Mathematics Education Research Journal

, Volume 21, Issue 2, pp 50–75

Structuring numbers 1 to 20: Developing facile addition and subtraction

  • David Ellemor-Collins
  • Robert (Bob) Wright
Articles

Abstract

The Numeracy Intervention Research Project (NIRP) aims to develop assessment and instructional tools for use with low-attaining 3rd- and 4th-graders. The NIRP approach to instruction in addition and subtraction in the range 1 to 20 is described. The approach is based on a notion of structuring numbers, which draws on the work of Freudenthal and the Realistic Mathematics Education program. NIRP involved 25 teachers and 300 students, 200 of whom participated in an intervention program of approximately thirty 25-minute lessons over 10 weeks. Data is drawn from case studies of two intervention students who made significant progress toward facile addition and subtraction. Pre- and post-assessment interviews and five lesson episodes are described, and data drawn from the activity of the students during the episodes are analysed. The discussion develops a detailed account of the progression of students’ learning of structuring numbers, and how this can result in significant level-raising of students’ arithmetical knowledge as it becomes more formalised and less context-dependent.

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References

  1. Australian Government (2008).National Numeracy Review Report. Canberra: Commonwealth of Australia.Google Scholar
  2. Bobis, J. (1996). Visualisation and the development of number sense with kindergarten children. In J. T. Mulligan & M. C. Mitchelmore (Eds.),Children’s number learning (pp. 17–33). Adelaide: Australian Association of Mathematics Teachers/Mathematics Education Research Group of Australasia.Google Scholar
  3. Bryant, D. P., Bryant, B. R., & Hammill, D. D. (2000). Characteristic behaviors of students with LD who have teacher-identified math weaknesses.Journal of Learning Disabilities, 33, 168–177, 199.CrossRefGoogle Scholar
  4. 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, 19–22.CrossRefGoogle Scholar
  5. Cobb, P. (1991). Reconstructing elementary school mathematics.Focus on Learning Problems in Mathematics, 13(3), 3–33.Google Scholar
  6. Cobb, P. (2003). Investigating students’ reasoning about linear measurement as a paradigm case of design research. In M. Stephan, J. Bowers, P. Cobb, & K. Gravemeijer (Eds.),Supporting students’ development of measuring conceptions: Analyzing students’ learning in social context, Journal for Research in Mathematics Education, Monograph No. 12, 1–16. Reston, VA: NCTM.Google Scholar
  7. Denvir, B., & Brown, M. (1986). Understanding of number concepts in low attaining 7–9 year olds: Part 1. Development of descriptive framework and diagnostic instrument.Educational Studies in Mathematics, 17, 15–36.CrossRefGoogle Scholar
  8. Ellemor-Collins, D., & Wright, R. J. (2008a). From counting by ones to facile higher decade addition: The case of Robyn. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. Sepúlveda (Eds.),Proceedings of the Joint Meeting of PME32 and PMENA XXX (Vol. 2, pp. 439–446). México: Cinvestav-UMSNH.Google Scholar
  9. Ellemor-Collins, D., & Wright, R. J. (2008b). Intervention instruction in Structuring Numbers 1 to 20: The case of Nate. In M. Goos, R. Brown & K. Makar (Eds.),Navigating currents and charting directions (Proceedings of the 31st annual conference of the Mathematics Education Research Group of Australasia, Brisbane, Vol. 1, pp. 179–186). Adelaide: MERGA.Google Scholar
  10. Ellemor-Collins, D., Wright, R. J., & Lewis, G. (2007). Documenting the knowledge of low-attaining 3rd- and 4th- graders: Robyn’s and Bel’s sequential structure and multidigit addition and subtraction. In J. Watson & K. Beswick (Eds.),Mathematics: Essential research, essential practice (Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia, Hobart,Vol. 1, pp. 265–274). Adelaide: MERGA.Google Scholar
  11. Freudenthal, H. (1983).Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Reidel.Google Scholar
  12. Freudenthal, H. (1991).Revisiting mathematics education. Dordrecht, The Netherlands: Kluwer.Google Scholar
  13. Fuson, K. C. (1988).Children’s counting and concepts of number. New York: Springer.Google Scholar
  14. Fuson, K. C. (1992). Research on whole number addition and subtraction. In D. A. Grouws (Ed.),Handbook of research on mathematics teaching and learning (pp. 243–275). New York: Macmillan.Google Scholar
  15. Gervasoni, A., Hadden, T., & Turkenburg, K. (2007). Exploring the number knowledge of children to inform the development of a professional learning plan for teachers in the Ballarat Diocese as a means of building community capacity. In J. Watson & K. Beswick (Eds.),Mathematics: Essential Research, Essential Practice (Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia, Hobart, Vol. 1, pp. 317–326). Adeaide: MERGA.Google Scholar
  16. Gravemeijer, K. P. E. (1991). An instruction-theoretical reflection on the use of manipulatives. In L. Streefland (Ed.),Realistic mathematics education in primary school (pp. 57–76). Utrecht, The Netherlands: Freudenthal Institute.Google Scholar
  17. Gravemeijer, K. P. E. (1994). Instructional design as a learning process. In K. P. E. Gravemeijer (Ed.),Developing Realistic Mathematics Education (pp. 17–54). Utrecht, The Netherlands: Freudenthal Institute.Google Scholar
  18. Gravemeijer, K. P. E., Cobb, P., Bowers, J. S., & Whitenack, J. W. (2000). Symbolizing, modeling and instructional design. In P. Cobb, E. Yackel, & K. J. McClain (Eds.),Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 225–273). Hillsdale, NJ: Lawrence ErlbaumGoogle Scholar
  19. Gray, E. (1991). An analysis of diverging approaches to simple arithmetic: Preference and its consequences.Educational Studies in Mathematics, 22, 551–574.CrossRefGoogle Scholar
  20. Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain.Journal for Research in Mathematics Education, 22, 170–218.CrossRefGoogle Scholar
  21. Heirdsfield, A. (2001). Integration, compensation and memory in mental addition and subtraction. In M. Van den Heuvel-Panhuizen (Ed.),Proceedings of the 25th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 129–136). Utrecht, The Netherlands: Program Committee.Google Scholar
  22. Hunting, R. P. (2003). Part-whole number knowledge in preschool children.Journal of Mathematical Behavior, 22, 217–235.CrossRefGoogle Scholar
  23. Mapping the territory: Primary students with learning difficulties. (2000). Canberra: Department of Education Training and Youth Affairs.Google Scholar
  24. McIntosh, A. J., Reys, B. J., & Reys, R. E. (1992). A proposed framework for examining basic number sense.For the Learning of Mathematics, 12, 2–8.Google Scholar
  25. Moon, B. (1986).The ‘new maths’ curriculum controversy: An international story. London: Falmer.Google Scholar
  26. Moser Opitz, E. (2001). Mathematical knowledge and progress in the mathematical learning of children with special needs in their first year of school. In M. Van den Heuvel-Panhuizen (Ed.),Proceedings of the 25th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 207–210). Utrecht, The Netherlands: Program Committee.Google Scholar
  27. 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: Program Committee.Google Scholar
  28. Olive, J. (2001). Children’s number sequences: An explanation of Steffe’s constructs and an extrapolation to rational numbers of arithmetic.The Mathematics Educator, 11(1), 4–9.Google Scholar
  29. Pearn, C. (1998). Is there a need for a mathematics intervention program in Grades 3 and 4? In C. Kanes, M. Goos, & E. Warren (Eds.),Teaching mathematics in new times (Proceedings of the 21st annual conference of the Mathematics Education Research Group of Australasia, Gold Coast, Vol. 2, pp. 444–451). Adelaide: MERGA.Google Scholar
  30. Pirie, S. E. B., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it?Educational Studies in Mathematics, 26, 165–190.CrossRefGoogle Scholar
  31. Principles and standards for school mathematics. (2000). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  32. Resnick, L. B. (1983). A developmental theory of number understanding. In H. P. Ginsburg (Ed.),The development of mathematical thinking (pp. 109–151). New York: Academic Press.Google Scholar
  33. Riley, M. S., Greeno, J. G., & Heller, J. I. (1983). Development of children’s problemsolving ability in arithmetic. In H. P. Ginsburg (Ed.),The development of mathematical thinking (pp. 153–196). New York: Academic Press.Google Scholar
  34. Rivera, D. P. (1998). Mathematics education and students with learning disabilities. In D. P. Rivera (Ed.),Mathematics education for students with learning disabilities (pp. 1–31). Austin, TX: Pro-Ed.Google Scholar
  35. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.Educational Studies in Mathematics, 22, 1–36.CrossRefGoogle Scholar
  36. Steffe, L. P., & Cobb, P. (1988).Construction of arithmetic meanings and strategies. New York: Springer.Google Scholar
  37. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. Kelly & R. Lesh (Eds.),Handbook of research design in mathematics and science education (pp. 267–306). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  38. Sugarman, I. (1997). Teachingfor strategies. In I. Thompson (Ed.),Teaching and learning early number (pp. 142–154). Milton Keynes, UK: Open University Press.Google Scholar
  39. The national numeracy project: An HMI evaluation. (1998). London, UK: Office for Standards in Education.Google Scholar
  40. 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
  41. Thornton, C. A. (1978). Emphasising thinking strategies in basic fact instruction.Journal for Research in Mathematics Education, 9, 214–227.CrossRefGoogle Scholar
  42. Threlfall, J. (2002). Flexible mental calculation.Educational Studies in Mathematics, 50, 29–47.CrossRefGoogle Scholar
  43. Treffers, A. (1987).Three dimensions: A model of goal and theory description in mathematics instruction-The Wiskobas Project. Dordrecht: Reidel.Google Scholar
  44. Treffers, A. (1991). Didactical background of a mathematics program for primary education. In L. Streefland (Ed.),Realistic mathematics education in primary school (pp. 21–56). Utrecht, The Netherlands: Freudenthal Institute.Google Scholar
  45. Treffers, A. (2001). Grade 1 (and 2): Calculation up to 20. In M. van den Heuvel-Panhuizen (Ed.),Children learn mathematics (pp. 43–60). Utrecht, The Netherlands: Freudenthal Institute.Google Scholar
  46. Treffers, A., & Beishuizen, M. (1999). Realistic mathematics education in the Netherlands. In I. Thompson (Ed.),Issues in teaching numeracy in primary schools (pp. 27–38). Milton Keynes, UK: Open University Press.Google Scholar
  47. Verschaffel, L., Greer, B., & Torbeyns, J. (2006). Numerical thinking. In A. Gutiérrez & P. Boero (Eds.),Handbook of research on the psychology of mathematics education. Rotterdam: Sense Publishers.Google Scholar
  48. van de Walle, J. A. (2004).Elementary and middle school mathematics: Teaching developmentally (5th ed.). Boston: Pearson.Google Scholar
  49. van Hiele, P.M. (1973).Begrip en inzicht [Understanding and insight]. Purmerend, The Netherlands: Muusses.Google Scholar
  50. Wright, R. J. (1994). A study of the numerical development of 5-year-olds and 6-year-olds.Educational Studies in Mathematics, 26, 25–44.CrossRefGoogle Scholar
  51. Wright, R. J., & Ellemor-Collins, D. (2008).Structuring numbers to 20: An important topic in early number learning. Manuscript submitted for publication.Google Scholar
  52. Wright, R. J., Ellemor-Collins, D., & Lewis, G. (2007). Developing pedagogical tools for intervention: Approach, methodology, and an experimental framework. In J. Watson & K. Beswick (Eds.),Mathematics: Essential Research, Essential Practice (Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia, Hobart, Vol. 2, pp. 843–852). Adelaide: MERGA.Google Scholar
  53. Wright, R. J., Martland, J., Stafford, A. K., & Stanger, G. (2006).Teaching number: Advancing children’s skills and strategies (2nd ed.). London: Chapman.Google Scholar
  54. Wright, R. J., Stanger, G., Stafford, A. K., & Martland, J. (2006).Teaching number in the classroom with 4-8 years-olds. London: Chapman.Google Scholar
  55. 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

Copyright information

© Mathematics Education Research Group of Australasia Inc. 2009

Authors and Affiliations

  • David Ellemor-Collins
    • 1
  • Robert (Bob) Wright
    • 1
  1. 1.Southern Cross UniversityLismoreAustralia

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