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Ann's fraction schemes

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Abstract

A longitudinal constructivist teaching experiment that lasted approximately one academic year was conducted with six third graders. The purpose of the teaching experiment was to analyze the itinerary of children's ways of operating while solving fraction tasks. Ann was one of the third graders who participated in the teaching experiment, and her case study presents the author's interpretation of the generation and evolution of Ann's fraction schemes.

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References

  • Behr, M. J., Lesh, R., Post, T. and Silver, E. A.: 1983, ‘Rational-number concepts’, in R. Lesh and M. Landau (eds.),Acquisition of mathematical concepts and processes. Academic Press, New York, pp. 91–126.

    Google Scholar 

  • Behr, M. J., Wachsmuth, I., Post, T. and Lesh, R.: 1984, ‘Order and equivalence of rational numbers: A clinical teaching experiment,’Journal for Research in Mathematics Education 15, 323–341.

    Google Scholar 

  • Behr, M. J., Wachsmuth, I. and Post, T. P.: 1985, ‘Construct a sum: A measure of children's understanding of fraction size’,Journal for Research in Mathematics Education 16, 120–131.

    Google Scholar 

  • Bergeron, J. and Herscovics, N.: 1987, ‘Unit fractions of a continuous whole’, in J. Bergeron, N. Herscovics and C. Chilean (Eds.),Proceedings of the Eleventh International Conference of Psychology of Mathematics Education 1, Montreal, 357–365.

  • Brown, C. A., Carpenter, T. P., Kouba, V. L., Lindquist, M. M., Silver, E. A. and Swafford, J. O.: 1988a, ‘Secondary school results for the fourth NAEP mathematics assessment: Discrete mathematics, data organization and interpretation, measurement, number and operations’,The Mathematics Teacher 81, 241–248.

    Google Scholar 

  • Brown, C. A., Carpenter, T. P., Kouba, V. L., Lindquist, M. M., Silver, E. A. and Swafford, J. O.: 1988b, ‘Secondary school results for the fourth NAEP mathematics assessment: Algebra, geometry, mathematical methods, and attitudes’,The Mathematics Teacher 81, 337–347.

    Google Scholar 

  • Courant, R. and Robbins, H.: 1969,What is Mathematics? An Elementary Approach to Ideas and Methods. Oxford University Press, New York.

    Google Scholar 

  • Dantzig, T.: 1954,Number: The Language of Science. The Free Press, New York.

    Google Scholar 

  • Green, G. A.: 1970, ‘A comparison of two approaches, area and finding a part of, and two instructional materials, diagrams and manipulative aids, on multiplication of fractional numbers in grade five’ (Doctoral Dissertation, University of Michigan, 1969).Dissertation Abstracts International, (University Microfilms No. 70-14533) 676A–677A.

  • Hunting, R.: 1980, ‘The role of the discrete quantity partition knowledge in the child's construction of fractional number’ (Doctoral dissertation, The University of Georgia).Dissertation Abstracts International, (University Microfilms No. 8107919) 4320-A.

  • Hunting, R.: 1983, ‘Alan: A case study of knowledge of units and performance with fraction’,Journal for Research in Mathematics Education 14, 182–197.

    Google Scholar 

  • Hunting, R.: 1986, ‘Rachel's schemes for constructing fraction knowledge’Educational Studies in Mathematics 17, 49–66.

    Google Scholar 

  • Kerslake, D.: 1986,Fractions: Children's Strategies and Errors. Windsor: NFER-NELSON.

    Google Scholar 

  • Kieren, T.: 1980, ‘The rational number construction: Its elements and mechanisms’, in T. Kieren (ed.),Recent Research on Number Learning. Columbus; ERIC/SMEAC, 125–150.

    Google Scholar 

  • Kieren, T.: 1988, ‘Personal knowledge of rational numbers: Its intuitive and formal development’, in J. Hiebert and M. Behr (eds.),Number Concepts and Operations in the Middle Grades. Reston; National Council of Teachers of Mathematics, 162–181.

    Google Scholar 

  • Kieren, T., Nelson, D. and Smith, G.: 1985, ‘Graphical algorithms in partitioning tasks’,The Journal of Mathematical Behavior 4, 25–36.

    Google Scholar 

  • McLellan, J. and Dewey, J.: 1908,The Psychology of Number. New York; D. Appleton.

    Google Scholar 

  • Muangnapoe, C.: 1975, ‘An investigation of the learning of the initial concept and oral/written symbols for fractional numbers in grades three and four’, (Doctoral dissertation, The University of Michigan).Dissertation Abstracts International, 1975, 1353A-1354A. (University Microfilms No. 75-20,415).

    Google Scholar 

  • Nik Pa, N. A.: 1987, ‘Children's fractional schemes’, (Doctoral dissertation, The University of Georgia).Dissertation Abstracts International, 1988, 2827A. (University Microfilms No. DA8800290).

  • Peck, D. M. and Jencks, S. M.: 1981, ‘Conceptual issues in the teaching and learning of fractions’;Journal for Research in Mathematics Education 12, 339–355.

    Google Scholar 

  • Piaget, J.: 1967,Six Psychological Studies. Random House; New York.

    Google Scholar 

  • Piaget, J.: 1970,Genetic Epistemology. Columbia University Press; New York.

    Google Scholar 

  • Piaget, J.: 1972,The Principles of Genetic Epistemology. Routledge and Kegan Paul; London; (Original work published 1970.)

    Google Scholar 

  • Piaget, J.: 1973,To Understand is to Invent: The Future of Education. Grossman Publishers; New York.

    Google Scholar 

  • Piaget, J., Inhelder, B. and Szeminska, A.: 1960,The child's Conception of Geometry. Routledge and Kegan Paul; London.

    Google Scholar 

  • Pothier, Y. and Sawada, D.: 1983, ‘Partitioning: The emergence of rational number ideas in young children’,Journal for Research in Mathematics Education 14, 307–317.

    Google Scholar 

  • Pothier, Y. and Sawada, D.: 1990, ‘Partitioning: An approach to fractions’,The Mathematics Teacher 38, 12–16.

    Google Scholar 

  • Sáenz-Ludlow, A.: 1990, ‘Children's fraction schemes: An elaboration of their number sequence’, (Doctoral dissertation, The University of Georgia).Dissertation Abstracts International, (University Microfilms No. DA9118148) 457A–458A.

  • Smith, D. E.: 1953,History of Mathematics (Vol. 2). Dover Publications, Inc. New York.

    Google Scholar 

  • Steffe, L. P.: 1983, ‘The teaching experiment methodology in a constructivist research program’, in M. Zweng, T. Green, J. Kilpatrick, H. Pollak and M. Suydam (eds.),Proceedings of the Fourth International Congress on Mathematical education. Birkhäuser, Boston.

    Google Scholar 

  • Steffe, L. P.: 1986, ‘Units and their constitutive operation in multiplicative contexts’, The University of Georgia. Unpublished paper.

  • Steffe, L. P.: 1988, ‘Children's construction of meaning for arithmetical words’, Paper presented at the conference on Implicit and Explicit Knowledge in Science and Mathematics, Tel Aviv University, Tel Aviv, Israel.

  • Steffe, L. P.: 1990, ‘Construction of multiplying and dividing schemes’, Unpublished paper, The University of Georgia.

  • Steffe, L. P.: 1991, April,Composite Units and Their Schemes. Paper presented at the 1991 Annual Meeting of the American Educational Research Association, Chicago.

  • Steffe, L. P. and Cobb, P.: 1988,Construction of Arithmetical Meanings and Strategies. Springer-Verlag, New York.

    Google Scholar 

  • Streefland, L.: 1979, ‘Young children (6–8): Ratio and proportion’,Educational Studies in Mathematics 10, 403–420.

    Google Scholar 

  • Streefland, L.: 1984, ‘Search for the roots of ratio: Some thoughts on the long term learning process’,Educational Studies in Mathematics 15, 327–348.

    Google Scholar 

  • Streefland, L.: 1990,Fractions in Realistic Mathematics Education: A Paradigm of Developmental Research. Kluwer, Dordrecht.

    Google Scholar 

  • von Glasersfeld, E.: 1980, ‘The concept of equilibration in a constructivist theory of knowledge’, in F. Benseler, P. M. Hejl and W. Kock (eds.),Autopoiesis, Communication and Society. Campus, New York, 75–85.

    Google Scholar 

  • Vygotsky, L. S.: 1986,Thought and Language (A. Kozulin, trans. and ed.). The MIT Press, Cambridge. (Original work published 1934).

    Google Scholar 

  • Waismann, F.: 1959,Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics. Harper and Brothers, New York.

    Google Scholar 

  • Wertsch, J. V. and Stone, A.: 1985, ‘The concept of internalization in Vygotsky's account of the genesis of higher mental functions’, in J. V. Wertsch (ed.),Culture, communication and cognition: Vygotskian perspectives (pp. 169–179). Cambridge University Press, New York.

    Google Scholar 

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Sáenz-Ludlow, A. Ann's fraction schemes. Educ Stud Math 28, 101–132 (1995). https://doi.org/10.1007/BF01295789

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