Engaging whole-number knowledge for rational-number learning using a computer-based tool

Journal for Research in Mathematics Education

Volumn 27, Issue 3, 1996, Pages 354-379

  Hunting, Robert P ^a   Davis, Gary ^b   Pearn, Catherine A ^a  

^a LA TROBE UNIVERSITY   (Australia)

^b UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 21344459627     PISSN: 00218251     EISSN: None     Source Type: Journal    
DOI: 10.2307/749369     Document Type: Article

References (59)

1. Behr, M. J., Wachsmuth, I., Post, T., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323-341.
2. Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296-333). New York: Macmillan.
3. Behr, M. J., Harel, G., Post, T., & Lesh, R. (1993). Rational numbers: Toward a semantic analysis - emphasis on the operator construct. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 13-47). Hillsdale, NJ: Erlbaum.
4. Braunfeld, P., & Wolfe, M. (1966). Fractions for low achievers. Arithmetic Teacher, 13, 647-655.
5. Carpenter, T. P. (1985). Learning to add and subtract: An exercise in problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 17-40). Hillsdale, NJ: Erlbaum.
6. Carpenter, T. P., Fennema, E., & Romberg, T. A. (1993). Toward a unified discipline of scientific inquiry. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 1-11). Hillsdale, NJ: Erlbaum.
7. Carpenter, T. P., & Moser, J. M. (1982). The development of addition and subtraction problem solving skills. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 9-24). New York: Academic Press.
8. Carraher, D. W. (1993, January). Building rational number concepts upon students' prior understanding. Paper presented at the International Conference on Rational Number Learning, University of Georgia, Athens, GA.
9. Charles, R., & Silver, E. (1989). The teaching and assessing of mathematical problem solving. Reston, VA: National Council of Teachers of Mathematics.
10. Cobb, P., Wood, T., & Yackel, E. (1990). Classrooms as learning environments for teachers and researchers. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the leaching and learning of mathematics. Journal for Research in Mathematics Education Monograph Series, Number 6 (pp. 125-146). Reston, VA: National Council of Teachers of Mathematics.
11. Cobb, P., Perlwitz, M., & Underwood, D. (1994). Individual construction, mathematical acculturation, and the classroom community. Revue des Sciences de l'Education, 28, 41-62.
12. Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14, 83-94.
13. Confrey, J. (1988). Multiplication and splitting: Their role in understanding exponential functions. Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Northern Illinois University: DeKalb, IL.
14. Davis, G. (1991). Fractions as operators and as cloning machines. In R. P. Hunting & G. Davis (Eds.), Early fraction learning (pp. 91-102). New York: Springer-Verlag.
15. Davis, G., & Hunting, R. P. (1991). On clinical methods for studying young children's mathematics. In R. P. Hunting & G. Davis (Eds.), Early fraction learning (pp. 209-224). New York: Springer-Verlag.
16. Davis, G., Hunting, R. P., & Pearn, C. (1993). What might a fraction mean to a child and how would a teacher know? Journal of Mathematical Behavior, 12, 63-76.
17. Dienes, Z. P. (1967). Fractions: An operational approach. Harlow, Essex, Great Britain: Educational Supply Association.
18. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, Netherlands: Reidel.
19. Fuson, K. (1988). Children's counting and concepts of number. New York: Springer-Verlag.
20. Gray, E. M. (1993, January). The transition from whole number to fraction: The need for flexible thinking. Paper presented at the International Conference on Rational Number Learning, University of Georgia, Athens, GA.
21. Hart, K. (1981). Children's understanding of mathematics 11-16. London: Murray.
22. Hilton, P. (1983). Do we still need to teach fractions? In M. Zweng (Ed.), Proceedings of the Fourth International Congress on Mathematical Education. Boston: Birkhauser.
23. Hunting, R. P. (1983). Emerging methodologies for understanding internal processes governing children's mathematical behaviour. Australian Journal of Education, 27(1), 45-61.
24. Hunting, R. P. (1986). Rachel's schemes for constructing fraction knowledge. Educational Studies in Mathematics, 17, 49-66.
25. Hunting, R. P., & Davis, G. (Eds.). (1991). Early fraction learning. New York: Springer-Verlag.
26. Hunting, R. P., Davis, G., & Bigelow, J. C. (1991). Higher order thinking in young children's engagements with a fraction machine. In R. P. Hunting, & G. Davis (Eds.), Early fraction learning (pp. 73-90). New York: Springer-Verlag.
27. Hunting, R. P., & Korbosky, R. K. (1990). Context and process in fraction learning. International Journal of Mathematical Education in Science and Technology, 21, 929-948.
28. Kaput, J. (1985). Multiplicative word problems and intensive quantities: An integrated software response (Technical Report). Harvard Graduate School of Education, Educational Technology Center.
29. Kieren, T. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh (Ed.), Number and measurement: Papers from a research workshop (pp. 101-144). Columbus, OH: ERIC/SMEAC.
30. Kieren, T. E. (1988). Personal knowledge of rational numbers - Its intuitive and formal development In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 162-181). Hillsdale, NJ: Erlbaum.
31. Kieren, T. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 49-84). Hillsdale, NJ: Erlbaum.
32. Kieren, T., & Nelson, D. (1978). The operator construct of rational numbers in childhood and adolescence - An exploratory study. Alberta Journal of Educational Research, 24, 22-30.
33. Kieren, T., & Southwell, B. (1979). The development in children and adolescents of the construct of rational numbers as operators. Alberta Journal of Educational Research, 25, 234-247.
34. Lamon, S. J. (1993a). 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 (pp. 131-156). Hillsdale, NJ: Erlbaum.
35. Lamon, S. J. (1993b). Ratio and proportion: Connecting content and children's thinking. Journal for Research in Mathematics Education, 24, 41-61.
36. Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, 16-32.
37. 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 (pp. 85-106). Hillsdale, NJ: Erlbaum.
38. Opper, S. (1977). Piaget's clinical method. Journal of Children's Mathematical Behavior, 1, 90-107.
39. Piaget, J. (1970). Genetic epistemology. (E. Duckworth, Trans.). New York: Columbia University Press.
40. Piaget, J. (1972). The principles of genetic epistemology. London: Routledge & Kegan Paul.
41. Pickert, G. (1968). Die Bruchrechnung als Operieren mit Abbildungen. [Treatment of elementary fractions by means of mappings.] Mathematisch-Physikalische Semesterberichte, Band XV, 32-47.
42. Post, T. R., Cramer, K. A., Behr, M., Lesh, R., & Harel, G. (1993). Curriculum implications of research on the learning, teaching and assessing of rational number concepts. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 327-362). Hillsdale, NJ: Erlbaum.
43. Resnick, L. B., & Singer, J. A. (1993). Protoquantitative origins of ratio reasoning. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 107-130). Hillsdale, NJ: Erlbaum.
44. Sáenz-Ludlow, A. (1992) Ann's strategies to add fractions. In W. Geeslin and K. Graham (Eds.), Proceedings of the Sixteenth Psychology of Mathematics Education Conference, Vol. 2, (pp. 266-273). Durham, NH: University of New Hampshire.
45. Sáenz-Ludlow, A. (1994). Michael's fraction schemes. Journal for Research in Mathematics Education, 25, 50-85.
46. Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 41-52). Hillsdale, NJ: Erlbaum.
47. Southwell, B. (1984). An analysis of the development of several subconstructs of rational numbers. In B. Southwell (Ed.), Proceedings of the Eighth International Conference for the Psychology of Mathematics Education (pp. 387-396). Sydney: Mathematical Association of New South Wales.
48. Steffe, L. P. (1984). The teaching experiment methodology in a constructivist research program. In M, Zweng (Ed.), Proceedings of the Fourth International Congress on Mathematical Education (pp. 469-471). Boston: Birkhauser.
49. Steffe, L. P. (1988). Children's construction of number sequences and multiplying schemes. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 119-140). Hillsdale, NJ: Erlbaum.
50. Steffe, L. P. (1993, January). Learning an iterative fraction scheme. Paper presented at the International Conference on Rational Number Learning, University of Georgia, Athens, GA.
51. Steffe, L. P., Cobb, P., and von Glasersfeld, E. (1988). Construction of arithmetical meanings and strategies. New York: Springer-Verlag.
52. Steffe, L. P., & Olive, J. (1993, January). Children's mathematical learning in computer microworlds. Paper presented at the International Conference on Rational Number Learning, University of Georgia, Athens, GA.
53. Steffe, L. P., & von Glasersfeld, E. (1985). Helping children to conceive of number. Recherches en Didactique des Mathématiques, 6(2-3), 269-303.
54. Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children's counting types: Philosophy, theory, and application. New York: Praegar Scientific.
55. Streefland, L. (1984). Unmasking N-Distracters as a source of failures in learning fractions. In B. Southwell (Ed.), Proceedings of the Eighth International Conference for the Psychology of Mathematics Education. Sydney: Mathematical Association of New South Wales.
56. Streefland, L. (1991). Fractions in realistic mathematics education. Dordrecht, Netherlands: Kluwer.
57. Tirosh, D., Wilson, J., Graeber, A., & Fischbein, E. (1993, January). Conceptual adjustments in progressing from whole to rational numbers. Paper presented at the International Conference on Rational Number Learning, University of Georgia, Athens, GA.
58. Usiskin, Z. (1979). The future of fractions. Arithmetic Teacher, 27, 18-20.
59. Wagner, S., & Kieren, C. (Eds.). (1989). Research issues in the learning and teaching of algebra. Reston, VA: National Council of Teachers of Mathematics.