In this article we consider issues relevant to the future of mathematics instruction and achievement for students with learning disabilities. The starting point for envisioning the future is the changing standards for mathematics learning and basic mathematical literacy. We argue that the shift from behaviorist learning theories to constructivist and social constructivist theories (see Rivera, this series) provides an opportunity to develop and implement a hybrid model of mathematics instruction. The hybrid model we propose embeds, or situates, important skill learning in meaningful contexts. We discuss some examples of instructional approaches to complex mathematical problem solving that make use of meaningful contexts. Evaluation data on these approaches have yielded positive and encouraging results for students with learning disabilities as well as general education students. Finally, we discuss various ways in which technology is important for realizing hybrid instructional models in mathematics.
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References
1.
Ackerman, P. T., Anhalt, J. M., & Dykman, R. A. (1986). Arithmetic automatization failure in children with attention and reading disorders: Associations and sequelae. Journal of Learning Disabilities, 19, 222–232.
2.
Anderson, J. R. (1982). Acquisition of cognitive skill. Psychological Review, 89, 369–406.
3.
Anderson, J. R. (1987). Skill acquisition: Compilation of weak-method problem solutions. Psychological Review, 94, 192–210.
4.
Babbitt, B. C., & Miller, S. P. (1996). Using hypermedia to improve the mathematics problem-solving skills of students with learning disabilities. Journal of Learning Disabilities, 29, 391–101, 412.
5.
Barron, B., Vye, N. J., Zech, L., Schwartz, D., Bransford, J. D., Goldman, S. R., Pellegrino, J., Morris, J., Garrison, S., & Kantor, R. (1995). Creating contexts for community-based problem solving: The Jasper Challenge Series. In C. N. Hedley, P. Antonacci, & M. Rabinowitz (Eds.), Thinking and literacy: The mind at work (pp. 47–71). Hillsdale, NJ: Erlbaum.
6.
Blumenfeld, P. C., Soloway, E., Marx, R. W., Krajcik, J. S., Guzdial, M., & Palincsar, A. (1991). Motivating project-based learning: Sustaining the doing, supporting the learning. Educational Psychologist, 26, 369–398.
7.
Bottge, B. A., & Hasselbring, T. S. (1993). A comparison of two approaches for teaching complex, authentic mathematics problems to adolescents with learning difficulties. Exceptional Children, 59, 556–566.
8.
Bransford, J. D., Franks, J. J., Morris, C. D., & Stein, B. S. (1979). Some general constraints on learning and research. In L. S. Cermak & F. I. M. Craik (Eds.), Levels of processing and human memory (pp. 331–354). Hillsdale, NJ: Erlbaum.
9.
Bransford, J. D., Goldman, S. R., & Hasselbring, T. S. (1995, April). Marrying constructivist and skills-based approaches: Could we, should we, and can technology help?Symposium presented at the annual meeting of the American Educational Research Association, San Francisco.
10.
Bransford, J. D., Goldman, S. R., & Vye, N. J. (1991). Making a difference in people's abilities to think: Reflections on a decade of work and some hopes for the future. In L. Okagaki & R. J. Sternberg (Eds.), Directors of development: Influences on children (pp. 147–180). Hillsdale, NJ: Erlbaum.
11.
Bransford, J. D., Sherwood, R. S., Vye, N. J., & Rieser, J. (1986). Teaching thinking and problem solving: Research foundations. American Psychologist, 41, 1078–1089.
12.
Bransford, J. D., Sharp, D. M., Vye, N. J., Goldman, S. R., Hasselbring, T. S., Goin, L., O'Banion, K., Livernois, J., Saul, E., & the Cognition and Technology Group at Vanderbilt. (1996). MOST environments for accelerating literacy development. In S. Vosniadou, E. de Corte, R. Glaser, & H. Mandl (Eds.), International perspectives on the psychological foundations of technology-based learning environments (pp. 223–256). Hillsdale, NJ: Erlbaum.
13.
Brown, A. L., & Campione, J. C. (1990). Communities of learning and thinking or a context by any other name. Human Development, 21, 108–125.
14.
Brown, A. L., & Campione, J. C. (1994). Guided discovery in a community of learners. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 229–272). Cambridge, MA: MIT Press.
15.
Brown, C., Carpenter, T., Kouba, V., Lindquist, M., Silver, E., & Swafford, J. (1988a, April). Secondary school results for the fourth NAEP assessment: Discrete mathematics, data organization and interpretation, measurement, number and operations. Mathematics Teachers, 81, 241–248.
16.
Brown, C., Carpenter, T., Kouba, V., Lindquist, M., Silver, E., & Swafford, J. (1988b, May). Secondary school results for the fourth NAEP assessment: Algebra, geometry, mathematical methods and attitudes. Mathematics Teachers, 81, 337–347, 397.
17.
Bruner, J. (1960). The process of education: A landmark in educational theory.Cambridge, MA: Harvard University.
18.
Carnine, D., Jones, E., & Dixon, R. (1994). Mathematics: Educational tools for diverse learners. School Psychology Review, 23, 406–427.
19.
Cawley, J. F., & Miller, J. H. (1989). Cross-sectional comparisons of the mathematical performance of children with learning disabilities: Are we on the right track toward comprehensive programming?. Journal of Learning Disabilities, 23, 250–254, 259.
20.
Charles R., & Silver, E. A. (Eds.). (1988). The teaching and assessing of mathematical problem solving.Hillsdale, NJ: Erlbaum.
21.
Cognition and Technology Group at Van-derbilt. (1990). Anchored instruction and its relationship to situated cognition. Educational Researcher, 19, (6), 2–10.
22.
Cognition and Technology Group at Van-derbilt. (1992a). An anchored instruction approach to cognitive skills acquisition and intelligent tutoring. In J. W. Regian & V. J. Shute (Eds.), Cognitive approaches to automated instruction (pp. 135–170). Hillsdale, NJ: Erlbaum.
23.
Cognition and Technology Group at Van-derbilt. (1992b). The Jasper experiment: An exploration of issues in learning and instructional design. Educational Technology Research and Development, 40, 65–80.
24.
Cognition and Technology Group at Van-derbilt. (1992c). The Jasper series as an example of anchored instruction: Theory, program description, and assessment data. Educational Psychologist, 27, 291–315.
25.
Cognition and Technology Group at Van-derbilt. (1993a). Anchored instruction and situated cognition revisited. Educational Technology, 33, 52–70.
26.
Cognition and Technology Group at Van-derbilt. (1993b). The Jasper series: Theoretical foundations and data on problem solving and transfer. In L. A. Penner, G. M. Batsche, H. M. Knoff, & D. L. Nelson (Eds.), The challenges in mathematics and science education: Psychology's response (pp. 113–152). Washington, DC: American Psychological Association.
27.
Cognition and Technology Group at Van-derbilt. (1994a). From visual word problems to learning communities: Changing conceptions of cognitive research. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 157–200). Cambridge, MA: MIT Press.
28.
Cognition and Technology Group at Van-derbilt. (1994b). Multimedia environments for developing literacy in at-risk students. In B. Means (Ed.), Technology and educational reform: The reality behind the promise (pp. 23–56). San Francisco: Jossey-Bass.
29.
Cognition and Technology Group at Van-derbilt. (1996a). Looking at technology in context: A framework for understanding technology and education. In D. C. Berliner & R. C. Calfee (Eds.), The handbook of educational psychology (pp. 807–840). New York: Macmillan.
30.
Cognition and Technology Group at Van-derbilt. (1996b). Multimedia environments for enhancing learning in mathematics. In S. Vosniadou, E. de Corte, R. Glaser, & H. Mandl (Eds.), International perspectives on the psychological foundations of technology-based learning environments (pp. 285–306). Hillsdale, NJ: Erlbaum.
31.
Cognition and Technology Group at Van-derbilt. (in press-a). The Jasper project: Lessons in curriculum, instruction, assessment, and professional development.Mahwah, NJ: Erlbaum.
32.
Cognition and Technology Group at Van-derbilt. (in press-b). The Jasper series: A design experiment in complex, mathematical problem-solving. In J. Hawkins & A. Collins (Eds.), Design experiments: Integrating technologies into schools.New York: Cambridge University.
33.
Collins, A., Hawkins, J., & Carver, S. M. (1991). A cognitive apprenticeship for disadvantaged students. In B. Means, C. Chelemer, & M. S. Knapp (Eds.), Teaching advanced skills to at-risk students (pp. 216–243). San Francisco: Jossey-Bass.
34.
Fleischner, J. E., Garnett, K., & Shepherd, M. (1982). Proficiency in arithmetic basic fact computation by learning disabled and nondisabled children. Focus on Learning Problems in Mathematics, 4, 47–55.
35.
Fuchs, L. S., Fuchs, D., & Hamlett, C. L. (1992). Computer applications to facilitate curriculum-based measurement. Teaching Exceptional Children, 24, (4), 58–60.
36.
Fuchs, L. S., Fuchs, D., Hamlett, C. L., & Stecker, P. M. (1991). Effects of curriculum-based measurement on teacher planning and student achievement in mathematics operations. American Educational Research Journal, 28, 617–641.
37.
Gagné, R. M. (1968). Contributions of learning to human development. Psychological Review, 75, 177–191.
38.
Ginsburg, H. (1977). Children's arithmetic: The learning process.New York: D. Van Nostrand.
39.
Goldman, S. R., Mertz, D. L., & Pellegrino, J. W. (1989). Individual differences in extended practice functions and solution strategies for basic addition facts. Journal of Educational Psychology, 81, 481–496.
40.
Goldman, S. R., & Pellegrino, J. W. (1986). Effective drill and practice on the microcomputer. Academic Therapy, 22, 133–140.
41.
Goldman, S. R., & Pellegrino, J. W. (1987). Information processing and educational microcomputer technology: Where do we go from here?. Journal of Learning Disabilities, 20, 144–154.
42.
Goldman, S. R., Pellegrino, J. W., & Mertz, D. L. (1988). Extended practice of basic addition facts: Strategy changes in learning disabled students. Cognition & Instruction, 5, 223–265.
Graham, S., & Harris, K. (1994). Implications of constructivism for teaching writing to students with special needs. The Journal of Special Education, 28, 275–289.
45.
Hasselbring, T. S., Goin, L. I., Alcantara, P., & Bransford, J. D. (1991, April). Developing mathematical fluency in children with learning handicaps. Paper presented at the annual meeting of the American Educational Research Association, Boston.
46.
Hasselbring, T. S., Goin, L., & Bransford, J. D. (1988). Developing math automaticity in learning handicapped children: The role of computerized drill and practice. Focus on Exceptional Children, 20, (6), 1–7.
47.
Hasselbring, T. S., Sherwood, R. D., Bransford, J. D., Mertz, J., Estes, B., Marsh, J., & Van Haneghan, J. (1991). An evaluation of specific videodisc courseware on student learning in a rural school environment (Technical Report). Nashville, TN: Van-derbilt University, Learning Technology Center.
48.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Erlbaum.
49.
Hofmeister, A. M. (1993). Elitism and reform in school mathematics. Remedial and Special Education, 14, (6), 8–13.
50.
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.
51.
Lesh, R. (1981). Applied mathematical problem solving. Educational Studies in Mathematics, 12, 235–264.
52.
Mercer, C. D., Harris, C. A., & Miller, S. P. (1993). Reforming reforms in mathematics. Remedial and Special Education, 24, (6), 14–19.
53.
Morris, C. D., Bransford, J. D., & Franks, J. J. (1979). Levels of processing versus transfer appropriate processing. Journal of Verbal Learning and Verbal Behavior, 16, 519–533.
54.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics.Reston, VA: Author.
55.
New Standards. (1995). New Standards portfolio field trial: 1995–96 workshop edition, high school.Rochester, NY: Author.
56.
Nickerson, R. S. (1988). On improving thinking through instruction. Review of Research in Education, 15, 3–57.
57.
Pellegrino, J. W., & Goldman, S. R. (1987). Information processing and elementary mathematics. Journal of Learning Disabilities, 20, 23–32, 57.
58.
Pellegrino, J. W., Hickey, D., Heath, A., Rewey, K., Vye, N. J., & Cognition and Technology Group at Vanderbilt. (1991). Assessing the outcomes of an innovative instructional program: The 1990–1991 implementation of the “Adventures of Jasper Woodbury” (Tech. Rep. No. 91–1). Nashville, TN: Vanderbilt University, Learning Technology Center.
59.
Peterson, P. L., Fennema, E., & Carpenter, T. (1991). Using children's mathematical knowledge. In B. Means, C. Chelemer, & M. S. Knapp (Eds.), Teaching advanced skills to at-risk students (pp. 68–111). San Francisco: Jossey-Bass.
60.
Porter, A. (1989). A curriculum out of balance: The case of elementary school mathematics. Educational Researcher, 18, 9–15.
61.
Resnick, L. (1987). Education and learning to think.Washington, DC: National Academy Press.
62.
Resnick, L. B., Bill, V. L., Lesgold, S. B., & Leer, M. N. (1991). Thinking in arithmetic class. In B. Means, C. Chelemer, & M. S. Knapp (Eds.), Teaching advanced skills to at-risk students (pp. 27–53). San Francisco: Jossey-Bass.
63.
Resnick, L. B., & Klopfer, L. E. (Eds.). (1989). Toward the thinking curriculum: Current cognitive research.Alexandria, VA: Association for Supervision and Curriculum Development.
64.
Rivera, D. (1993). Examining mathematics reform and the implications for students with mathematical disabilities. Remedial and Special Education, 14, (6), 24–27.
65.
Russell, R. L., & Ginsburg, H. P. (1984). Cognitive analysis of children's mathematics difficulties. Cognition & Instruction, 1, 217–244.
66.
Scardamalia, M., & Bereiter, C. (1991). Higher levels of agency for children in knowledge building: A challenge for the design of new knowledge media. Journal of the Learning Sciences, 1, 37–68.
67.
Schoenfeld, A. H. (1989). Teaching mathematical thinking and problem solving. In L. B. Resnick & L. E. Klopfer (Eds.), Toward the thinking curriculum: Current cognitive research (pp. 83–103). Alexandria, VA: ASCD.
68.
Silver, E. A. (1986). Using conceptual and procedural knowledge: A focus on relationships. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 181–189). Hillsdale, NJ: Erlbaum.
69.
U.S. Department of Labor. (1992). Secretary's Commission on Achieving Necessary Skills report for America 2000.Washington, DC: Author.
70.
Vye, N. J., Sharp, D. M., McCabe, K., & Bransford, J. D. (1991). Commentary: Thinking in arithmetic class. In B. Means, C. Chelemer, & M. S. Knapp (Eds.), Teaching advanced skills to at-risk students (pp. 54–67). San Francisco: Jossey-Bass.
71.
Whitehead, A. N. (1929). The aims of education.New York: Macmillan.
