The Type of Fractional Quotient and Consequential Development of Children's Quotient Subconcept of Rational Numbers

다국어 초록 (Multilingual Abstract)

This paper investigated the conceptual schemes four children constructed as they related division number sentences to various types of fraction: Proper fractions, improper fractions, and mixed numbers in both contextual and abstract symbolic forms.
Methods followed those of the constructivist teaching experiment. Four fifth-grade students from an inner city school in the southwest United States were interviewed eight times: Pre-test clinical interview, six teaching / semi-structured interviews, and a final post-test clinical interview.
Results showed that for equal sharing situations, children conceptualized division in two ways: For mixed numbers, division generated a whole number portion of quotient and a fractional portion of quotient. This provided the conceptual basis to see improper fractions as quotients. For proper fractions, they tended to see the quotient as an instance of the multiplicative structure: a×b=c ; a÷c=1/b ; b÷c=1/a.
Results suggest that first, facility in recall of multiplication and division fact families and understanding the multiplicative structure must be emphasized before learning fraction division. Second, to facilitate understanding of the multiplicative structure children must be fluent in representing division in the form of number sentences for equal sharing word problems. If not, their reliance on long division hampers their use of syntax and their understanding of divisor and dividend and their relation to the concepts of numerator and denominator.

참고문헌 (Reference)

1 Charles, K, "Young children’s partitioning strategies" 43 (43): 191-221, 2000

2 Middleton, J. A, "The emergence of quotient understandings in a fifth-grade classroom: A classroom teaching experiment" 2001

3 Piaget, J, "The child’s conception of geometry" Basic Books 1960

4 Lamon, S. J, "Teaching fractions and ratios for understanding" Lawrence Erlbaum Associates 1999

5 Steffe, L. P, "Teaching experiment methodology: Underlying principles and essential elements, In Handbook of research design in mathematics and science education" Erlbaum 267-307, 2000

6 Wearne, D, "Results from the seventh mathematics assessment of the National Assessment of Educational Progress" National Council of Teachers of Mathematics 2000

7 Behr, M. J, "Rational-number concepts, In Acquisition of mathematics concepts and processes" Academic Press 91-126, 1983

8 Lamon, S. J, "Rational numbers and proportional reasoning, In The second handbook of research on mathematics teaching and learning" Information Age Publishing 629-667, 2007

9 Pothier, Y, "Partitioning: The emergence of rational number ideas in young children" 14 (14): 307-317, 1983

10 Kieren, T, "On the mathematical, cognitive, and instructional foundations of rational numbers, In Number and measurement" Ohio State University 104-144, 1976

11 Vergnaud, G, "Multiplicative structures, In Acquisition of mathematics concepts and processes" Academic Press 127-174, 1983

12 Hiebert, J, "Mathematical, cognitive, and instructional analysis of decimal fractions, In Analysis of arithmetic for mathematics teaching" Lawence Erlbaum Associates 283-322, 1992

13 Ohlsson, S, "Mathematical meaning and applicational meaning in the semantics of fractions and related concepts, In Number concepts and operations in the middle grades" National Council of Teachers of Mathematics 53-92, 1988

14 Sinicrope, R, "Interpretations of fraction division, In Making sense of fractions, ratios, and proportions: 2002 yearbook" National Council of Teachers of Mathematics 153-161, 2002

15 Streefland, L, "Fractions in realistic mathematics education: a paradigm of developmental research" Kluwer Academic Publishers 1991

16 Huberman, A. M, "Data management and analysis methods, In Handbook of qualitative research" sage publications 428-444, 1994

17 Carpenter, T. P, "Children’s mathematics" National Council of Teachers of Mathematics 1999

18 Toluk, Z, "Children’s conceptualizations of the quotient subconstruct of rational numbers" Arizona State University 1999