From sharing to dividing: Young children's understanding of division

Developmental Science

Volumn 5, Issue 4, 2002, Pages 452-466

  Squire, Sarah ^a   Bryant, Peter ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036853429     PISSN: 1363755X     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-7687.00240     Document Type: Article

References (69)

1. Adelson, B. (1984). When novices surpass experts: the difficulty of a task may increase with expertise. Journal of Experimental Psychology: Learning, Memory and Cognition, 10, 483-495.
2. Baroody, A.J., Gannon, K.E., Berent, R., & Ginsburg, H.P. (1984). The development of basic formal mathematics abilities. Acta Paedologica, 1, 133-150.
3. Bourgeois, R., & Nelson, D. (1977). Young children's behavior in solving division problems. Alberta Journal of Educational Research, 23, 178-185.
4. Bradley, L., & Bryant, P. (1985). Rhyme and reason in reading and spelling. IARLD Monographs. Ann Arbor: University of Michigan Press.
5. Brown, M. (1981). Number operations. In K.M. Hart (Ed.), Children's understanding of mathematics (pp. 11-16). Oxford: John Murray.
6. Bryant, P. (1997). Mathematical understanding in the nursery school years. In T. Nunes & P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 53-67). Hove: Psychology Press.
7. Carpenter, T.P., Ansell, E., Franke, M.L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: a study of kindergarten Children's problem-solving processes. Journal for Research in Mathematics Education, 24, 428-441.
8. Carpenter, T.P., & Fennema, E. (1992). Cognitively guided instruction: building on the knowledge of students and teachers. International Journal of Research in Education, 17, 457-470.
9. Carpenter, T.P., Fennema, E., & Franke, M.L. (1996). Cognitively guided instruction: a knowledge base for reform in primary mathematics instruction. Elementary School Journal, 97, 3-20.
10. Carpenter, T.P., Franke, M.L., Jacobs, V.R., Fennema, E., & Empson, S.B. (1997). A longitudinal study of invention and understanding in children's multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3-20.
11. Carpenter, T.P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Fennema & T.A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 19-32). Studies in mathematical thinking and learning series. Mahwah, NJ: Lawrence Erlbaum Associates.
12. Carpenter, T.P., & Moser, J.M. (1982). The development of addition and subtraction problem solving skills. In T.P. Carpenter, J.M. Moser & T. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 10-24). Hillsdale, NJ: Lawrence Erlbaum Associates.
13. Carraher, T.N., Carraher, D.W., & Schliemann, A.D. (1987). Written and oral mathematics. Journal for Research in Mathematics Education, 18, 83-97.
14. Chi, M.T.H., Feltovich, P.J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121-152.
15. Confrey, J. (1994). Splitting, similarity, and rate of change: a new approach to multiplication and exponential functions. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 291-330). Albany, NY: State University of New York Press.
16. Correa, J. (1994). Young children's understanding of the division concept. University of Oxford: Unpublished PhD thesis.
17. Correa, J., Nunes, T., & Bryant, P. (1998). Young children's understanding of division: the relationship between division terms in a noncomputational task. Journal of Educational Psychology, 90, 321-329.
18. De Abreu, G., Bishop, A.J., & Pompeu, G. Jr. (1997). What children and teachers count as mathematics. In T. Nunes & P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 233-265). Hove, UK: Psychology Press.
19. De Corte, E., Greer, B., & Verschaffel, L. (1996). Mathematics. In D. Berliner & R. Calfee (Eds.), Handbook of educational psychology (pp. 491-549). New York: Macmillan.
20. Department for Education and Employment (1999). The national numeracy strategy: Mathematical vocabulary. Watford: GPS Printworks.
21. Desforges, A., & Desforges, G. (1980). Number-based strategies of sharing in young children. Educational Studies, 6, 97-109.
22. Fennema, E., Carpenter, T.P., Franke, M.L., Levi, L., Jacobs, V.R., & Empson, S.B. (1996). A longitudinal study of learning to use children's thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403-434.
23. Fennema, E., Sowder, J., & Carpenter, T.P. (1999). Creating classrooms that promote understanding. In E. Fennema & T.A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 185-199). Mahwah, NJ: Lawrence Erlbaum Associates.
24. Fischbein, R., Deri, M., Nello, M., & Marino, M. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16, 3-17.
25. Frydman, O., & Bryant, P.E. (1988). Sharing and the understanding of number equivalence by young children. Cognitive Development, 3, 323-339.
26. Goswami, U. (1989). Relational complexity and the development of analogical reasoning. Cognitive Development, 4, 251-268.
27. Gravemeijer, K. (1997). Mediating between concrete and abstract. In T. Nunes & P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 315-345). Hove, UK: Psychology Press.
28. Gunderson, A.G. (1955). Thought-patterns of young children in learning multiplication and division. Elementary School Journal, 55, 453-461.
29. Hayes, J.R., & Simon, H.A. (1974). Understanding written problem instructions. In L.W. Gregg (Eds.), Knowledge and cognition (pp. 167-200). Hillsdale, NJ: Erlbaum.
30. Hayes, J.R., & Simon, H.A. (1977). Psychological differences among problem isomorphs. In N.J. Castellan, D.B. Pisoni & G. Potts (Eds.), Cognitive theory (pp. 21-44). Hillsdale, NJ: Erlbaum.
31. Hiebert, J. (Ed.) (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Erlbaum.
32. Hiebert, J., & Carpenter, T.P. (1992). Learning and teaching with understanding. In D.A. Grouws (Eds.), Handbook of research on mathematics teaching and learning (pp. 65-97). New York: Macmillan.
33. Hughes, M. (1986). Children and number. Oxford: Blackwell.
34. Huttenlocher, J., Jordan, N.C., & Levine, S.C. (1994). A mental model for early arithmetic. Journal of Experimental Psychology: General, 123, 284-296.
35. Johnson-Laird, P.N. (1983). Mental models. Cambridge: Cambridge University Press.
36. Kornilaki, E. (1999). Young children's understanding of multiplicative concepts: a psychological approach. University of London: Unpublished PhD thesis.
37. Kotovsky, K., Hayes, J.R., & Simon, H.A. (1985). Why are some problems hard? Evidence from Tower of Hanoi. Cognitive Psychology, 17, 248-294.
38. Kotovsky, K., & Simon, H.A. (1990). What makes some problems really hard: explorations in the problem space of difficulty. Cognitive Psychology, 22, 143-183.
39. Lago, M.O., Rodriguez, P., Zamora, A., & Madrono, L. (1999). Influencia de los modelos intuitivos en la comprehension de la multiplication y la division [The influence of intuitive models on the comprehension of multiplication and division]. Anuario de Psicologia, 30, 71-89.
40. Levine, S.C., Jordan, N.C., & Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of Experimental Child Psychology, 53, 72-103.
41. Lewis, M.W., & Anderson, J.R. (1985). Discrimination of operator schemata in problem solving. Learning from examples. Cognitive Psychology, 17, 26-65.
42. Mack, N.K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26, 422-441.
43. Marton, F., & Neuman, D. (1996). Phenomenography and children's experience of division. In L.R. Steffe, P. Nosher, P. Cobb, G.A. Goldin & B. Greer (Eds.), Theories of mathematical learning (pp. 315-333). Mahwah, NJ: Lawrence Erlbaum Associates.
44. Medin, D.L., & Ortony, A. (1989). Psychological essentialism. In S. Vosniaou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 179-195). NewYork: Cambridge University Press.
45. Miller, K. (1984). The child as the measurer of all things: measurement procedures and the development of quantitative concepts. In C. Sophian (Ed.), Origins of cognitive skills (pp. 193-228). Hillsdale, NJ: Erlbaum.
46. Mix, K.S., Levine, S., & Huttenlocher, J. (1999). Early fraction calculation ability. Developmental Psychology, 35, 164-174.
47. Murray, H., Olivier, A., & Human, P. (1992). The development of young students' division strategies. Proceedings of the Sixteenth International Conference for the Psychology of Mathematics Education, 2, 152-159.
48. National Council for Teachers of Mathematics (2000). Principles and standards for school mathematics [http://standards.nctm.org/document/prepost/cover.htm].
49. Novick, L.R. (1988). Analogical transfer, problem similarity and expertise. Journal of Experimental Psychology: Learning, Memory and Cognition, 14, 510-520.
50. Nunes, T. (1999). Mathematics learning as the socialisation of the mind. Mind, Culture, and Activity, 6, 33-52.
51. Nunes, T., & Bryant, P. (1996). Children doing mathematics. Oxford: Blackwell.
52. Nunes, T., Schliemann, A.L., & Carraher, D. (1993). Street mathematics and school mathematics. New York: Cambridge University Press.
53. Piaget, J. (1972). The psychology of intelligence (M. Piercy & D.E. Berltyne, trans.). Totowa, NJ: Littlefield, Adams & (Co. Original work published in 1947).
54. Piaget, J. (2001). Studies in reflecting abstraction (R.L. Campbell, trans.). Hove: Psychology Press. (Original work published in 1977).
55. Reed, S.K., Ernst, G.W., & Banerji, R. (1974). The role of analogy in transfer between similar problem states. Cognitive Psychology, 6, 436-450.
56. Riley, M.S., Greeno, J.G., & Heller, J.I. (1983). Development of children's problem-solving ability in arithmetic. In H. Ginsburg (Ed.), The development of mathematical thinking (pp. 153-196). New York: Academic Press.
57. Rittle-Johnson, B., & Siegler, R.S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: a review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75-110). Studies in developmental psychology. Hove: Psychology Press.
58. Schoenfeld, A.H. (1991). On mathematics and sense-making: an informal attack on the unfortunate divorce of formal and informal mathematics. In J.F. Voss & D.N. Perkins (Eds.), Informal reasoning and education (pp. 311-343). Hillsdale, NJ: Lawrence Erlbaum Associates.
59. Silver, E.A., Shapiro, L.J., & Deutsch, A. (1993). Sense making and the solution of division problems involving remainders: an examination of middle school students' solution processes and their interpretations of solutions. Journal for Research in Mathematics Education, 24, 117-135.
60. Simon, M.A. (1993). prospective elementary teachers' knowledge of division. Journal for Research in Mathematics Education, 24, 233-259.
61. Sophian, C., Garyantes, D., & Chang, C. (1997). When three is less than two: early developments in children's understanding of fractional quantities. Developmental Psychology, 33, 731-744.
62. Spiro, R.J., Feltovich, P.J., Coulson, R.L., & Anderson, D.K. (1989). Multiple analogies for complex concepts: antidotes for analogy-induced misconception in advanced knowledge acquisition. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 498-531). New York: Cambridge University Press.
63. Squire, S., & Bryant, P. (2002). The influence of sharing on children's initial concept of division. Journal of Experimental Child Psychology, 81, 1-43.
64. Steffe, L. (1994). Children's multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 3-40). Albany, NY: State University of New York Press.
65. Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: the case of division of fractions. Journal for Research in Mathematics Education, 31, 5-25.
66. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisitions of mathematics concepts and procedures (pp. 127-174). New York: Academic Press.
67. Vergnaud, G. (1985). Conceptes et schemes dans une théorie opératoire de la représentation [Concepts and schemes in an operational theory of representation]. Psychologie Francaise, 30, 246-252.
68. Verschaffel, L., & de Corte, E. (1997). Word problems: a vehicle for promoting authentic mathematical understanding and problem solving in the primary school? In T. Nunes & P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 69-97). Hove: Psychology Press.
69. Zweng, M.J. (1964). Division problems and the concept of rate. Arithmetic Teacher, 11, 547-556.