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Educational Studies in Mathematics

, Volume 79, Issue 2, pp 193–214 | Cite as

Procedural and conceptual changes in young children’s problem solving

  • Chronoula VoutsinaEmail author
Article

Abstract

This study analysed the different types of arithmetic knowledge that young children utilise when solving a multiple-step addition task. The focus of the research was on the procedural and conceptual changes that occur as children develop their overall problem solving approach. Combining qualitative case study with a micro-genetic approach, clinical interviews were conducted with ten 5–6-year-old children. The aim was to document how children combine knowledge of addition facts, calculation procedures and arithmetic concepts when solving a multiple-step task and how children’s application of different types of knowledge and overall solving approach changes and develops when children engage with solving the task in a series of problem solving sessions. The study documents children’s pathways towards developing a more effective and systematic approach to multiple-step tasks through different phases of their problem solving behaviour. The analysis of changes in children’s overt behaviour reveals a dynamic interplay between children’s developing representation of the task, their improved procedures and gradually their more explicit grasp of the conceptual aspects of their strategy. The findings provide new evidence that supports aspects of the “iterative model” hypothesis of the interaction between procedural and conceptual knowledge and highlight the need for educational approaches and tasks that encourage and trigger the interplay of different types of knowledge in young children’s arithmetic problem solving.

Keywords

Arithmetic Addition Problem solving Procedural knowledge Conceptual knowledge Iterative model 

Notes

Acknowledgements

This study was supported by a grant from the Economic and Social Research Council (PTA-026-27-0697). I am very grateful to the children who participated in this project. I would also like to thank my colleagues Keith Jones and Lianghuo Fan as well as the three anonymous reviewers for their very constructive comments on this article.

References

  1. Alibali, M. W. (1999). How children change their minds: Strategy change can be gradual or abrupt. Developmental Psychology, 35(1), 127–145.CrossRefGoogle Scholar
  2. Alibali, M. W., Phillips, K. O. M., & Fischer, A. D. (2009). Learning new problem solving strategies leads to changes in problem representation. Cognitive Development, 24, 89–101.CrossRefGoogle Scholar
  3. Baroody, A. J. (1985). Mastery of basic number combinations: Internalisation of relationships or facts? Journal for Research in Mathematics Education, 16, 83–98.CrossRefGoogle Scholar
  4. Baroody, A. J. (1987). Children’s mathematical thinking. New York: Teachers College Press.Google Scholar
  5. Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The interaction of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. 1–33). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  6. Baroody, A. J., & Dowker, A. (2003). The development of arithmetic concepts and skills: Constructing adaptive expertise. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  7. 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
  8. Baroody, A. J., & Gannon, K. E. (1984). The development of the commutativity principle and economical addition strategies. Cognition and Instruction, 1, 321–339.CrossRefGoogle Scholar
  9. Baroody, A. J., & Ginsburg, H. P. (1986). The relationship between initial and mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of arithmetic (pp. 75–112). Hillsdale: Erlbaum.Google Scholar
  10. 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
  11. Bjorklund, D. F., & Rosenblum, K. E. (2002). Context effects in children’s selection and use of simple arithmetic strategies. Journal of Cognition and Development, 3(2), 225–242.CrossRefGoogle Scholar
  12. Briars, D., & Siegler, R. S. (1984). A featural analysis of preschoolers’ counting knowledge. Developmental Psychology, 20, 607–618.CrossRefGoogle Scholar
  13. Bryant, P., & Nunes, T. (2004). Children’s understanding of mathematics. In U. Goswami (Ed.), Childhood cognitive development (pp. 412–439). Oxford: Blackwell.Google Scholar
  14. Byrnes, J. P. (1992). The conceptual basis of procedural learning. Cognitive Development, 7, 235–257.CrossRefGoogle Scholar
  15. Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptual knowledge in mathematical procedural skill. Developmental Psychology, 27(5), 777–786.CrossRefGoogle Scholar
  16. Canobi, K. H., Reeve, R. A., & Pattison, P. (1998). The role of conceptual understanding in children’s addition problem solving. Developmental Psychology, 34(5), 882–891.CrossRefGoogle Scholar
  17. Carpenter, T. P. (1986). Conceptual knowledge as a foundation for procedural knowledge: Implications from research on the initial learning of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 113–132). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  18. Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15(3), 179–202.CrossRefGoogle Scholar
  19. Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 547–589). N. Jersey: Lawrence Erlbaum Associates.Google Scholar
  20. 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
  21. De Corte, E., & Verschaffel, L. (2006). Mathematical thinking and learning. In K. A. Renninger & I. E. Sigel (Eds.), Handbook of child psychology (pp. 103–152). New Jersey: Wiley.Google Scholar
  22. Delazer, M. (2003). Neurospychological findings on conceptual knowledge of arithmetic. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills. Constructing adaptive expertise (pp. 385–407). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  23. DfES (Department for Education and Skills). (2006). Primary framework for literacy and mathematics. Norwich, U.K.: DfESGoogle Scholar
  24. Dowker, A. (1998). Individual differences in normal arithmetical development. In C. Donlan (Ed.), The development of mathematical skills (pp. 275–302). Hove: Psychology Press.Google Scholar
  25. Dowker, A. (2003). Young children’s estimates for addition: The zone of partial knowledge and understanding. In A. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills (pp. 243–266). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  26. Fuson, K. C. (1982). An analysis of the counting-on procedure in addition. In T. P. Carpenter, J. Moser, & T. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 67–81). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  27. Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer.CrossRefGoogle Scholar
  28. Geary, D. C. (2003). Learning disabilities in arithmetic: Problem-solving differences and cognitive deficits. In H. L. Swanson, K. R. Harris, & S. Graham (Eds.), Handbook of learning disabilities (pp. 199–212). New York: The Guilford Press.Google Scholar
  29. 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.CrossRefGoogle Scholar
  30. 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.CrossRefGoogle Scholar
  31. Gilmore, C. K., & Bryant, P. (2006). Individual differences in children’s understanding of inversion and arithmetic skills. The British Journal of Educational Psychology, 76, 309–331.CrossRefGoogle Scholar
  32. Ginsburg, H. (1981). The clinical interview in psychological research on mathematical thinking: Aims, rationales, techniques. For the Learning of Mathematics, 1(3), 4–11.Google Scholar
  33. Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). N. Jersey: Lawrence Erlbaum Associates.Google Scholar
  34. Gray, E. (2009). Compressing the counting process: Strength from the flexible interpretation of symbols. In I. Thompson (Ed.), Teaching and learning early number (pp. 82–93). Maidenhead: Open University Press.Google Scholar
  35. Gray, E., Pinto, M., Pitta, D., & Tall, D. (1999). Knowledge construction and diverging thinking in elementary & advanced mathematics. Educational Studies in Mathematics, 38, 111–133.CrossRefGoogle Scholar
  36. Gray, E., & Tall, D. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 25, 116–140.CrossRefGoogle Scholar
  37. Groth, R. E., & Bergner, J. A. (2006). Preservice elementary teachers’ conceptual and procedural knowledge of mean, median, mode. Mathematical Thinking and Learning, 8(1), 37–63.CrossRefGoogle Scholar
  38. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  39. 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: Lawrence Erlbaum Associates.Google Scholar
  40. Karmiloff-Smith, A. (1992). Beyond modularity: A developmental perspective on cognitive science. A Bradford Book. London: The MIT Press.Google Scholar
  41. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learning mathematics. Washington, D.C.: National Academy Press.Google Scholar
  42. Kuhn, D. (1995). Microgenetic study of change: What has it told us? Psychological Science, 6(3), 133–139.CrossRefGoogle Scholar
  43. Lee, K., & Karmiloff-Smith, A. (2002). Macro- and microdevelopmental research: Assumptions, research strategies, constraints, and utilities. In N. Granott & J. Parziale (Eds.), Microdevelopment. Transition processes in development and learning (pp. 243–265). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  44. Luwel, K., Siegler, R. S., & Verschaffel, L. (2008). A microgenetic study of insightful problem solving. Journal of Experimental Child Psychology, 99, 210–232.CrossRefGoogle Scholar
  45. Miller, P. H., & Coyle, T. R. (1999). Developmental change: Lessons from microgenesis. In E. K. Scholnick, K. Nelson, S. A. Gelman, & P. H. Miller (Eds.), Conceptual Development. Piaget’s Legacy (pp. 209–239). N. Jersey: Lawrence Erlbaum Associates.Google Scholar
  46. Nührenbörger, M., & Steinbring, H. (2009). Forms of mathematical interaction in different social settings: Examples from students’, teachers’ and teacher–students’ communication about mathematics. Journal of Mathematics Teacher Education, 12, 111–132.CrossRefGoogle Scholar
  47. Nunes, T., Bryant, P., & Watson, A. (2009). Key understandings in mathematics learning. London: Nuffield Foundation.Google Scholar
  48. Putnam, R. T., deBettencourt, L. U., & Leinhardt, G. (1990). Understanding of derived-fact strategies in addition and subtraction. Cognition and Instruction, 7(3), 245–285.CrossRefGoogle Scholar
  49. Riley, M., Greeno, J. G., & Heller, J. I. (1983). Development of children’s problem solving ability in arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 153–196). New York: Academic Press.Google Scholar
  50. Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge in mathematics: Does one lead to the other? Journal of Educational Psychology, 91, 175–189.CrossRefGoogle Scholar
  51. Rittle-Johnson, B., Kalchman, M., Czarnocha, B., & Baker, W. (2002). An integrated approach to the procedural/conceptual debate. Discussion group PME-NA XXIV, Georgia Centre for Continuing Education, Athens GA. (http://www.pmena.org/2002/).
  52. Rittle-Johnson, B., & Siegler, R. S. (1998). The relationship between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75–110). Hove: Psychology Press.Google Scholar
  53. 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.CrossRefGoogle Scholar
  54. Robinson, K. M., & Dubé, A. K. (2008). A microgenetic study of simple division. Canadian Journal of Experimental Psychology, 62(3), 156–162.CrossRefGoogle Scholar
  55. Robinson, K. M., & Dubé, A. K. (2009). Children’s understanding of addition and subtraction concepts. Journal of Experimental Child Psychology, 103, 532–545.CrossRefGoogle Scholar
  56. Saxe, G. B., Gearhart, M., & Nasir, N. I. S. (2001). Enhancing students’ understanding of mathematics: A study of three contrasting approaches to professional support. Journal of Mathematics Teacher Education, 4, 55–79.CrossRefGoogle Scholar
  57. Siegler, R. S. (1987). The perils of averaging data over strategies: An example from children’s addition. Journal of Experimental Psychology. General, 116, 250–264.CrossRefGoogle Scholar
  58. Siegler, R. S. (1991). In young children’s counting, procedures precede principles. Educational Psychology Review, 3(2), 127–135.CrossRefGoogle Scholar
  59. Siegler, R. S. (1995). Children’s thinking: How does change occur? In F. E. Weinert & W. Schneider (Eds.), Memory performance and competencies: Issues in growth and development (pp. 405–430). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  60. Siegler, R. S. (1996). Emerging Minds. The process of change in children’s thinking. Oxford: Oxford University Press.Google Scholar
  61. Siegler, R. S. (2000). The rebirth of children’s learning. Child Development, 71(1), 26–35.CrossRefGoogle Scholar
  62. Siegler, R. S. (2001). Cognition, instruction, and the quest for meaning. In S. M. Carver & D. Klahr (Eds.), Cognition and instruction (pp. 195–203). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  63. Siegler, R. S., & Crowley, K. (1991). The microgenetic method: A direct means for studying cognitive development. The American Psychologist, 46(6), 606–620.CrossRefGoogle Scholar
  64. Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  65. Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), Origins of cognitive skills (pp. 229–293). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  66. Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: A microgenetic analysis. Journal of Experimental Psychology. General, 127, 377–397.CrossRefGoogle Scholar
  67. Sophian, C. (1997). Beyond competence: The significance of performance for conceptual development. Cognitive Development, 12, 281–303.CrossRefGoogle Scholar
  68. Star, R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38(2), 132–135.Google Scholar
  69. Thompson, I. (1995). The role of counting in the idiosyncratic mental calculation algorithms of young children. European Early Childhood Education Research Journal, 3(1), 5–16.CrossRefGoogle Scholar
  70. 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
  71. Torbeyns, J., Verschaffel, L., & Ghesquiere, P. (2004). Strategic aspects of simple addition and subtraction: The influence of mathematical ability. Learning and Instruction, 14, 177–195.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.School of EducationUniversity of SouthamptonSouthamptonUK

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