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Abstract
A concrete–semiconcrete–abstract (CSA) instructional approach derived from discovery learning (DIS) was embedded in a direct instruction (DI) methodology to teach eight elementary students with math disabilities. One-minute abstract-level probes were the primary metric used to assess student performance on subtraction problems (minuends 0–9). A single-subject, multiple-baseline across-participants design was used to identify the differential effects of discontinuing instruction at crossover (i.e., the point when the correct response rate exceeded the incorrect response rate). Results indicate that the commonly accepted practice of teaching an entire CSA unit of lessons may not be the most efficacious approach for classroom teachers. Instead, through daily data collection and application of the “crossover decision rule” (discontinue rule), teachers can selectively target those students appropriate for additional concrete- and/or semiconcrete-level instruction and those students for whom continued practice at the abstract level is more appropriate. Implications for teaching computational skills are examined.
References
Section:
|
Allsopp, D., Lovin, L., Green, G., Savage-Davis, E. (2003). Why students with special needs have difficulty with learning mathematics and what teachers can do to help. Mathematics Teaching in the Middle School, 8, 308. Google Scholar | |
|
Ausubel, D. P. (1968). Educational psychology: A cognitive view. New York, NY: Holt, Rinehart & Winston. Google Scholar | |
|
Ball, D., Ferrini-Munday, J., Kilpatrick, J., Milgram, J., Schmid, W., Schaar, R. (2005). Reaching for common ground in K-12 mathematics education. Notices of the American Mathematical Society, 52, 1055–1058. Google Scholar | |
|
Barlow, D. H., Hersen, M. (1984). Single case experimental design: Strategies for studying behavior change (2nd ed.). New York, NY: Pergamon. Google Scholar | |
|
Battista, M. C. (1999). The mathematical miseducation of America’s youth: Ignoring research and scientific study in education. Phi Delta Kappan, 80, 424–433. Google Scholar | ISI | |
|
Berch, D., Mazzocco, M. (2007). Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities. Baltimore, MD: Brookes. Google Scholar | |
|
Best, W., Kahn, J. V. (1998). Research in education (8th ed.). Boston, MA: Allyn & Bacon. Google Scholar | |
|
Bley, N. S. (1994). Accommodating special needs. In Thornton, C. A., Bley, N. S. (Eds.), Windows of opportunity: Mathematics for students with special needs (pp. 137–163). Reston, VA: National Council for Teachers of Mathematics. Google Scholar | |
|
Bley, N. S., Thornton, C. A. (1995). Teaching mathematics to students with learning disabilities (3rd ed.). Austin, TX: PRO-ED. Google Scholar | |
|
Bouck, E. C., Kulkarni, G., Johnson, L. (2011). Mathematical performance of students with disabilities in middle school standards-based and traditional curricula. Remedial and Special Education, 32, 429–443. Google Scholar | SAGE Journals | ISI | |
|
Brophy, J., Good, T. L. (1986). Teacher behavior and student achievement. In Whittrock, M. C. (Ed.), Handbook of research on teaching (3rd ed., pp. 328–375). New York, NY: Macmillan. Google Scholar | |
|
Bruner, J. S. (1961). The act of discovery. Harvard Educational Review, 32, 21–32. Google Scholar | |
|
Bruner, J. S. (2004). The process of education. Cambridge, MA: Harvard University Press. Google Scholar | |
|
Campbell, P. F., Rowan, T. E., Suarez, A. R. (1998). What criteria for student-invested algorithms? In Morrow, L. J. (Ed.), The teaching and learning of algorithms in school mathematics (pp. 49–55). Reston, VA: National Council of Teachers of Mathematics. Google Scholar | |
|
Carpenter, T., Fennema, E., Peterson, P., Chiang, C., Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 6, 499–531. Google Scholar | SAGE Journals | |
|
Cates, G. L., Skinner, C. H., Watson, S. T., Meadows, T. J., Weaver, A., Jackson, B. (2003). Instructional effectiveness and instructional efficiency as considerations for data-based decision making: An evaluation of integrating procedures. School Psychology Review, 32, 601–616. Google Scholar | ISI | |
|
Crawford, D., Engelmann, K. E., Engelmann, S. E. (2009). Direct instruction. In Anderman, E. M., Anderman, L. H. (Eds.), Psychology of classroom learning: An encyclopedia (pp. 326–330). New York, NY: Macmillan. Google Scholar | |
|
Dewey, J. (1997). How we think. Mineola, NY: Dover. Google Scholar | |
|
Dienes, Z. D. (1961). On abstraction and generalization. Harvard Educational Review, 31, 281–301. Google Scholar | ISI | |
|
Engelmann, S. E. (1968). Relating operant techniques to programming and teaching. Journal of School Psychology, 6, 89–98. Google Scholar | Crossref | ISI | |
|
Englert, C. S. (1984). Effective direct instruction practices in special education settings. Remedial and Special Education, 5, 38–47. Google Scholar | SAGE Journals | |
|
Fosnot, C., Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division. Portsmouth, NH: Heinemann. Google Scholar | |
|
Fuchs, D., Fuchs, L. S. (2006). Introduction to response to intervention: What, why and how valid is it? Reading Research Quarterly, 41, 93–99. Google Scholar | Crossref | ISI | |
|
Fuchs, L. S., Fuchs, D. (2001). Principles for the prevention and intervention of mathematics difficulties. Learning Disabilities Research & Practice, 16, 85–95. Google Scholar | Crossref | |
|
Funkhouser, C. (1995). Developing number sense and basic computational skills in students with special needs. School Science and Mathematics, 95, 236–239. Google Scholar | Crossref | |
|
Fuson, K. C., Grandau, L., Sugiyama, P. A. (2001). Achievable numerical understanding for all young children. Teaching Children Mathematics, 7, 522–526. Google Scholar | |
|
Gall, M. D., Borg, W. R., Gall, J. P. (1996). Educational research: An introduction (6th ed.). White Plains, NY: Longman. Google Scholar | |
|
Gersten, R., Carnine, D., Woodward, J. (1987). Direct instruction research: The third decade. Remedial and Special Education, 8, 48–56. Google Scholar | SAGE Journals | ISI | |
|
Griffin, S. (2003). Laying the foundations for computational fluency in early childhood. Teaching Children Mathematics, 9, 306–309. Google Scholar | |
|
Hamilton, L., Halverson, R., Jackson, S., Mandinach, E., Supovitz, J., Wayman, J. (2009). Using student achievement data to support instructional decision making (NCEE 2009-4067). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Science, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/pdf/practice_guides/dddm_pg_092909.pdf Google Scholar | |
|
Hendrickson, J. M., Frank, A. R. (1992). Engagement and performance feedback: Enhancing the classroom achievement of students with mild mental disabilities. In Gable, R. A., Warren, S. F. (Eds.), Strategies for teaching students with mild to severe mental retardation (pp. 11–47). Baltimore, MD: Brookes. Google Scholar | |
|
Hudson, P. J., Peterson, S. K., Mercer, C. D., McLeod, P. (1988). Place value instruction. Teaching Exceptional Children, 20, 72–73. Google Scholar | SAGE Journals | |
|
Individuals With Disabilities Education Improvement Act of 2004 , 20 U.S.C. § 1400 et seq. (2004) (reauthorization of the Individuals With Disabilities Education Act of 1990) Google Scholar | |
|
Jitendra, A. K., Salmento, M., Haydt, L. (1999). A case study of subtraction analysis in basal mathematics programs: Adherence to important instructional design criteria. Learning Disabilities Research & Practice, 14, 69–79. Google Scholar | Crossref | |
|
Jones, J. C. (2012). Visualizing elementary and middle school mathematics methods. Hoboken, NJ: Wiley. Google Scholar | |
|
Joyce, B., Weil, M., Callahan, E. (2006). Models of teaching. Boston, MA: Allyn & Bacon. Google Scholar | |
|
Kamii, C., Kirkland, L., Lewis, B. A. (2001). Representation and abstraction in young children’s numerical reasoning. In Cuoco, A. A., Curcio, F. R. (Eds.), The roles of representation in school mathematics (pp. 24–34). Reston, VA: National Council of Teachers of Mathematics. Google Scholar | |
|
Kamii, C., Rummelsburg, J. (2008). Arithmetic for first graders lacking number concepts. Teaching Children Mathematics, 14, 389–394. Google Scholar | |
|
Kanu, C. K., Dominick, A. (1998). To teach or not to teach the algorithms. Journal of Mathematical Behavior, 16, 51–62. Google Scholar | |
|
Karp, K. S., Voltz, D. L. (2000). Weaving mathematical instructional strategies into inclusive settings. Intervention in School and Clinic, 35, 206–215. Google Scholar | SAGE Journals | ISI | |
|
Kavale, K. A., Spalding, L. S. (2008). Is response to intervention good policy for specific learning disability? Learning Disabilities Research & Practice, 2, 169–179. Google Scholar | Crossref | |
|
Keislar, G. R., Shulman, L. S. (1966). Learning by discovery: A critical appraisal. Chicago, IL: Rand McNally Education Series. Google Scholar | |
|
Kilpatrick, J., Swafford, J., Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Research Council. Google Scholar | |
|
Kirschner, P. A., Sweller, J., Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41, 75–86. Google Scholar | Crossref | ISI | |
|
Kroesbergen, E. H., Van Luit, J. E. H. (2003). Mathematical interventions for children with special educational needs. Remedial and Special Education, 24, 97–114. Google Scholar | SAGE Journals | ISI | |
|
Maccini, P., Gagnon, J. C. (2000). Best practices for teaching mathematics to secondary students with special needs. Focus on Exceptional Children, 32, 1–22. Google Scholar | Crossref | |
|
Maccini, P., Gagnon, J. C. (2002). Perceptions and application of NCTM Standards by special and general education teachers. Exceptional Children, 68, 325–344. Google Scholar | SAGE Journals | ISI | |
|
Mayer, R. (2004). Should there be a three-strikes rule against pure discovery learning? The case for guided methods of instruction. American Psychologist, 59, 14–19. Google Scholar | Crossref | Medline | ISI | |
|
Mercer, C. D., Mercer, A. R. (2005). Teaching students with learning problems (7th ed.). Upper Saddle River, NJ: Merrill/Prentice Hall. Google Scholar | |
|
Mercer, C. D., Mercer, A. R., Pullen, P. (2011). Teaching students with learning problems (8th ed.). Upper Saddle River, NJ: Pearson. Google Scholar | |
|
Mercer, C. D., Miller, S. P. (1992). Teaching students with learning problems in math to achieve, understand, and apply basic math facts. Remedial and Special Education, 13, 19–35. Google Scholar | SAGE Journals | ISI | |
|
Moreno, R. (2004). Decreasing cognitive load for novice students: Effects of explanatory versus corrective feedback in discovery-based multimedia. Instructional Science, 32, 99–113. Google Scholar | Crossref | ISI | |
|
Mullins, I. V., Martin, M. O., Gonzalez, E. J., Gregory, K. D., Garden, R. A., O’Connor, K. M. (2000). TIMSS 1999 international mathematics report: Findings from IEA’s report of the third international mathematics and science study at the eighth grade. Chestnut Hill, MA: Boston College. Google Scholar | |
|
Murphy, J., Weil, M., McGreal, T. C. (1986). The basic practice model of instruction. Elementary School Journal, 87, 83–95. Google Scholar | Crossref | ISI | |
|
National Assessment of Educational Progress. ( 2009, September). Retrieved from http://nces.ed.gov/nationsreportcard Google Scholar | |
|
National Council of Supervisors of Mathematics . (1998). Twelve components of essential mathematics. Minneapolis, MN: Author. Google Scholar | |
|
National Council of Teachers of Mathematics . (2000, July). Principles and standards for school mathematics (2000). Availablefrom www.nctm.org Google Scholar | |
|
National Research Council . (2001). Adding it up: Helping children learn mathematics. Washington, DC: Author. Google Scholar | |
|
No Child Left Behind Act of 2001 , 20 U.S.C. 70 § 6319 et seq. (2002). Google Scholar | |
|
O’Brien, T. C. (1999). Parrot math. Phi Delta Kappan, 80, 434–438. Google Scholar | ISI | |
|
Ormrod, J. E. (2008). Human learning (5th ed.). Upper Saddle River, NJ: Pearson/Prentice Hall. Google Scholar | |
|
Piaget, J. (1973). To understand is to invent. New York, NY: Grossman. Google Scholar | |
|
Rebar, M. (2007). Academic acceleration in first grade using the Direct Instruction model (Report No. 2007-1). Cheney, WA: Eastern Washington University. Google Scholar | |
|
Reys, R., Lindquist, M., Lamdin, D., Smith, N. (2012). Helping children learn mathematics (10th ed.). Hoboken, NJ: Wiley. Google Scholar | |
|
Rogers, C. (1969). Freedom to learn: A view of what education might become. Columbus, OH: Charles Merrill. Google Scholar | |
|
Scandura, J. (1964). An analysis of exposition and discovery modes of problem solving instruction. Journal of Experimental Education, 33, 149–159. Google Scholar | Crossref | ISI | |
|
Sealander, K. (2003). Single subject experimental research: An overview for practitioners. In deMarrais, K., Lapan, S. (Eds.), Foundations for research: Methods of inquiry in education and social sciences (pp. 303–307). Mahwah, NJ: Erlbaum. Google Scholar | |
|
Silbert, I., Carnine, D., Stein, M. (1981). Direct instructional mathematics. Columbus, OH: Charles E. Merrill. Google Scholar | |
|
Slavin, R. E. (1991). Synthesis of research on cooperative learning. Educational Leadership, 48, 71–82. Google Scholar | ISI | |
|
Stevens, R., Rosenshine, B. (1981). Advances in research on teaching. Exceptional Education Quarterly, 4, 40–51. Google Scholar | |
|
Swanson, H. L. (2001). Searching for the best model for instructing students with learning disabilities. Focus on Exceptional Children, 34, 1–15. Google Scholar | |
|
Taba, H. (1966). Teaching strategies and cognitive functioning in elementary school children (Cooperative Research Project 2404). San Francisco, CA: San Francisco State University. Google Scholar | |
|
Tamir, P. (1995). Discovery learning and teaching. In Anderson, L. (Ed.), International encyclopedia of teaching and teacher education (2nd ed., pp. 149–155). Oxford, UK: Elsevier Science. Google Scholar | |
|
Tipps, S., Johnson, A., Kennedy, L. (2011). Guiding children’s learning of mathematics (12th ed.). Belmont, CA: Wadsworth. Google Scholar | |
|
Tuovinen, J. E., Sweller, J. (1999). A comparison of cognitive load associated with discovery learning and worked examples. Journal of Educational Psychology, 91, 334–341. Google Scholar | Crossref | ISI | |
|
Van de Walle, J. (2004). Elementary and middle school mathematics: Teaching developmentally. New York, NY: Addison-Wesley/Longman. Google Scholar | |
|
Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). Boston, MA: Allyn & Bacon. Google Scholar | |
|
Villasenor, A., Kepner, H. (1993). Arithmetic from a problem-solving perspective: An urban implementation. Journal for Research in Mathematics Education, 24, 62–69. Google Scholar | Crossref | ISI | |
|
Witzel, B., Mercer, C. D., Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities: Research & Practice, 18, 121–131. Google Scholar | Crossref | |
|
Yang, E. F., Liao, C. Y., Ching, E., Chang, T., Chan, T. (2010). The effectiveness of inductive discovery learning in the mathematics classroom. In Ogata, H., Liu, C.-C., Kinshuk, Biswas, Chee, Y. S., Wong, S. L., Yu, . . . F.-Y. (Eds.), Proceedings of the 18th International Conference on Computers in Education (pp. 743–747). Putrajaya, Malaysia: Asia-Pacific Society for Computers in Education. Google Scholar |
