Students' whole number multiplicative concepts: A critical constructive resource for fraction composition schemes

Journal of Mathematical Behavior

Volumn 28, Issue 1, 2009, Pages 1-18

  Hackenberg, Amy J ^a   Tillema, Erik S ^b  

^a INDIANA UNIVERSITY   (United States)

^b IU School of Education at Indianapolis   (United States)

Author keywords

Fraction multiplication; Fractions; Multiplicative concepts; Schemes; Students' mathematical learning

Indexed keywords

EID: 67349215710     PISSN: 07323123     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmathb.2009.04.004     Document Type: Article

References (47)

1. Armstrong B.E., and Bezuk N. Multiplication and division of fractions: The search for meaning. In: Sowder J.T., and Schappelle B.P. (Eds). Providing a foundation for teaching mathematics in the middle grades (1995), State University of New York Press, Albany, NY 85-119
2. Ball D.L., and Bass H. Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In: Boaler J. (Ed). Multiple perspectives on mathematics teaching and learning (2000), Ablex, Westport, CT 81-104
3. Ball D., Lubienski S., and Mewborn D. Research on teaching mathematics: The unsolved problem of teachers' mathematical knowledge. In: Richardson V. (Ed). Handbook of research on teaching. 4th ed. (2001), American Educational Research Association, Washington, DC
4. Behr M.J., Harel G., Post T.R., and Lesh R. Rational number, ratio, and proportion. In: Grouws D.A. (Ed). Handbook of research on mathematics teaching and learning (1992), Macmillan, New York 296-333
5. Behr M.J., Harel G., Post T.R., and Lesh R. Rational numbers: Toward a semantic analysis-emphasis on the operator construct. In: Carpenter T.P., Fennema E., and Romberg T.A. (Eds). Rational numbers: An integration of research (1993), Lawrence Erlbaum Associates, Hillsdale, NJ 13-47
6. Behr M.J., Wachsmuth I., Post T.R., and Lesh R. Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education 15 5 (1984) 323-341
7. Biddlecomb B.D. Theory-based development of computer microworlds. Journal of Research in Early Childhood Education 8 2 (1994) 87-98
8. Biddlecomb B.D. Numerical knowledge as enabling and constraining fraction knowledge: An example of the reorganization hypothesis. Journal of Mathematical Behavior 21 (2002) 167-190
9. Biddlecomb, B., Olive, J. (2000). JavaBars [Computer software]. Retrieved June 4, 2002 from http://jwilson.coe.uga.edu/olive/welcome.html.
10. Carpenter T.P., Fennema E., Franke M.L., Levi L., and Empson S.B. Children's mathematics: Cognitively guided instruction (1999), Heinemann, Portsmouth, NH
11. Clement J. Analysis of clinical interviews: Foundations and model viability. In: Lesh R., and Kelly A.E. (Eds). Handbook of research design in mathematics and science education (2000), Erlbaum, Hillsdale, NJ 547-589
12. Cobb P., and Gravemeijer K. Experimenting to support and understand learning processes. In: Kelly A.E., Lesh R.A., and Baek J.Y. (Eds). Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (2008), Routledge, New York 68-95
13. Confrey J., and Lachance A. Transformative teaching experiments through conjecture-driven research design. In: Lesh R., and Kelly A.E. (Eds). Handbook of research design in mathematics and science education (2000), Erlbaum, Hillsdale, NJ 231-265
14. Davis G., Hunting R.P., and Pearn C. What might a fraction mean to a child and how would a teacher know?. Journal of Mathematical Behavior 12 (1993) 63-76
15. Empson S.B., Junk D., Dominguez H., and Turner E. Fractions as the coordination of multiplicatively related quantities: A cross-sectional study of children's thinking. Educational Studies in Mathematics 63 (2006) 1-28
16. Fosnot C.T., and Dolk M. Young mathematicians at work: Constructing multiplication and division (2001), Heinemann, Portsmouth, NH
17. Greer B. Multiplication and division as models of situations. In: Grouws D.A. (Ed). Handbook of research on mathematics teaching and learning (1992), Macmillan, New York 276-295
18. Hackenberg, A. J. (2005). Construction of algebraic reasoning and mathematical caring relations. Unpublished doctoral dissertation, University of Georgia.
19. Hackenberg A.J. Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior 26 (2007) 27-47
20. Hackenberg, A. J. (submitted for publication). Students' reversible multiplicative reasoning with fractions. Cognition and Instruction.
21. Hunting R.P., Davis G., and Pearn C. Engaging whole-number knowledge for rational-number learning using a computer-based tool. Journal for Research in Mathematics Education 27 3 (1996) 354-379
22. Izsak A. Mathematical knowledge for teaching fraction multiplication. Cognition and Instruction 26 (2008) 95-143
23. Izsak A., Tillema E., and Tunc-Pekkan Z. Teaching and learning fraction addition on number lines. Journal for Research in Mathematics Education 39 1 (2008) 33-62
24. Kamii C., and Housman L.B. Young children reinvent arithmetic. 2nd ed. (2000), Teachers College Press, New York
25. Kieren T.E. The rational number construct: Its elements and mechanisms. In: Kieren T.E. (Ed). Recent research on number learning (1980), ERIC/SMEAC, Columbus, OH 125-150
26. Kieren T.E. Creating spaces for learning fractions. In: Sowder J.T., and Schappelle B.P. (Eds). Providing a foundation for teaching mathematics in the middle grades (1995), State University of New York Press, Albany, NY 31-65
27. Lappan G., Fey J.T., Fitzgerald W., Friel S.N., and Phillips E.D. Connected mathematics 2 (2006), Prentice Hall, Boston, MA
28. Mack N.K. Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education 26 5 (1995) 422-441
29. Mack N.K. Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education 32 3 (2001) 267-295
30. Olive J. From fractions to rational numbers of arithmetic: A reorganization hypothesis. Mathematical Thinking and Learning 1 4 (1999) 279-314
31. Olive J., and Steffe L.P. The construction of an iterative fractional scheme: The case of Joe. Journal of Mathematical Behavior 20 (2002) 413-437
32. Steffe L.P. The constructivist teaching experiment: Illustrations and implications. In: von Glasersfeld E. (Ed). Radical constructivism in mathematics education (1991), Kluwer Academic, Boston 177-194
33. Steffe L.P. Schemes of action and operation involving composite units. Learning and Individual Differences 4 3 (1992) 259-309
34. Steffe L.P. Children's multiplying schemes. In: Harel G., and Confrey J. (Eds). The development of multiplicative reasoning in the learning of mathematics (1994), State University of New York Press, Albany, NY 3-39
35. Steffe L.P. A new hypothesis concerning children's fractional knowledge. Journal of Mathematical Behavior 20 (2002) 267-307
36. Steffe L.P. Fractional commensurate, composition, and adding schemes: Learning trajectories of Jason and Laura: Grade 5. Journal of Mathematical Behavior 22 (2003) 237-295
37. Steffe L.P. On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning 6 2 (2004) 129-162
38. Steffe L.P., and Cobb P. Construction of arithmetical meanings and strategies (1988), Springer-Verlag, New York
39. Steffe L.P., and Thompson P.W. Teaching experiment methodology: Underlying principles and essential elements. In: Lesh R., and Kelly A.E. (Eds). Handbook of research design in mathematics and science education (2000), Erlbaum, Hillsdale, NJ 267-306
40. Steffe L.P., and Tzur R. Interaction and children's mathematics. In: Ernest P. (Ed). Constructing mathematical knowledge: Epistemology and mathematics education vol. 4 (1994), Falmer, London 8-32
41. In: Steffe L.P., von Glasersfeld E., Richards J., and Cobb P. (Eds). Children's counting types: Philosophy, theory, and application (1983), Praeger Scientific, New York
42. Streefland L. Fractions in realistic mathematics education (1991), Kluwer Academic, Dordrecht, The Netherlands
43. Thompson P.W., and Saldanha L.A. Fractions and multiplicative reasoning. In: Kilpatrick J., Martin W.G., and Schifter D. (Eds). Research companion to the principles and standards for school mathematics (2003), National Council of Teachers of Mathematics, Reston, VA 95-113
44. Tzur R. An integrated study of children's construction of improper fractions and the teacher's role in promoting that learning. Journal for Research in Mathematics Education 30 4 (1999) 390-416
45. Tzur R. Teachers' and students' joint production of a reversible fraction conception. Journal of Mathematical Behavior 23 (2004) 93-114
46. Vergnaud G. Multiplicative structures. In: Behr M.J., and Hiebert J. (Eds). Number concepts and operations in the middle grades (1988), National Council of Teachers of Mathematics, Reston, VA 141-161
47. von Glasersfeld E. Radical constructivism: A way of knowing and learning vol. 6 (1995), Falmer, London