Competent Reasoning With Rational Numbers

Cognition and Instruction

Volumn 13, Issue 1, 1995, Pages 3-50

  Smith, John P ^a  

^a MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0003114978     PISSN: 07370008     EISSN: 1532690X     Source Type: Journal    
DOI: 10.1207/s1532690xci1301_1     Document Type: Article

References (51)

1. Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323–341.
2. Bolster, L. C., Crown, W., & Lindquist, M. M. (1985). Invitation to mathematics (Teacher Texts, Grades 3–8). Glenview, IL: Scott, Foresman.
3. Boyer, C. B. (1991). A history of mathematics (2nd ed.). New York: Wiley.
4. Bright, G. W., Behr, M. J., Post, T. R., & Wachsmuth, I. (1988). Identifying fractions on number lines. Journal for Research in Mathematics Education, 19, 215–232.
5. Carpenter, T. P., & Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades one through three. Journal for Research in Mathematics Education, 15, 179–202.
6. Carraher, T. N., Schliemann, A. D., & Carraher, D. W. (1988). Mathematical concepts in daily life. In G. B. Saxe & M. Gearhart (Eds.), Children's mathematics (pp. 71–88). San Francisco: Jossey-Bass.
7. Corbitt, M. K. (Ed.). (1981). Results from the Second Mathematics Assessment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics.
8. Crosswhite, F. J. (1986). Cognitive science and mathematics education: A mathematics educator's perspective. In A. H. Schoenfeld (ed.), Cognitive science and mathematics education (pp. 265–277). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
9. Dowker, A. D. (1992). Computational estimation strategies of professional mathematicians. Journal for Research in Mathematics Education, 23, 45–55.
10. Eicholz, R. E., O’Daffer, P. G., & Fleenor, C. R. (1985). Addison-Wesley mathematics. Reading, MA: Addison-Wesley.
11. Ginsburg, H. P., Kossan, N. E., Schwartz, R., & Swanson, D. (1983). Protocol methods in research in mathematical thinking. In H. P. Ginsburg (ed.), The development of mathematical thinking (pp. 7–47). New York: Academic.
12. Greeno, J. G. (1983). Conceptual entities. In D. Gentner & A. L. Stevens (Eds.), Mental models (pp. 227–252). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
13. Groen, G. J., & Resnick, L. B. (1977). Can preschool children invent addition algorithms? Journal of Educational Psychology, 69, 645–652.
14. Hart, K. M. (Ed.). (1981). Children's understanding of mathematics; 11–16. London: Murray.
15. Hiebert, J., & Weame, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 197–223). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
16. Hope, J. A., & Sherrill, J. M. (1987). Characteristics of skilled and unskilled mental calculators. Journal for Research in Mathematics Education, 18, 98–111.
17. Karplus, R., Pulos, S., & Stage, E. K. (1983). Proportional reasoning in early adolescents. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 45–90). New York: Academic.
18. Kieren, T. E. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh & D. Bradbard (Eds.), Number and measurement: Papers from a research workshop (pp. 101–144). Columbus, OH: ERIC/SMEAC.
19. Kieren, T. E. (1980). Knowing rational numbers: Ideas and symbols. In M. Lindquist (ed.), Selected issues in mathematics education (pp. 69–81). Berkeley, CA: McCutchan.
20. Kieren, T. E. (1983, April). Constructive mechanisms and informal symbolism in rational number knowledge building: Partitioning. Paper presented at the annual meeting of the American Educational Research Association, Montreal.
21. Kieren, T. E., & Southwell, B. (1979). The development in children and adolescents of the construct of rational numbers as operators. The Alberta Journal of Educational Research, 25, 234–247.
22. Kouba, V. L., Brown, C. A., Carpenter, T. P., Lindquist, M. M., Silver, E. A., & Swafford, J. O. (1988). Results of the fourth NAEP Assessment of Mathematics: Number, operations, and word problems. Arithmetic Teacher, 35 (8), 14–19.
23. McKnight, C. C., Travers, K. J., Crosswhite, F. J., & Swafford, J. O. (1985). Eighth-grade mathematics in U.S. schools: A report from the Second International Mathematics Study. Arithmetic Teacher, 32 (8), 20–26.
24. Murray, H., & Olivier, A. (1989). A model of understanding two-digit numeration and computation. In G. Vergnaud (ed.), Proceedings of the Thirteenth International Conference for the Psychology of Mathematics Education (Vol. 3, pp. 3–10). Paris: CNRS-Paris V.
25. National Assessment of Educational Progress. (1983). The third National Mathematics Assessment: Results, trends, and issues. Denver, CO: Educational Commission of the States.
26. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
27. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.
28. Noelting, G. (1980 a). The development of proportional reasoning and the ratio concept. Part I— Differentiation of stages. Educational Studies in Mathematics, 11, 217–253.
29. Noelting, G. (1980 b). The development of proportional reasoning and the ratio concept. Part II— Problem structure at successive stages: Problem-solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, 11, 331–363.
30. Nurit, H., Hershkowitz, R., & Friedlander, A. (n.d.). Links: Fractions. Rehovot, Israel: Department of Science Teaching, Weizmann Institute of Science.
31. Ohlsson, S. (1988). Mathematical meaning and application meaning in the semantics of fractions and related concepts. In J. Hiebert & M. Behr (Eds.), Number operations and concepts in the middle grades (pp. 53–92). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
32. Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child's concept of geometry. New York: Basic Books.
33. Post, T. R. (1981). Fractions: Results and implications from National Assessment. Arithmetic Teacher, 28 (9), 26–31.
34. Post, T. R., Behr, M. J., & Lesh, R. (1986). Research-based observations about learning of rational number concepts. Focus on Learning Problems in Mathematics, 8 (1), 39–48.
35. Post, T. R., & Cramer, K. (1987). Children's strategies in ordering rational numbers. Arithmetic Teacher, 35 (2), 33–35.
36. Post, T. R., Harel, G., Behr, M. J., & Lesh, R. (1988). Intermediate teachers' knowledge of rational number concepts. In L. Fennema, T. Carpenter, & S. Lamon (Eds.), Integrating research on teaching and learning mathematics (pp. 194–219). Madison, WI: Center for Educational Research.
37. Post, T. R., Wachsmuth, I., Lesh, R., & Behr, M. J. (1985). Order and equivalence of rational numbers: A cognitive analysis. Journal for Research in Mathematics Education, 16, 18–36.
38. Resnick, L. B. (1983). A developmental theory of number understanding. In H. P. Ginsburg (ed.), The development of mathematical thinking (pp. 109–151). New York: Academic.
39. Resnick, L. B. (1987). Constructing knowledge in school. In L. S. Liben (ed.), Development and learning: Conflict or congruence? (pp. 19–50). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
40. Resnick, L. B., 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.
41. Saxe, G. B. (1991). Culture and cognitive development: Studies in mathematical understanding. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
42. Siegler, R. S., & Jenkins, E. A. (1989). How children discover new strategies. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
43. Smith, J. (1987 a). Qualitative reasoning in the development of fraction conceptual knowledge. Unpublished manuscript, Michigan State University.
44. Smith, J. (1987 b, April). What is fraction conceptual knowledge? Paper presented at the annual meeting of the American Educational Research Association, Washington, DC.
45. Smith, J. P. (1990). Learning rational number. Unpublished doctoral dissertation, University of California, Berkeley.
46. Smith, J. P., diSessa, A. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3, 115–163.
47. Sowder, J. T. (1992). Making sense of numbers in school mathematics. In G. Leinhardt, R. T. Putnam, & R. A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 1–51). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
48. Vance, J. H. (1986). Ordering decimals and fractions: A diagnostic study. Focus on Learning Problems in Mathematics, 8 (2), 51–59.
49. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127–174). New York: Academic.
50. Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number operations and concepts in the middle grades (pp. 162–181). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
51. Williams, D. E. (1984). By way of introduction. Arithmetic Teacher, 31(6), 1.