Abstract
An exploratory study of77 high ability Grade 5 and 6 children investigated links between their understanding of the numeration system and their representations of the counting sequence 1–100. Analysis of children’s explanations, and pictorial and notational recordings of the numbers 1–100 revealed three dimensions of external representation—pictorial, ikonic, or notational characteristics—thus providing evidence of creative structural development of the number system, and evidence for the static or dynamic nature of the internal representation. Our observations indicated that children used a wide variety of internal images of which about 30% were dynamic internal representations. Children with a high level of understanding of the numeration system showed evidence of both structure and dynamic imagery in their representations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bednarz, N., & Janvier, B. (1988). A constructivist approach to numeration in primary school: results of a three year intervention with the same group of children.Educational Studies in Mathematics, 19, 299–331.
Bishop, A. J. (1989). Review of research on visualisation in mathematics education.Focus on Learning Problems in Mathematics, 11(1), 7–15.
Bobis, J. (1993). Visualisation and the development of mental computation. In B. Atweh, C. Kanes, M. Carss & G. Booker (Eds.),Contexts in mathematics education (pp. 117–122). Brisbane: Queensland University of Technology.
Boulton-Lewis, G. M. (1994). An analysis of the relation between sequence counting and knowledge of place value in the early years of school.Mathematics Education Research Journal, 5(2), 94–106.
Brown, M. (1981). Place value and decimals. In Hart, K. M. (Ed.),Children’s understanding of mathematics: 11-16 (pp. 48–65). London: John Murray.
Brown, D., & Presmeg, N. (1993). Types of imagery used by elementary and secondary school students in mathematical reasoning. In Hirabayashi, N. Nohda, K. Shigematsu & F.-L. Lin (Eds.),Proceedings of the 17th International Conference of the Group for the Psychology of Mathematics Education (Vol. II, pp. 137–144). Tsukuba, Japan: International Group for the Psychology of Mathematics Education.
Brown, D. L., & Wheatley, G. H. (1990). The role of imagery in mathematical reasoning. In G. Booker, P. Cobb & T. de Mendicutti (Eds.),Proceedings of the 14th Annual PME Conference (p. 217). Oaxtepec, Mexico: International Group for the Psychology of Mathematics Education.
Cobb, P., & Wheatley, G. (1988). Children’s initial understandings of ten.Focus on Learning Problems in Mathematics, 10(3), 1–28.
Davis, R. B., & Maher, C. (1993). Children’s development of methods of proof. InProceedings of the 17th Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 105–112). Tsukuba, Japan: International Group for the Psychology of Mathematics Education.
Davis, R. B., Maher, C. A., & Noddings, N. (Eds.). (1990).Journal for Research in Mathematics Education, Monograph Number 4: Constructivist views on the teaching and learning of mathematics. Reston, VA: National Council of Teachers of Mathematics.
Denvir, B., & Brown, M. (1986a). Understanding of number concepts in low attaining 7–9 year-olds: Part I. Development of descriptive framework and diagnostic instrument.Educational Studies in Mathematics, 17, 15–36.
Denvir, B., & Brown, M. (1986b). Understanding of number concepts in low attaining 7–9 year-olds: Part II. The teaching studies.Educational Studies in Mathematics, 17, 143–164.
Fuson, K. (1990). Issues in place-value and multidigit addition and subtraction learning and teaching.Journal for Research in Mathematics Education, 21, 4, 273–280.
Goldin, G. A. (1987a). Levels of language in mathematical problem solving. Cognitive representational systems for mathematical problem solving. In C. Janvier (Ed.),Problems of representation in the teaching and learning of mathematics (pp. 59–65). Hillsdale: NJ: Erlbaum.
Goldin, G. A. (1987b). Cognitive representational systems for mathematical problem solving. In C. Janvier (Ed.),Problems of representation in the teaching and learning of mathematics (pp. 125–145). Hillsdale: NJ: Erlbaum.
Goldin, G. A. (1988). The Development of a model for competence in mathematical problem solving. In A. Borbas, (Ed.),Proceedings of the 12th International Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 358–368). Vészprem, Hungry: International Group for the Psychology of Mathematics Education.
Goldin, G. A. (1992). On developing a unified model for the psychology of mathematical learning and problem solving. In W. Geeslin & K. Graham (Eds.),Proceedings of the 16th International Group for the Psychology of Mathematics Education (Vol. 3, pp. 235–261). Durham, NH: International Group for the Psychology of Mathematics Education.
Goldin, G. A. (1993). Observing mathematical problem solving: perspectives on structured, task-based interviews. In B. Atweh, C. Kanes, M. Carss & G. Booker (Eds.),Proceedings of the 16th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 303–309). Brisbane: Queensland University of Technology.
Goldin, G.A., & Herscovics, N. (1991). The conceptual-representational analysis of children’s early arithmetic, In R. G. Underhill (Ed.),Proceedings of the 13th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 118–125). Blacksburg, VA: International Group for the Psychology of Mathematics Education, North American Chapter.
Goldin, G. A., & McClintock, C. E. (Eds.). (1979),Task variables in mathematical problem solving. Philadelphia: Franklin Institute Press (now Lawrence Erlbaum Associates, Hillsdale, NJ). ERIC Clearinghouse for Science, Mathematics and Environmental Education, Columbus, Ohio.
Hershkowitz, R. (1993). “Seeing geometry”: — with the eyes? with the mind’s eyes? In A. Baturo (Ed.),Teaching geometry and visual thinking, Third Annual Conference presented by the Centre for Mathematics and Science Education, Queensland University of Technology, Queensland.
Hershkowitz, R., & Markovits, Z. (1992). Conquer Mathematics Concepts by Developing Visual Thinking.Arithmetic Teacher, 39(9), 38–41.
Hiebert, J., & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (Ed.),Conceptual and procedural knowledge: The case of mathematics (pp. 199–223). Hillsdale, NJ: Erlbaum.
Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade.Journal for Research in Mathematics Education, 23(2), 98–122.
Hughes, M. (1986).Children’s minds. Oxford: Basil Blackwell.
Kamii, C. K. (1986). Place value: an explanation of its difficulty and educational implications for the primary grades.Journal of Research in Childhood Education, 1(2), 75–86.
Kamii, C. K. (1989).Young children continue to reinvent arithmetic. New York: Teachers College Press.
Lean, G. A. (1994).Counting systems of Papua New Guinea and Oceania. Unpublished PhD thesis, Papua New Guinea University of Technology, Lae, PNG.
Maher, C. A., Davis, R. B. & Alsatian, A. (1992). A teacher’s struggle to assess student cognitive growth. In Lesh, R. & Lamon, S. (Eds.),Assessment of authentic performance in school mathematics (pp. 249–264). Washington, DC: AAAS Press.
Mason, J. (1992). Doing and construing mathematics in screen-space. In B. Southwell, B. Perry, & K. Owens (Eds.),Proceedings of the 15th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 1–17). Sydney: Mathematics Education Research Group of Australasia.
Owens, K. (1994).Visual imagery employed by young students in spatial problem solving. Paper presented at annual conference of the Australian Association for Research in Education, University of Newcastle.
Owens, K., & Clements, M. A. (1993).Visualising, conceptualising, and the constructing of mental models. Paper presented at the conference on Constructivism, Deakin University, Geelong.
Pengelly, H. (1988). Representing mathematics: From materials to symbols. In D. Plummer (Ed.),Planning for thinkers and learners: The early years (pp. 89–107). Melbourne: Australian Reading Association.
Presmeg, N. C. (1986). Visualisation in high school mathematics.For the Learning of Mathematics, 6(3), 42–46.
Presmeg, N. C. (1992); Prototypes, metaphors, metonymies and imaginative rationality in High School Mathematics.Educational Studies in Mathematics, 23, 595–610.
Resnick, L., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. (1989). Conceptual bass of arithmetic errors: The case of decimal fractions.Journal for Research in Mathematics Education, 20(1), 8–27.
Ross, S. (1989). Parts, wholes, and place value: A developmental view.Arithmetic Teacher, 36(6), 47–51.
Ross, S. (1990). Children’s acquisition of place-value numeration concepts: the roles of cognitive development and instruction.Focus on Learning Problems in Mathematics, 12(1), 1–17.
Rubin, A., & Russell, S. (1992). Children’s developing concepts of landmarks in the number system. In W. Geeslin & K. Graham (Eds.),Proceedings of the 16th Conference of the International Group for the Psychology of Mathematics Education (Vol. III, p. 136). Durham, NH: International Group for the Psychology of Mathematics Education.
Steffe, L. P., Cobb, P., & Richards, J. (1988).Construction of arithmetical meanings and strategies. New York: Springer-Verlag.
Steffe, L. P. (1991).Epistemological foundations of mathematical experience. New York: Springer-Verlag.
Thomas, N. (1992). An analysis of children’s understanding of numeration. In B. Southwell, B. Perry, & K. Owens (Eds.),Proceedings of the 15th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 521–540). Nepean, Australia: University of Western Sydney.
Thomas, N., Mulligan, J., & Goldin, G. A. (1994). Children’s Representations of the counting sequence 1–100: Study and theoretical interpretation. In J. P. da Ponte & J. F. Matos (Eds.),Proceedings of the 18th Annual Conference of the International Group for the Psychology of Mathematics Education (Vol III, pp. 1–8). Lisbon, Portugal: International Group for the Psychology of Mathematics Education.
Wright, R. J. (1991). The role of counting in children’s numerical development.The Australian Journal of Early Childhood, 16(2), 43–48.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Thomas, N., Mulligan, J. Dynamic imagery in children’s representations of number. Math Ed Res J 7, 5–25 (1995). https://doi.org/10.1007/BF03217273
Issue Date:
DOI: https://doi.org/10.1007/BF03217273