Linking meaning of symbols of fractions to problem situations

HAJIME YOSHIDA; KAZUHIRO KURIYAMA

doi:10.4992/psycholres1954.37.229

分数の記号の意味と問題状況との関連づけ

吉田甫, 栗山和広

著者情報

ジャーナルフリー

1995 年 37 巻 4 号 p. 229-239

DOI https://doi.org/10.4992/psycholres1954.37.229

詳細

発行日: 1995/12/25 受付日: 1994/02/03 J-STAGE公開日: 2009/02/24 受理日: 1994/09/03 早期公開日: - 改訂日: -
訂正情報
訂正日: 2009/02/24 訂正理由: - 訂正箇所: 論文抄録訂正内容: Wrong : The present study aimed to extract knowledge on both knowing of the meaning of symbols of fractions and linking it to problem situations. Fourth and fifth graders were interviewed individually. Three kinds of meanings of symbols were found; appropriate meaning, inappropriate meaning, and lack of meaning. Four kinds of ways of linking were identified; appropriate linking, linking by inappropriate fractional knowledge, linking by whole number concepts, and linking by quotient. It was found that the most important factor in judging magnitudes of fractions was not knowledge for the symlbols of fractions but one for linking it to the problem situations.
Right : The present study aimed to extract knowledge on both knowing of the meaning of symbols of fractions and linking it to problem situations. Fourth and fifth graders were interviewed individually. Three kinds of meanings of symbols were found; appropriate meaning, inappropriate meaning, and lack of meaning. Four kinds of ways of linking were identified; appropriate linking, linking by inappropriate fractional knowledge, linking by whole number concepts, and linking by quotient. It was found that the most important factor in judging magnitudes of fractions was not knowledge for the symbols of fractions but one for linking it to the problem situations.

訂正日: 2009/02/24 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. 1984 Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323-341.
Behr, M. J., Wachsmuth, I., & Post, T. R. 1985 Constructs a sum: A measure of children's understanding of fraction size. Journal for Research in Mathematics Education, 16, 120-131.
Brown, J. S., Collins, A., & Duguid, P. 1989 Situated cognition and the culture of learning. Educational Researcher, 18, 32-42.
Ginsburg. H. P. 1982 Children's arithmetic. Austin, TX: Pro-Ed.
Hant, K. M. 1988 Ratio and operation. In J. Hiebert & M. J. Behn (Eds.), Number concepts and operations in the middle grade. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp. 198-219.
Heller, P. M., Post, T. R.. Behr, M., & Lesh, R. 1990 Qualitative and numerical reasoning about fractions and rates by seventh- and eighth-grade students. Journal for Research in Mathematics Education, 21, 388-402.
Hiebert, J. 1988 A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355.
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. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp. 1-23.
Hunting, R. P., & Sharpiey, C. F. 1988 Fraction knowledge in preschool children. Journal for Research in Mathematics Education, 19, 175-180.
Lamon, S. J. 1993 Ratio and proportion: Children's cognitive and metacognitive processes. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp. 131-156.
Leinhardt, G. 1988 Getting to know: Tracing students' mathematical knowledge from intuition to competence. Educational Psychologist. 23, 119-144.
Mack, N. K. 1993 Learning rational numbers with understanding: The case of informal knowledge. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp. 85-106.
Philipp, R. A., & Sowder, J. T. 1993 Conceptualizing rate: Four teachers' struggle. In K. Hirabayashi et al. (Eds.), Proceedings of the seventeenth PME. Vol. 3. Pp. 107-114.
Piaget, J. 1960 The psychology of intelligence. Totowa, NJ: Littlefiel, Adams.
Pothier, Y., & Sawada, D. 1983 Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Ecuation, 14, 307-317.
Resnick, L. B. 1982 Syntax and semantics in learning to subtract. In T. P. Carpenter, J. M. Moser & T. A. Romberg, (Eds.), Addition and subtraction: A cognitive perspective. Hillsdale, NJ: Lawrence Erlbaum Associates.
Resnick, L., Cauzinille-Marmeche, E., & Mathieu, J. 1987 Understanding algebra. In J. A. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics. Oxford: Clarendon Press. Pp. 169-203.
Resnick, L., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. 1989 Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20, 8-27.
Riley, M. S., 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. New York: Academic Press. Pp. 153-196.
Schwarz, J. L. 1988 Intensive quantity and referent transforming arithmetic operations. Hillsdale, HJ: Lawrence Erlbaum Associates. Pp. 41-52.
Yoshida, H., & Kuriyama, K. 1988 Process of the acquisition of knowledge for understanding fractions. A paper present at the Third JapanGermany Science Seminar, held in Tokyo.
Yoshida, H., & Kuriyama, K. 1991 A developmental analysis on the acquisition process of the concept of fractions. Japanese Journal of Educational Psychology, 39, 382-391. (In Japanese with English abstract)

Right : Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. 1984 Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323-341.
Behr, M. J., Wachsmuth, I., & Post, T. R. 1985 Constructs a sum: A measure of children's understanding of fraction size. Journal for Research in Mathematics Education, 16, 120-131.
Brown, J. S., Collins, A., & Duguid, P. 1989 Situated cognition and the culture of learning. Educational Researcher, 18, 32-42.
Ginsburg. H. P. 1982 Children's arithmetic. Austin, TX: Pro-Ed.
Hant, K. M. 1988 Ratio and operation. In J. Hiebert & M. J. Behn (Eds.), Number concepts and operations in the middle grade. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.198-219.
Heller, P. M., Post, T. R. Behr, M., & Lesh, R. 1990 Qualitative and numerical reasoning about fractions and rates by seventh- and eighth-grade students. Journal for Research in Mathematics Education, 21, 388-402.
Hiebert, J. 1988 A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355.
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.
Hillsdale, NJ: Lawrence Erlbaum Associates. pp.1-23.
Hunting, R. P., & Sharpley, C. F. 1988 Fraction knowledge in preschool children, Journal for Research in Mathematics Education, 19, 175-180.
Lamon, S. J. 1993 Ratio and proportion: Children's cognitive and metacognitive processes. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.131-156.
Leinhardt, G. 1988 Getting to know: Tracing students' mathematical knowledge from intuition to competence. Educational Psychologist. 23, 119-144.
Mack, N. K. 1993 Learning rational numbers with understanding: The case of informal knowledge. In T. P. Carpenter, E. Fennema, & T. A.
Romberg (Eds.), Rational numbers: An integration of research. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.85-106.
Philipp, R. A., & Sowder, J. T. 1993 Conceptualizing rate: Four teachers' struggle. In K. Hirabayashi et al.(Eds.), Proceedings of the seventeenth PME. Vol.3. Pp.107-114.
Piaget, J. 1960 The psychology of intelligence. Totowa, NJ: Littlefiel, Adams.
Pothier, Y., & Sawada, D. 1983 Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Ecuation, 14, 307-317.
Resnick, L. B. 1982 Syntax and semantics in learning to subtract. In T. P. Carpenter, J. M. Moser & T. A. Romberg, (Eds.), Addition and subtraction: A cognitive perspective. Hillsdale, NJ: Lawrence Erlbaum Associates.
Resnick, L., Cauzinille-Marmeche, E., & Mathieu, J. 1987 Understanding algebra. In J. A. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics.Oxford: Clarendon Press. Pp.169-203.
Resnick, L., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. 1989 Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20, 8-27.
Riley, M. S., 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. New York: Academic Press. Pp.153-196.
Schwarz, J. L. 1988 Intensive quantity and referent transforming arithmetic operations. Hillsdale, Hi: Lawrence Erlbaum Associates. Pp.41-52.
Yoshida, H., & Kuriyama, K. 1988 Process of the acquisition of knowledge for understanding fractions. A paper present at the Third Japan-Germany Science Seminar, held in Tokyo.
Yoshida, H., & Kuriyama, K. 1991 A developmental analysis on the acquisition process of the concept of fractions. Japanese Journal of Educational Psychology, 39, 382-391.(In Japanese with English abstract)

訂正日: 2009/02/24 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

訂正日: 2009/02/24 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. 1984 Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323-341.
Behr, M. J., Wachsmuth, I., & Post, T. R. 1985 Constructs a sum: A measure of children's understanding of fraction size. Journal for Research in Mathematics Education, 16, 120-131.
Brown, J. S., Collins, A., & Duguid, P. 1989 Situated cognition and the culture of learning. Educational Researcher, 18, 32-42.
Ginsburg. H. P. 1982 Children's arithmetic. Austin, TX: Pro-Ed.
Hant, K. M. 1988 Ratio and operation. In J. Hiebert & M. J. Behn (Eds.), Number concepts and operations in the middle grade. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.198-219.
Heller, P. M., Post, T. R. Behr, M., & Lesh, R. 1990 Qualitative and numerical reasoning about fractions and rates by seventh- and eighth-grade students. Journal for Research in Mathematics Education, 21, 388-402.
Hiebert, J. 1988 A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355.
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.
Hillsdale, NJ: Lawrence Erlbaum Associates. pp.1-23.
Hunting, R. P., & Sharpley, C. F. 1988 Fraction knowledge in preschool children, Journal for Research in Mathematics Education, 19, 175-180.
Lamon, S. J. 1993 Ratio and proportion: Children's cognitive and metacognitive processes. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.131-156.
Leinhardt, G. 1988 Getting to know: Tracing students' mathematical knowledge from intuition to competence. Educational Psychologist. 23, 119-144.
Mack, N. K. 1993 Learning rational numbers with understanding: The case of informal knowledge. In T. P. Carpenter, E. Fennema, & T. A.
Romberg (Eds.), Rational numbers: An integration of research. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.85-106.
Philipp, R. A., & Sowder, J. T. 1993 Conceptualizing rate: Four teachers' struggle. In K. Hirabayashi et al.(Eds.), Proceedings of the seventeenth PME. Vol.3. Pp.107-114.
Piaget, J. 1960 The psychology of intelligence. Totowa, NJ: Littlefiel, Adams.
Pothier, Y., & Sawada, D. 1983 Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Ecuation, 14, 307-317.
Resnick, L. B. 1982 Syntax and semantics in learning to subtract. In T. P. Carpenter, J. M. Moser & T. A. Romberg, (Eds.), Addition and subtraction: A cognitive perspective. Hillsdale, NJ: Lawrence Erlbaum Associates.
Resnick, L., Cauzinille-Marmeche, E., & Mathieu, J. 1987 Understanding algebra. In J. A. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics.Oxford: Clarendon Press. Pp.169-203.
Resnick, L., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. 1989 Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20, 8-27.
Riley, M. S., 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. New York: Academic Press. Pp.153-196.
Schwarz, J. L. 1988 Intensive quantity and referent transforming arithmetic operations. Hillsdale, Hi: Lawrence Erlbaum Associates. Pp.41-52.
Yoshida, H., & Kuriyama, K. 1988 Process of the acquisition of knowledge for understanding fractions. A paper present at the Third Japan-Germany Science Seminar, held in Tokyo.
Yoshida, H., & Kuriyama, K. 1991 A developmental analysis on the acquisition process of the concept of fractions. Japanese Journal of Educational Psychology, 39, 382-391.(In Japanese with English abstract)

Right : Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. 1984 Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323-341.
Behr, M. J., Wachsmuth, I., & Post, T. R. 1985 Constructs a sum: A measure of children's understanding of fraction size. Journal for Research in Mathematics Education, 16, 120-131.
Brown, J. S., Collins, A., & Duguid, P. 1989 Situated cognition and the culture of learning. Educational Researcher, 18, 32-42.
Ginsburg. H. P. 1982 Children's arithmetic. Austin, TX: Pro-Ed.
Hant, K. M. 1988 Ratio and operation. In J. Hiebert & M. J. Behn (Eds.), Number concepts and operations in the middle grade. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.198-219.
Heller, P. M., Post, T. R., Behr, M., & Lesh, R., 1990 Qualitative and numerical reasoning about fractions and rates by seventh- and eighth-grade students. Journal for Research in Mathematics Education, 21, 388-402.
Hiebert, J. 1988 A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355.
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.
Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.1-23.
Hunting, R. P., & Sharpley, C. F. 1988 Fraction knowledge in preschool children. Journal for Research in Mathematics Education, 19, 175-180.
Lamon, S. J. 1993 Ratio and proportion: Children's cognitive and metacognitive processes. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.131-156.
Leinhardt, G. 1988 Getting to know: Tracing students' mathematical knowledge from intuition to competence. Educational Psychologist. 23, 119-144.
Mack, N. K. 1993 Learning rational numbers with understanding: The case of informal knowledge. In T. P. Carpenter, E. Fennema, & T. A.
Romberg (Eds.), Rational numbers: An integration of research. Hillsdale, NJ: Lawrence Erlbaum Associates. Pp.85-106.
Philipp, R. A., & Sowder, J. T. 1993 Conceptualizing rate: Four teachers' struggle. In K. Hirabayashi et al.(Eds.), Proceedings of the seventeenth PME. Vol.3. Pp.107-114.
Piaget, J. 1960 The psychology of intelligence. Totowa, NJ: Littlefiel, Adams.
Pothier, Y., & Sawada, D. 1983 Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Ecuation, 14, 307-317.
Resnick, L. B. 1982 Syntax and semantics in learning to subtract. In T. P. Carpenter, J. M. Moser & T. A. Romberg,(Eds.), Addition and subtraction: A cognitive perspective. Hillsdale, NJ: Lawrence Erlbaum Associates.
Resnick, L., Cauzinille-Marmeche, E., & Mathieu, J. 1987 Understanding algebra. In J. A. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics. Oxford: Clarendon Press. Pp.169-203.
Resnick, L., Nesher, P., Leonard, F., Magone, M., Omanson, S., & Peled, I. 1989 Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20, 8-27.
Riley, M. S., 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. New York: Academic Press. Pp.153-196.
Schwarz, J. L. 1988 Intensive quantity and referent transforming arithmetic operations. Hillsdale, HJ: Lawrence Erlbaum Associates. Pp.41-52.
Yoshida, H., & Kuriyama, K. 1988 Process of the acquisition of knowledge for understanding fractions. A paper present at the Third Japan-Germany Science Seminar, held in Tokyo.
Yoshida, H., & Kuriyama, K. 1991 A developmental analysis on the acquisition process of the concept of fractions. Japanese Journal of Educational Psychology, 39, 382-391.(In Japanese with English abstract)

抄録

The present study aimed to extract knowledge on both knowing of the meaning of symbols of fractions and linking it to problem situations. Fourth and fifth graders were interviewed individually. Three kinds of meanings of symbols were found; appropriate meaning, inappropriate meaning, and lack of meaning. Four kinds of ways of linking were identified; appropriate linking, linking by inappropriate fractional knowledge, linking by whole number concepts, and linking by quotient. It was found that the most important factor in judging magnitudes of fractions was not knowledge for the symbols of fractions but one for linking it to the problem situations.

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