This article describes constructivist principles of learning in geometry specific to children's imaginal anticipations in line measurement and the related area of fractions. To assist with this discussion, seven sixth-grade students classified as learning disabled (LD) and receiving some type of special education service for mathematics were individually tested in tasks that investigated their imaginal anticipations of space and their representations of this understanding in the fraction symbol. Additionally, the investigation examined the effectiveness of constructivist teaching techniques in extending student thinking. All students perceived the most static elements of the line and could represent a metric unit. In the related area of fractions, they interpreted simple fractions as operational units. However, the majority of students were unable to coordinate and, thus, imagine the second-order nested hierarchies that could be inferred from the line. Similarly, they had difficulty coordinating the higher-order nested relationships necessary to interpret equivalent fractions intelligently. The students benefited from instruction that questioned their way of knowing and from manipulatives that served to support their reflections.
References
1.
BallD. L. (1993). Halves, pieces, twoths: Constructing and using representational contexts in teaching fractions. InCarpenterT. P.FennemaE.RombergT. A. (Eds.), Rational numbers: An integration of research (pp. 157–196). Hillsdale, NJ: Erlbaum.
2.
BehrM. J.HarelG.PostT.LeshR. (1993). Rational numbers: Toward a semantic analysis—Emphasis on the operator construct. InCarpenterT. P.FennemaE.RombergT. A. (Eds.), Rational numbers: An integration of research (pp. 13–48). Hillsdale, NJ: Erlbaum.
3.
BleyN. S.ThorntonC. A. (1995). Teaching mathematics to students with learning disabilities. Austin, TX: PRO-ED.
4.
CarnineD.JitendraA. K.SilbertJ. (1997). A descriptive analysis of mathematics curricular materials from a pedagogical perspective: A case of fractions. Remedial and Special Education, 18, 66–81.
5.
ClementsD. H.BattistaM. T. (1992). Geometry and spatial reasoning. InGrouwsD. A. (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). New York: Macmillan.
6.
DeanA. L. (1990). The development of mental imagery: A comparison of Piagetian and cognitive psychological perspectives. Annals of Child Development, 7, 105–144.
7.
DeweyJ. (1933). How we think: A restatement of the relation of reflective thinking to the educational process. New York: D. C. Heath & Co.
8.
DeweyJ. (1934/1980). Art as experience. New York: Perigee Books.
9.
EngelmannS.CarnineD.SteelyD. G. (1991). Making connections in mathematics. Journal of Learning Disabilities, 24, 292–303.
10.
GrobeckerB. (1998). The new science of life and learning differences. Learning Disability Quarterly, 21, 207–227.
11.
GrobeckerB.De LisiR. (2000). A investigation of spatial-geometrical understanding in students with learning disabilities. Learning Disability Quarterly, 23(1), 7–22.
12.
GrossenB.CarnineD. (1996, Summer). Considerate instruction helps students with disabilities achieve world class standards. Teaching Exceptional Children, 77–81.
13.
KamiiC.ClarkF. B. (1995). Equivalent fractions: Their difficulty and educational implications. Journal of Mathematical Behavior, 14, 365–378.
14.
KamiiC.ClarkF. B. (1997). Measurement of length: The need for a better approach to teaching. School Science and Mathematics, 97, 116–121.
15.
KameenuiE. J.CarnineE. W. (1998). Effective teaching strategies that accommodate diverse learners. Upper Saddle River, NJ: Merrill.
16.
KierenT. E. (1993). Rational and fractional numbers: From quotient fields to recursive understanding. InCarpenterT. P.FennemaE.RombergT. A. (Eds.), Rational numbers: An integration of research (pp. 49–84). Hillsdale, NJ: Erlbaum.
17.
LamonS. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal for Research in Mathematics Education, 27, 170–191.
18.
MackN. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, 16–32.
19.
MackN. K. (1993). Learning rational numbers with understanding: The case of informal knowledge. InCarpenterT. P.FennemaE.RombergT. A. (Eds.), Rational numbers: An integration of research (pp. 85–105). Hillsdale, NJ: Erlbaum.
20.
MackN. K. (1998). Building a foundation for understanding the multiplication of fractions. Teaching Children Mathematics, 5, 34–38.
21.
ParmarR. S.FrazitaR.CawleyJ. F. (1996). Mathematics assessment for students with disabilities: An exploration of content validity. Learning Disability Quarterly, 19, 127–136.
22.
PiagetJ. (1980). Adaption and intelligence: Organic selection and phenocopy. Chicago: University of Chicago Press.
23.
PiagetJ. (1981). Intelligence and affectivity: Their relationship during child development. Palo Alto, CA: Annual Reviews Inc.
24.
PiagetJ. (1985). The equilibration of cognitive structures. Chicago: University of Chicago Press.
25.
PiagetJ.Ackermann-ValladaoE.NoschisK. (1987). Necessities involved in length measures. InPiagetJ. (Ed.), Possibility and necessity (Vol. 2) (pp. 49–61). Minneapolis: University of Minnesota Press.
26.
PiagetJ.InhelderB. (1956). The child's conception of space. New York: Norton.
27.
PiagetJ.InhelderB. (1971). Mental imagery in the child. New York: Basic Books.
28.
PiagetJ.InhelderB. (1975). The origin of the idea of chance in children. New York: W. W. Norton.
29.
PiagetJ.InhelderB.SzeminskaA. (1960). The child's conception of geometry. New York: Harper Torchbooks.
30.
PiagetJ.MartiE.CollC. (1987). Sectioning a square. InPiagetJ. (Ed.), Possibility and necessity (Vol. 1) (pp. 39–53). Minneapolis: University of Minnesota Press.
31.
PothierY.SawadaD. (1983). Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14, 307–317.
32.
RiveraD. P. (1997). Mathematics education and students with learning disabilities: Introduction to the special series. Journal of Learning Disabilities, 30, 2–19, 68.
33.
StreeflandL. (1993). Fractions: A realistic approach. InCarpenterT. P.FennemaE.RombergT. A. (Eds.), Rational numbers: An integration of research (pp. 289–326). Hillsdale, NJ: Erlbaum.
34.
ThorntonC. A.LangrallC. W.JonesG. A. (1997). Mathematics instruction for elementary students with learning disabilities. Journal of Learning Disabilities, 30, 143–150.
35.
WheatleyG.CobbP. (1990). Analysis of young children's spatial constructions. InSteffeL. P.WoodT. (Eds.), Transforming early childhood mathematics education: International perspectives (pp. 161–173). Hillsdale, NJ: Erlbaum.
