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Educational Studies in Mathematics

, Volume 93, Issue 2, pp 175–193 | Cite as

Oral counting sequences: a theoretical discussion and analysis through the lens of representational redescription

  • Chronoula Voutsina
Original Paper

Abstract

Empirical research has documented how children’s early counting develops into an increasingly abstract process, and initial counting procedures are reified as children develop and use more sophisticated counting. In this development, the learning of different oral counting sequences that allow children to count in steps bigger than one is seen as an essential skill that supports children’s mental calculation strategies. This paper proposes that the reification or refinement of the counting process that results to increased-in-sophistication use of counting is underlaid by the process of knowledge explicitation that the model of representational redescription postulates. The paper uses a case study to provide insight into the pathway that a 6-year-old child followed from learning how to verbally count in twos and tens to being able to use this knowledge for calculation purposes. The proposal is that knowledge of oral sequences is redescribed in more explicit and accessible formats before children are able to connect their knowledge of the verbal counting with the goal of using the sequence for calculation. The discussion presented here queries the notion of spontaneity as an inherent element of the theory and discusses the role that social interaction may play in supporting knowledge redescription. If it is the case that children’s knowledge of oral counting sequences is redescribed into increasingly explicit formats before it can be applied for calculation, then children need to be provided early in their education with structured activities that trigger knowledge redescription and support the necessary connections between counting, number structure and calculation.

Keywords

Counting sequences Oral counting Representational redescription Skip counting Addition Arithmetic 

Notes

Acknowledgments

I am thankful to my daughter for sharing with me her knowledge and thoughts about numbers and allowing me to write about our math discussions in this paper. I would like to express my thanks to my colleague Keith Jones for his feedback on an earlier draft of the paper, to the editors and the four anonymous reviewers for their very constructive comments and suggestions on how this work can be taken forward.

References

  1. Anghileri, J. (2008). Uses of counting in multiplication and division. In I. Thompson (Ed.), Teaching and learning early number (pp. 110–121). Maidenhead: Open University Press.Google Scholar
  2. Aubrey, C. (2003). “When we were very young”: The foundations for mathematics. In I. Thompson (Ed.), Enhancing primary mathematics teaching (pp. 43–43). Maidenhead: Open University Press.Google Scholar
  3. Baroody, A. (1995). The role of the number-after rule in the invention of computational short cuts. Cognition and Instruction, 13, 189–219.CrossRefGoogle Scholar
  4. Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualisation of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38(2), 115–131.Google Scholar
  5. Baroody, A. J., & Tiilikainen, S. H. (2003). Two perspectives on addition development. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. 75–125). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  6. Beishuizen, M., & Anghileri, J. (1998). Which mental strategies in the early number curriculum? A comparison of British ideas and Dutch views. British Educational Research Journal, 24(3), 519–538.CrossRefGoogle Scholar
  7. Bialystok, E., & Codd, J. (1997). Cardinal limits: Evidence from language awareness and bilingualism for developing concepts of number. Cognitive Development, 12, 85–106.Google Scholar
  8. Bruce, B., & Threlfall, J. (2004). One, two, three and counting. Educational Studies in Mathematics, 55, 3–26.CrossRefGoogle Scholar
  9. Bruner, J. (2009). Culture, mind and education. In K. Illeris (Ed.), Contemporary theories of learning. Learning theorists… in their own words (pp. 159–168). London: Routledge.Google Scholar
  10. Bryant, P., & Nunes, T. (2004). Children’s understanding of mathematics. In U. Goswami (Ed.), Childhood cognitive development (pp. 412–439). Oxford: Blackwell Publishing.Google Scholar
  11. Clements, D. H., & Sarama, J. (2009). Learning and teaching early math. Oxon: Routledge.Google Scholar
  12. Cowan, R. (2003). Does it all add up? Changes in children’s knowledge of addition combinations, strategies and principles. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. 35–74). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  13. DfE (Department for Education). (2013). Mathematics programmes of study: Key stages 1 and 2. National curriculum in England (Report No. DFE-00180-2013). London: Department for Education. Retrieved from https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf. Accessed 18 Dec 2015.
  14. DfES (Department for Education and Skills). (2006). Primary framework for literacy and mathematics. Norwich: DfES.Google Scholar
  15. Dixon, J.A., & Bangert, A.S. (2002). The prehistory of discovery: Precursors of representational change in solving gear system problems. Developmental Psychology, 38, 6, 918–933.Google Scholar
  16. Dixon, J. A., & Bangert, A. S. (2005). From regularities to concepts: The development of children’s understanding of mathematical relation. Cognitive Development, 20, 65–86.Google Scholar
  17. Dowker, A., Bala, S., & Lloyd, D. (2008). Linguistic influences on mathematical development: How important is the transparency of the counting system? Philosophical Psychology, 21(4), 523–538.Google Scholar
  18. Ellemor-Collins, D., & Wright, R. (2007). Assessing pupil knowledge of the sequential structure of numbers. Educational and Child Psychology, 24(2), 54–63.Google Scholar
  19. Ellemor-Collins, D., & Wright, R. (2011). Developing conceptual place value: Instructional design for intensive intervention. Australian Journal of Learning Difficulties, 16(1), 41–63.Google Scholar
  20. Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer.CrossRefGoogle Scholar
  21. Fuson, K. C. (1992a). Relationships between counting and cardinality from age 2 to 8. In J. Bideau, C. Meljac, & J.-P. Fisher (Eds.), Pathways to number: Children’s developing numerical abilities (pp. 127–149). Hillsdale, New Jersey: Lawrence Erlbaum Associates.Google Scholar
  22. Fuson, K. C. (1992b). Research on whole number addition and subtraction. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243–275). New York: Macmillan.Google Scholar
  23. Gelman, R., & Greeno, J. G. (1989). On the nature of competence: Principles for understanding in a domain. In L. B. Resnick (Ed.), Knowing, learning and instruction: Essays in honor of Robert Glaser (pp. 125–186). Hillsdale, New Jersey: Erlbaum.Google Scholar
  24. Gray, E. (2008). Compressing the counting process: Strengths from the flexible interpretation of symbols. In I. Thompson (Ed.), Teaching and learning early number (pp. 82–93). Maidenhead: Open University Press.Google Scholar
  25. Gray, E., & Tall, D. (1992). Success and failure in mathematics: The flexible meaning of symbols as process and concept. Mathematics Teaching, 142, 6–10.Google Scholar
  26. Gray, E., & Tall, D. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal of Research in Mathematics Education, 26(2), 115–141.Google Scholar
  27. Greeno, J. G., Riley, M. S., & Gelman, R. (1984). Conceptual competence and children’s counting. Cognitive Psychology, 16, 94–143.CrossRefGoogle Scholar
  28. Griffiths, R. (2008). The family counts. In I. Thompson (Ed.), Teaching and learning early number (pp. 47–58). Maidenhead: Open University Press.Google Scholar
  29. Karmiloff-Smith, A. (1979). Micro- and macro-developmental changes in language acquisition and other representational systems. Cognitive Science, 3, 91–118.CrossRefGoogle Scholar
  30. Karmiloff-Smith, A. (1984). Children’s problem solving. In M. E. Lamb, A. L. Brown, & B. Rogoff (Eds.), Advances in developmental psychology (pp. 39–90). Hillsdale, New Jersey: Lawrence Erlbaum Associates.Google Scholar
  31. Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. London: The MIT Press.Google Scholar
  32. Karmiloff-Smith, A. (1993). What’s special about the development of the human mind/brain? Mind & Language, 8(4), 569–581.CrossRefGoogle Scholar
  33. Karmiloff-Smith, A. (1994). Précis of beyond modularity: A developmental perspective on cognitive science. Behavioral and Brain Sciences, 17, 696–745.Google Scholar
  34. LeFevre, J., Clarke, T., & Stringer, A. (2002). Influences of language and parental involvement on the development of counting skills: Comparisons of French- and English-speaking Canadian children. Early Child Development and Care, 172, 283–300.Google Scholar
  35. Montague-Smith, A., & Price, A. (2012). Mathematics in early years education. Abington, Oxon: Routledge.Google Scholar
  36. Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49.CrossRefGoogle Scholar
  37. Munn, P. (2008). Children’s beliefs about counting. In I. Thompson (Ed.), Teaching and learning early number (pp. 19–33). Berkshire: Open University Press.Google Scholar
  38. Resnick, L. (1992). From proroquantities to operators: Building mathematical competence on a foundation of everyday knowledge. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 373–425). Hillsdale, New Jersey: Lawrence Erlbaum Associates.Google Scholar
  39. Rittle-Johnson, B., & Koedinger, K. R. (2009). Iterating between lessons on concepts and procedures can improve mathematics knowledge. British Journal of Educational Psychology, 79, 483–500.CrossRefGoogle Scholar
  40. 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.Google Scholar
  41. Siegler, R. S. (2006). Microgenetic analyses of learning. In D. Kuhn & R. S. Siegler (Eds.), Handbook of child psychology (Vol. 2, pp. 464–510). New Jersey: John Wiley & Sons.Google Scholar
  42. Siegler, R. S., & Robinson, M. (1982). The development of numerical understandings. In H. W. Reese & L. P. Lipsitt (Eds.), Advances in child development and behavior (Vol. 16, pp. 241–312). London: Academic.Google Scholar
  43. Slavit, D. (1999). The role of operation sense in transitions from arithmetic to algebraic thought. Educational Studies in Mathematics, 37, 251–274.CrossRefGoogle Scholar
  44. Sophian, C. (1997). Beyond competence: The significance of performance for conceptual development. Cognitive Development, 12, 281–303.Google Scholar
  45. Sophian, C. (2009). Numerical knowledge in early childhood. In R. E. Tremblay, R. G. Barr, R. D. V. Peters, & M. Boivin (Eds.), Encyclopedia of early childhood development (pp. 1–7). Montreal, Quebec: Centre of Excellence for Early Childhood Development. retrieved from http://www.child-encyclodedia.com/documents/SophianANGxp.pdf. Accessed 23 Nov 2015.
  46. Sophian, C., Wood, A., & Vong, K. I. (1995). Making numbers count: The early development of numerical inferences. Developmental Psychology, 31, 263–273.CrossRefGoogle Scholar
  47. Star, R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38(2), 132–135.Google Scholar
  48. Star, R., & Stylianides, G. J. (2013). Procedural and conceptual knowledge: Exploring the gap between knowledge type and knowledge quality. Canadian Journal of Science, Mathematics, and Technology Education, 13(2), 169–181.Google Scholar
  49. Steffe, L. P., & Olive, J. (1996). Symbolizing as a constructive activity in a computer microworld. Journal of Educational Computing Research, 14(2), 113–138.Google Scholar
  50. Tall, D., Gray, E., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., Yusof, Y. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics, and Technology Education, 1(1), 81–104.Google Scholar
  51. Thompson, I. (2008). From counting to deriving number facts. In I. Thompson (Ed.), Teaching and learning early number (pp. 97–109). Maidenhead: Open University Press.Google Scholar
  52. Thompson, I., & Bramald, R. (2002). An investigation of the relationship between young children’s understanding of the concept of place value and their competence at mental addition. Newcastle upon Tyne, UK: University of Newcastle upon Tyne. Report of the Nuffield Foundation.Google Scholar
  53. Threlfall, J. (2008). Development in oral counting, enumeration and counting for cardinality. In I. Thompson (Ed.), Teaching and learning early number (pp. 61–71). Maidenhead: Open University Press.Google Scholar
  54. Threlfall, J., & Bruce, B. (2007). Just counting: Young children’s oral counting and enumeration. European Early Childhood Education Research Journal, 13(2), 63–77.Google Scholar
  55. Voutsina, C. (2012). A micro-developmental approach to studying young children’s problem solving behavior in addition. The Journal of Mathematical Behavior, 31, 366–381.CrossRefGoogle Scholar
  56. Vygotsky, L. S. (1978). Mind in society. Cambridge, MA: Harvard University Press.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Southampton Education SchoolUniversity of SouthamptonSouthamptonUK

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