen for difficulties in mathematics: Preliminary findings. Assessment for Effective Intervention, 30(2), 3–14.CrossRefGoogle Scholar
  20. Clarke, B., & Shinn, M. (2004). A preliminary investigation into the identification and development of early mathematics curriculum-based measurement. School Psychology Review, 33(2), 234–248.Google Scholar
  21. Clements, D. (1999). Subitizing. What is it? Why teach it? Teaching Children Mathematics, 5(7), 400–405.Google Scholar
  22. Clements, D. (2007). Curriculum research: Toward a framework for “research-based curricula”. Journal for Research in Mathematics Education, 38(1), 35–70.Google Scholar
  23. Clements, D., & Sarama, J. (2007). Early childhood mathematics learning. In F. Lester (Ed.), Handbook of research on teaching and learning mathematics (pp. 461–555). Greenwich, CT: Information Age.Google Scholar
  24. Clements, D., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York: Routledge.Google Scholar
  25. Conderman, G., Jung, M., & Hartman, P. (2014). Subitizing and early mathematics standards: A winning combination. Kappa Delta Pi Record, 50(1), 18–23.CrossRefGoogle Scholar
  26. Cowan, R., & Renton, M. (1996). Do they know what they are doing? Children’s use of economical addition strategies and knowledge of commutativity. Educational Psychology, 16(4), 407–420.CrossRefGoogle Scholar
  27. Dale, A., Arber, S., & Procter, M. (1988). Doing secondary analysis. London: Unwin Hyman.Google Scholar
  28. De Smedt, B., Noël, M.-P., Gilmore, C., & Ansari, D. (2013). How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children’s mathematical skills? A review of evidence from brain and behavior. Trends in Neuroscience and Education, 2(2), 48–55.CrossRefGoogle Scholar
  29. De Smedt, B., Verschaffel, L., & Ghesquière, P. (2009). The predictive value of numerical magnitude comparison for individual differences in mathematics achievement. Journal of Experimental Child Psychology, 103(4), 469–479.CrossRefGoogle Scholar
  30. Dehaene, S. (2001). Précis of the number sense. Mind & Language, 16(1), 16–36.CrossRefGoogle Scholar
  31. Desoete, A., Ceulemans, A., De Weerdt, F., & Pieters, S. (2012). Can we predict mathematical learning disabilities from symbolic and non-symbolic comparison tasks in kindergarten? Findings from a longitudinal study. British Journal of Educational Psychology, 82(1), 64–81.CrossRefGoogle Scholar
  32. Desoete, A., Stock, P., Schepens, A., Baeyens, D., & Roeyers, H. (2009). Classification, seriation, and counting in grades 1, 2, and 3 as two-year longitudinal predictors for low achieving in numerical facility and arithmetical achievement? Journal of Psychoeducational Assessment, 27(3), 252–264.CrossRefGoogle Scholar
  33. Faulkner, V. (2009). The components of number sense. Teaching Exceptional Children, 41(5), 24–30.Google Scholar
  34. Faulkner, V., & Cain, C. (2013). Improving the mathematical content knowledge of general and special educators: Evaluating a professional development module that focuses on number sense. Teacher Education and Special Education: The Journal of the Teacher Education Division of the Council for Exceptional Children, 36(2), 115–131.CrossRefGoogle Scholar
  35. Fayol, M., Barrouillet, P., & Marinthe, C. (1998). Predicting arithmetical achievement from neuro-psychological performance: A longitudinal study. Cognition, 68(2), B63–B70.CrossRefGoogle Scholar
  36. Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314.CrossRefGoogle Scholar
  37. Fuson, K. (1988). Children’s counting and concept of number. New York: Springer.CrossRefGoogle Scholar
  38. Gallistel, C., & Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4(2), 59–65.CrossRefGoogle Scholar
  39. Geary, D. (2011). Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study. Developmental Psychology, 47(6), 1539–1552.CrossRefGoogle Scholar
  40. Geary, D. (2013). Early foundations for mathematics learning and their relations to learning disabilities. Current Directions in Psychological Science, 22(1), 23–27.CrossRefGoogle Scholar
  41. Geary, D., Bailey, D., & Hoard, M. (2009). Predicting mathematical achievement and mathematical learning disability with a simple screening tool. Journal of Psychoeducational Assessment, 27(3), 265–279.CrossRefGoogle Scholar
  42. Geary, D., Hamson, C., & Hoard, M. (2000). Numerical and arithmetical cognition: A longitudinal study of process and concept deficits in children with learning disability. Journal of Experimental Child Psychology, 77(3), 236–263.CrossRefGoogle Scholar
  43. Gelman, R., & Tucker, M. (1975). Further investigations of the young child’s conception of number. Child Development, 46(1), 167–175.CrossRefGoogle Scholar
  44. Gersten, R., Jordan, N., & Flojo, J. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38(4), 293–304.CrossRefGoogle Scholar
  45. Gracia-Bafalluy, M., & Noël, M. P. (2008). Does finger training increase young children’s numerical performance? Cortex, 44(4), 368–375.CrossRefGoogle Scholar
  46. Griffin, S. (2004). Building number sense with number worlds: A mathematics program for young children. Early Childhood Research Quarterly, 19(1), 173–180.CrossRefGoogle Scholar
  47. Griffin, S., Case, R., & Siegler, R. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 24–49). Cambridge, MA: MIT Press.Google Scholar
  48. Heaton, J. (2008). Secondary analysis of qualitative data: An overview. Historical Social Research/Historische Sozialforschung, 33(3) (125), 33–45.Google Scholar
  49. Holloway, I., & Ansari, D. (2009). Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement. Journal of Experimental Child Psychology, 103(1), 17–29.CrossRefGoogle Scholar
  50. Howell, S., & Kemp, C. (2005). Defining early number sense: A participatory Australian study. Educational Psychology, 25(5), 555–571.CrossRefGoogle Scholar
  51. Hunting, R. (2003). Part-whole number knowledge in preschool children. Journal of Mathematical Behavior, 22(3), 217–235.CrossRefGoogle Scholar
  52. Ivrendi, A. (2011). Influence of self-regulation on the development of children’s number sense. Early Childhood Education Journal, 39(4), 239–247.CrossRefGoogle Scholar
  53. Jordan, N., Huttenlocher, J., & Levine, S. (1992). Differential calculation abilities in young children from middle- and low-income families. Developmental Psychology, 28(4), 644–653.CrossRefGoogle Scholar
  54. Jordan, N., Kaplan, D., Locuniak, M., & Ramineni, C. (2007). Predicting first-grade math achievement from developmental number sense trajectories. Learning Disabilities Research & Practice, 22(1), 36–46.CrossRefGoogle Scholar
  55. Jordan, N., Kaplan, D., Nabors Oláh, L., & Locuniak, M. (2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematics difficulties. Child Development, 77(1), 153–175.CrossRefGoogle Scholar
  56. Jordan, N., Kaplan, D., Ramineni, C., & Locuniak, M. (2009). Early math matters: Kindergarten number competence and later mathematics outcomes. Developmental Psychology, 45(3), 850–867.CrossRefGoogle Scholar
  57. Jordan, N., & Levine, S. (2009). Socioeconomic variation, number competence, and mathematics learning difficulties in young children. Developmental Disabilities Research Reviews, 15(1), 60–68.CrossRefGoogle Scholar
  58. Jung, M. (2011). Number relationships in a preschool classroom. Teaching Children Mathematics, 17(9), 550–557.Google Scholar
  59. Jung, M., Hartman, P., Smith, T., & Wallace, S. (2013). The effectiveness of teaching number relationships in preschool. International Journal of Instruction, 6(1), 165–178.Google Scholar
  60. Kalchman, M., Moss, J., & Case, R. (2001). Psychological models for the development of mathematical understanding: Rational numbers and functions. In S. Carver & D. Klahr (Eds.), Cognition and instruction: Twenty-five years of progress (pp. 1–38). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  61. Koontz, K., & Berch, D. (1996). Identifying simple numerical stimuli: Processing inefficiencies exhibited arithmetic learning disabled children. Mathematical Cognition, 2(1), 1–23.CrossRefGoogle Scholar
  62. Krajewski, K., & Schneider, W. (2009). Early development of quantity to number-word linkage as a precursor of mathematical school achievement and mathematical difficulties: Findings from a four-year longitudinal study. Learning and Instruction, 19(6), 513–526.CrossRefGoogle Scholar
  63. Kroesbergen, E., Van Luit, J., Van Lieshout, E., Van Loosbroek, E., & Van de Rijt, B. (2009). Individual differences in early numeracy. Journal of Psychoeducational Assessment, 27(3), 226–236.CrossRefGoogle Scholar
  64. Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basic numerical capacities: A study of 8-9-year-old students. Cognition, 93(2), 99–125.CrossRefGoogle Scholar
  65. LeFevre, J.-A., Smith-Chant, B., Fast, L., Skwarchuk, S.-L., Sargla, E., Arnup, J., et al. (2006). What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through Grade 2. Journal of Experimental Child Psychology, 93(4), 285–303.CrossRefGoogle Scholar
  66. Lembke, E., & Foegen, A. (2009). Identifying early numeracy indicators for kindergarten and first-grade students. Learning Disabilities Research & Practice, 24(1), 12–20.CrossRefGoogle Scholar
  67. Levine, S., Jordan, N., & Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53(1), 72–103.CrossRefGoogle Scholar
  68. Libertus, M. E., Feigenson, L., & Halberda, J. (2011). Preschool acuity of the approximate number system correlates with school math ability. Developmental Science, 14(6), 1292–1300.CrossRefGoogle Scholar
  69. Lipton, J., & Spelke, E. (2005). Preschool children’s mapping of number words to nonsymbolic numerosities. Child Development, 76(5), 978–988.CrossRefGoogle Scholar
  70. Locuniak, M., & Jordan, N. (2008). Using kindergarten number sense to predict calculation fluency in second grade. Journal of Learning Disabilities, 41(5), 451–459.CrossRefGoogle Scholar
  71. Lyons, I., & Beilock, S. (2011). Numerical ordering ability mediates the relation between number-sense and arithmetic competence. Cognition, 121(2), 256–261.CrossRefGoogle Scholar
  72. Malofeeva, E., Day, J., Saco, X., Young, L., & Ciancio, D. (2004). Construction and evaluation of a number sense test with Head Start children. Journal of Educational Psychology, 96(4), 648–659.CrossRefGoogle Scholar
  73. Mazzocco, M. M., Feigenson, L., & Halberda, J. (2011). Preschoolers’ precision of the approximate number system predicts later school mathematics performance. PLoS One, 6(9), e23749.CrossRefGoogle Scholar
  74. McIntosh, A., Reys, B., & Reys, R. (1992). A proposed framework for examining basic number sense. For the Learning of Mathematics, 12(3), 2–8.Google Scholar
  75. Moeller, K., Neuburger, S., Kaufmann, L., Landerl, K., & Nuerk, H. C. (2009). Basic number processing deficits in developmental dyscalculia: Evidence from eye tracking. Cognitive Development, 24(4), 371–386.CrossRefGoogle Scholar
  76. Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49.CrossRefGoogle Scholar
  77. Mulligan, J., Mitchelmore, M., & Prescott, A. (2006). Integrating concepts and processes in early mathematics: The Australian pattern and structure mathematics awareness project (PASMAP). In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 209–216). Prague, Czech Republic: PME.Google Scholar
  78. Mundy, E., & Gilmore, C. (2009). Children’s mapping between symbolic and nonsymbolic representations of number. Journal of Experimental Child Psychology, 103(4), 490–502.CrossRefGoogle Scholar
  79. Murphy, D. (2014). Issues with PISA’s use of its data in the context of international education policy convergence. Policy Futures in Education, 12(7), 893–916.CrossRefGoogle Scholar
  80. Nan, Y., Knösche, T., & Luo, Y. J. (2006). Counting in everyday life: Discrimination and enumeration. Neuropsychologica, 44(7), 1103–1113.CrossRefGoogle Scholar
  81. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.Google Scholar
  82. Noël, M. P. (2005). Finger gnosia: A predictor of numerical abilities in children? Child Neuropsychology, 11(5), 413–430.CrossRefGoogle Scholar
  83. Obersteiner, A., Reiss, K., & Ufer, S. (2013). How training on exact or approximate mental representations of number can enhance first-grade students’ basic number processing and arithmetic skills. Learning and Instruction, 23, 125–135.CrossRefGoogle Scholar
  84. Okamoto, Y., & Case, R. (1996). Exploring the microstructure of children’s central conceptual structures in the domain of number. In R. Case, Y. Okamoto, G. Sharon, A. McKeough, C. Bleiker, B. Henderson, K. Stephenson, R. Siegler, & D. Keating (Eds.), The role of central conceptual structures in the development of children’s thought (pp. 27–58). Wiley on behalf of the Society for Research in Child Development.Google Scholar
  85. Passolunghi, M., Vercelloni, B., & Schadee, H. (2007). The precursors of mathematics learning: Working memory, phonological ability and numerical competence. Cognitive Development, 22(2), 165–184.CrossRefGoogle Scholar
  86. Penner-Wilger, M., Fast, L., LeFevre, J., Smith-Chant, B., Skwarchuk, S., Kamawar, D., et al. (2007). The foundations of numeracy: Subitizing, finger gnosia, and fine-motor ability. In D. McNamara & J. Trafton (Eds.), Proceedings of the 29th Annual Cognitive Science Society (pp. 1385–1390). Austin, TX: Cognitive Science Society.Google Scholar
  87. Richardson, K. (2004). Making sense. In D. Clements & J. Sarama (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics (pp. 321–324). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  88. Robinson, C., Menchetti, B., & Torgesen, J. (2002). Toward a two-factor theory of one type of mathematics disabilities. Learning Disabilities Research & Practice, 17(2), 81–89.CrossRefGoogle Scholar
  89. Sadler, F. (2009). Help! They still don’t understand counting. Teaching Exceptional Children Plus, 6(1), 1–12.Google Scholar
  90. Sayers, J., & Andrews, P. (2015). Foundational number sense: Summarising the development of an analytical framework. Paper presented to the Ninth Congress of European Research in Mathematics Education (CERME9), Prague.Google Scholar
  91. Siegler, R., & Booth, J. (2004). Development of numerical estimation in young children. Child Development, 75(2), 428–444.CrossRefGoogle Scholar
  92. Stock, P., Desoete, A., & Roeyers, H. (2010). Detecting children with arithmetic disabilities from kindergarten: Evidence from a 3-year longitudinal study on the role of preparatory arithmetic abilities. Journal of Learning Disabilities, 43(3), 250–268.CrossRefGoogle Scholar
  93. Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory. London: Sage.Google Scholar
  94. Thomas, N., Mulligan, J., & Goldin, G. (2002). Children’s representation and structural development of the counting sequence 1-100. The Journal of Mathematical Behavior, 21(1), 117–133.CrossRefGoogle Scholar
  95. Van de Rijt, B., Van Luit, J., & Pennings, A. (1999). The construction of the Utrecht early mathematical competence scale. Educational and Psychological Measurement, 59(2), 289–309.CrossRefGoogle Scholar
  96. Van Luit, J., & Schopman, E. (2000). Improving early numeracy of young children with special educational needs. Remedial and Special Education, 21(1), 27–40.CrossRefGoogle Scholar
  97. Van Nes, F., & De Lange, J. (2007). Mathematics education and neurosciences: Relating spatial structures to the development of spatial sense and number sense. The Montana Mathematics Enthusiast, 4(2), 210–229.Google Scholar
  98. Van Nes, F., & Doorman, M. (2011). Fostering young children’s spatial structuring ability. International Electronic Journal of Mathematics Education, 6(1), 27–39.Google Scholar
  99. Van Nes, F., & Van Eerde, D. (2010). Spatial structuring and the development of number sense: A case study of young children working with blocks. The Journal of Mathematical Behavior, 29(2), 145–159.Google Scholar
  100. Yang, D. C., & Li, M. N. (2008). An investigation of 3rd-grade Taiwanese students’ performance in number sense. Educational Studies, 34(5), 443–455.CrossRefGoogle Scholar
  101. Young-Loveridge, J. (2002). Early childhood numeracy: Building an understanding of part-whole relationships. Australian Journal of Early Childhood, 27(4), 36–42.Google Scholar
  102. Zur, O., & Gelman, R. (2004). Young children can add and subtract by predicting and checking. Early Childhood Research Quarterly, 19(1), 121–137.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Judy Sayers
    • 1
    Email author
  • Paul Andrews
    • 1
  • Lisa Björklund Boistrup
    • 1
  1. 1.Stockholm UniversityStockholmSweden

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