Reform in mathematics as advocated by the National Council of Teachers of Mathematics (NCTM) has been met with much criticism by special educators. Specifically, researchers are arguing that NCTM reports have neglected the needs of students with learning differences (LD) by failing to attend to the remedial techniques that have demonstrated effectiveness with this population. However, rather than to dismiss the recommendations made, some of the teaching methodologies have been incorporated in traditional, explicitly based instructional programs. This practice is unfortunate because constructivist theory related to the development of logical-mathematical structures of thought holds a completely different philosophical premise than the information-processing perspective. This article will examine constructivist beliefs regarding the evolution of logical-mathematical thought structures so that its potential in meeting the needs of children with LD can be more honestly evaluated.
Get full access to this article
View all access options for this article.
References
1.
AllinderR. (1996). An examination of the relationship between teacher efficacy and curriculum-based measurement and student achievement. Remedial and Special Education, 16, 247–254.
2.
BaroodyA. J. (1987). Children's mathematical thinking. New York: Teacher's College.
3.
BaroodyA. J. (1996). An investigative approach to the mathematics instruction of children classified as learning disabled. In ReidD. K.HreskoW. P.SwansonH. L. (Eds.), Cognition approaches to learning disabilities (3rd ed., pp. 545–615). Austin, TX: PRO-ED.
4.
BaroodyA. J.GinsburgH. P. (1986). The relationship between initial meaningful and mechanical knowledge of arithmetic. In HiebertJ. (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 75–112). Hillsdale, NJ: Erlbaum.
5.
BaroodyA. J.GinsburgH. P. (1991). A cognitive approach to assessing the mathematical difficulties of children labeled as learning disabled. In SwansonH. L. (Ed.), Handbook on the assessment of learning disabilities: Theory, research, and practice (pp. 177–228). Austin, TX: PRO-ED.
6.
BauersfeldH. (1988). Interaction, construction, and knowledge: Alternative perspectives for mathematics education. In GrouwsD. A.CooneyR. J.JonesD. (Eds.), Perspectives on research on effective mathematics teaching (pp. 27–46). Reston, VA: National Council of Teachers of Mathematics.
7.
BauersfeldH. (1995). The structuring of the structures: Development and function of mathematizing as a social practice. In SteffeL. P.GaleJ. (Eds.), Constructivism in education (pp. 137–158). Hillsdale, NJ: Erlbaum.
8.
BleyN. S.ThorntonC. A. (1995). Teaching mathematics to students with learning disabilities. Austin, TX: PRO-ED.
9.
BovetM. C. (1981). Learning research within Piagetian lines. In AxelrodP.ElkindD. (Eds.), Topics in learning and learning disabilities (Vol. 1) (pp. 1–9). Rockville, MD: Aspen Systems Corporation.
10.
BrophyA. J. (1986). Teaching and learning mathematics: Where research should be going. Journal for Research in Mathematics Education, 17, 323–346.
11.
BrophyJ.GoodT. L. (1986). Teacher behavior and student achievement. In WittrockM. C. (Ed.), Handbook of research on teaching, 3rd ed. (pp. 328–375). New York: MacMillian.
12.
BryantB. R.RiveraD. P. (1997). Educational assessment of mathematics skills and abilities. Journal of Learning Disabilities, 30, 57–68.
13.
CapraF. (1982). The turning point: Science, society and the rising culture. New York: Bantam Books.
14.
CarnineD. (1991). Curricular interventions for teaching higher order thinking to all students: Introduction to the special series. Journal of Learning Disabilities, 24, 261–269.
15.
CarnineD. W. (1993, October). Effective teaching for higher cognitive functioning. Educational Technology, 29–33.
16.
CarnineC. W. (1997). Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130–141.
17.
CaseL. P.HarrisK. R.GrahamS. (1992). Improving the mathematical problem-solving skills of students with learning disabilities: Self-regulated strategy development. The Journal of Special Education, 26, 1–19.
18.
ChapmanM. (1988). Constructive evolution: Origins and development of Piaget's thought. New York: Cambridge University Press.
19.
ClarkF. B.KamiiC. (1996). Identification of multiplicative thinking in children in grades 1–5. Journal for Research in Mathematics Education, 27, 41–51.
20.
CobbP.WoodT.YackelE. (1991). Analogies from the philosophy and sociology of science for understanding classroom life. Science Education, 75, 23–44.
21.
CobbP.YackelE.WoodT. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2–33.
22.
ConfreyJ. (1986). A critique of teacher effectiveness research in mathematics education. Journal for Research in Mathematics Education, 17, 347–360.
23.
ConfreyJ. (1994). Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions. In HarelG.ConfreyJ. (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 293–332). Albany: SUNY Press.
24.
ConfreyJ.SmithE. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26, 66–86.
25.
DixonB. (1994). Research guidelines for selecting mathematics curriculum. Effective School Practices, 13, 47–55.
26.
DixonR.CarnineD. (1994). Ideologies, practices, and their implications for special education. The Journal of Special Education, 28, 356–367.
27.
DuckworthE. (1996). The having of wonderful ideas and other essays on teaching and learning. New York: Teachers College Press.
28.
EngelmannS.CarnineD.SteelyD. G. (1991). Making connections in mathematics. Journal of Learning Disabilities, 24, 292–303.
29.
EnglertC. S.TarrantK. L.MariageT. V. (1992). Defining and redefining instructional practice in special education: Perspectives on good teaching. Teacher Education and Special Education, 15, 62–86.
30.
FennemaE.FrankeM. L.CarpenterT. P.CareyD. A. (1993). Using children's mathematical knowledge in instruction. American Educational Research Journal, 30, 555–583.
31.
FuchsL. S.FuchsD.KarnsK.HamlettC. L.DutkaS.KatzaroffM. (1996). The relation between student ability and the quality and effectiveness of explanations. American Educational Research Journal, 33, 631–664.
32.
GearyD. C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362.
33.
GinsburgH. P. (1997). Mathematics learning disabilities: A view from developmental psychology. Journal of Learning Disabilities, 30, 20–33.
34.
GiordanoG. (1993). Fourth invited response: The NCTM standards: A consideration of the benefits. Remedial and Special Education, 14, 28–32.
35.
GrantS. G.PetersonP. L.Shojgreen-DownerA. (1996). Learning to teach mathematics in the context of systematic reform. American Educational Research Journal, 33, 509–541.
36.
GrobeckerB. (1997). Partitioning and unitizing in children with learning differences. Learning Disability Quarterly, 20, 317–335.
37.
GrobeckerB. (1998). The evolution of proportional structures in children with and without learning differences. Poster session presented at the 1998 American Educational Research Association, San Diego, CA.
38.
HatanoG. (1988). Social and motivational basis for mathematical understanding. In SaxeG. B.GearhartM. (Eds.), Children's mathematics (pp. 55–70). San Francisco: Jossey-Bass.
39.
HofmeisterA. M. (1993). Elitism and reform in school mathematics. Remedial and Special Education, 14, 8–13.
40.
HutchinsonN. L. (1993). Second invited response: Students with disabilities and mathematics education reform—Let the dialogue begin. Remedial and Special Education, 14, 20–23.
41.
InhelderB.SinclairH.BovetM. (1974). Learning and the development of cognition. London: Routledge & Kegan Paul.
42.
JitendraA. K.HoffD. (1996). The effects of schema-based instruction on the mathematical word-problem-solving performance of students with learning disabilities. Journal of Learning Disabilities, 29, 422–431.
43.
JitendraA.XinY. P. (1997). Mathematics word-problem-solving instruction for students with mild disabilities and students at risk for math failure: A research synthesis. The Journal of Special Education, 30, 412–438.
44.
KamiiC. (1985). Young children reinvent arithmetic: Implications of Piaget's theory. New York: Teachers College Press.
45.
KamiiC. (1994). Young children continue to reinvent arithmetic: Implications of Piaget's theory. New York: Teachers College Press.
46.
KameenuiE. J.CarnineE. W. (1998). Effective teaching strategies that accommodate diverse learners. Upper Saddle River, NJ: Merrill.
47.
KuhnT. S. (1962). The structure of scientific revolutions. Chicago: University of Chicago Press.
48.
LamonS. J. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal for Research in Mathematics Education, 27, 170–193.
49.
MackN. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education, 21, 16–32.
50.
MackN. K. (1993). Learning rational numbers with understanding: The case of informal knowledge. In CarpenterT. P.FennemaE.RombergT. A. (Eds.), Rational numbers: An integration of research (pp. 85–105). Hillsdale, NJ: Erlbaum.
51.
MeltzerL.ReidD. K. (1994). New directions in the assessment of students with special needs: The shift toward a constructivist approach. The Journal of Special Education, 28, 338–355.
52.
MercerC. D.HarrisC. A.MillerS. P. (1993). First invited response: Reforming reforms in mathematics. Remedial and Special Education, 14, 14–19.
53.
MercerC. D.JordanL.MillerS. P. (1994). Implications of constructivism for teaching math to students with moderate to mild disabilities. The Journal of Special Education, 28, 290–306.
54.
MillerS. P.MercerC. D. (1997). Educational aspects of mathematics disabilities. Journal of Learning Disabilities, 30, 48–56.
55.
MurphyL. A. (1997). Math reform in practice for higher- and lower-achieving children: Perspectives of children, parents, and teachers. Paper presented at the American Education Research Association, Chicago.
56.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
57.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.
58.
National Council of Teachers of Mathematics. (1995). Assessment standards for teaching mathematics. Reston, VA: Author.
59.
ParmarR. S.FrazitaR.CawleyJ. F. (1996). Mathematics assessment for students with mild disabilities: An exploration of content validity. Learning Disability Quarterly, 19, 127–136.
60.
Pascual-LeoneJ. (1976). On learning and development, Piagetian style: A reply to Lafebvre-Pinard. Canadian Psychological Review, 17, 270–288.
61.
PiagetJ. (1971). Biology and knowledge. Chicago: University of Chicago Press.
62.
PiagetJ. (1973). To understand is to invent: The future of education. New York: Grossman.
63.
PiagetJ. (1965). The child's conception of number. New York: W. W. Norton & Co.
64.
PiagetJ. (1980). Adaption and intelligence: Organic intelligence and phenocopy. Chicago: University of Chicago Press.
65.
PiagetJ. (1985). The equilibration of cognitive structures. Chicago: University of Chicago Press.
66.
PiagetJ. (1987a). Possibility and necessity (vol. 1): The role of possibility in cognitive development. Minneapolis: University of Minnesota Press.
67.
PiagetJ. (1987b). Possibility and necessity (vol. 2): The role of necessity in cognitive development. Minneapolis: University of Minnesota Press.
68.
PiagetJ.CollC.MartiE. (1987). Associativity of lengths. In PiagetJ. (Ed.), Possibility and necessity: The role of necessity in cognitive development (pp. 62–77). Minneapolis: University of Minnesota Press.
69.
PostT. R.CramerK. A.BehrM.LeshR.HarelG. (1993). Curriculum implications of research on the learning, teaching, and assessing of rational number concepts. In CarpenterT. P.FennemaE.RombergT. A. (Eds.), Rational numbers: An integration of research (pp. 327–362). Hillsdale, NJ: Erlbaum.
70.
PothierY.SawadaD. (1983). Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14, 307–317.
71.
RiveraD. M. (1993). Third invited response: Examining mathematics reform and the implications for students with mathematics disabilities. Remedial and Special Education, 14, 24–27.
72.
SaxeG. B. (1988). The mathematics of street vendors. Child Development, 59, 1415–1425.
73.
SawadaP.CaleyM. T. (1985). Dissipative structures: New metaphor for becoming in education. Educational Researcher, 14, 13–19.
74.
SchifterD. (1997). Developing operation sense as a foundation for algebra. Paper presented at the American Educational Research Association, Chicago.
75.
SchoenfeldA. H. (1994). A discourse on methods. Journal for Research in Mathematics Education, 25, 697–710.
76.
SherinM. G. (1997). When teaching becomes learning. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago.
77.
SherinM. G.MendezE. P.LouisD. A. (1997). A discipline apart: Mathematics as a challenge for FCL teachers. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago.
78.
SinclairH. (1990). Learning: The interactive reaction of knowledge. In SteffeL. P.WoodT. (Eds.), Transforming children's mathematics education: International perspectives (pp. 19–29). Hillsdale, NJ: Erlbaum.
79.
SinclairH.SinclairA. (1986). Children's mastery of written numerals and the construction of basic number concepts. In HiebertJ. A. (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 59–74). Hillsdale, NJ: Erlbaum.
80.
SteffeL. (1988). Children's construction of number sequences and multiplying schemes. In HiebertJ.BehrM. (Eds.), Number concepts and operations in the middle grades (vol. 2) (pp. 119–140). Reston, VA: The National Council of Teachers of mathematics.
81.
SteffeL. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4, 259–309.
82.
SteffeL. (1994). Children's multiplying schemes. In HarelG.ConfreyJ. (Eds.), The development of multiplicative schemes in the learning of mathematics (pp. 3–39). Albany: State University of New York.
83.
SwansonH. L. (1993). An information processing analysis of learning disabled children's problem solving. American Educational Research Journal, 30, 861–893.
84.
VergnaudG. (1982). A classification of cognitive tasks and operations of thought involved in addition and subtraction problems. In CarpenterT. P.MoserJ. M.RombergT. A. (Eds.), Addition and subtraction: A cognitive perspective (pp. 39–59). Hillsdale, NJ: Erlbaum.
85.
VergnaudG. (1983). Multiplicative structures. In LeshR.LandauM. (Eds.), Acquisition of mathematical concepts and processes (pp. 128–175). New York: Academic Press.
86.
VergnaudG. (1988). Multiplicative structures. In HiebertJ.BehrM. (Eds.), Number concepts and operations in the middle grades (vol. 2) (pp. 141–161). Reston, VA: National Council of Teachers of Mathematics, Inc.
87.
Von GlasersfeldE. (1989). Cognition, construction of knowledge, and teaching. Synthese, 80, 121–140.
88.
Von GlasersfeldE. (1990). Environment and communication. In SteffeL. P.WoodT. (Eds.), Transforming children's mathematics education: International perspectives (pp. 30–38). Hillsdale, NJ: Erlbaum.
89.
Von GlasersfeldE. (1995). A constructivist approach to teaching. In SteffeL. P.GaleJ. (Eds.), Constructivism in education (pp. 3–15). Hillsdale, NJ: Erlbaum.
90.
WansartW. L. (1990). Learning to solve a problem: A micro-analysis of the solution strategies of children with learning disabilities. Journal of Learning Disabilities, 23, 164–170.
91.
YackelE.CobbP.WoodT. (1991). Small-group interactions as a source of learning opportunities in second-grade mathematics. Journal for Research in Mathematics Education, 22, 390–408.
