The current study demonstrates how conceptual and procedural knowledge can be used as a heuristic to better understand student math difficulties in order to develop interventions and lay the groundwork for future research. Math interventions were implemented with two elementary students using a nonexperimental single-case design. One student demonstrated acceptable conceptual understanding but low procedural knowledge, and the other student demonstrated low conceptual understanding in addition to difficulties with procedural knowledge. The mismatched intervention (e.g., procedural for a student who needs a conceptual intervention) was implemented first for both students, followed by the appropriate intervention. The intervention that was identified as the most appropriate was more effective than the mismatched intervention for both students. The mean percentage of nonoverlapping data was 100% for the matched intervention and 16.5% for the mismatched intervention. Suggestions for future research are included.

Alberto, P., Troutman, A. C. (2003). Applied behavior analysis for teachers (6th ed.). Upper Saddle River, NJ: Merrill Prentice Hall.
Google Scholar
Baker, S., Gersten, R., Lee, D. S. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. Elementary School Journal, 103, 5173.
Google Scholar | Crossref | ISI
Bloom, B. S., Hastings, J. T., Madaus, G. F. (1971). Handbook on formative and summative evaluation of student learning. New York, NY: McGaw-Hill.
Google Scholar
Burns, M. K. (2004). Empirical analysis of drill ratio research: Refining the instructional level for drill tasks. Remedial and Special Education, 25, 167175.
Google Scholar | SAGE Journals | ISI
Burns, M. K. (2005). Using incremental rehearsal to practice multiplication facts with children identified as learning disabled in mathematics computation. Education and Treatment of Children, 28, 237249.
Google Scholar
Burns, M. K., Boice, C. H. (2009). Comparison of the relationship between words retained and intelligence for three instructional strategies among students with low IQ. School Psychology Review, 38, 284292.
Google Scholar
Burns, M. K., Codding, R. S., Boice, C. H., Lukito, G. (2010). Meta-analysis of acquisition and fluency math interventions with instructional and frustration level skills: Evidence for a skill by treatment interaction. School Psychology Review, 39, 6983.
Google Scholar | ISI
Burns, M. K., Dean, V. J., Foley, S. (2004). Preteaching unknown key words with incremental rehearsal to improve reading fluency and comprehension with children identified as reading disabled. Journal of School Psychology, 42, 303314.
Google Scholar | Crossref | ISI
Burns, M. K., Klingbeil, D. A. (2010). Assessment of academic skills in math within a problem solving model. In Ervin, R., Gimpel, G., Daly, E., Merrell, K. (Eds.), Practical handbook of school psychology (pp. 8698). New York, NY: Guilford.
Google Scholar
Burns, M. K., VanDerHeyden, A. M., Jiban, C. (2006). Assessing the instructional level for mathematics: A comparison of methods. School Psychology Review, 35, 401418.
Google Scholar | ISI
Canobi, K. C., Reeve, R. A., Pattison, P. E. (2002). Young children’s understanding of addition concepts. Educational Psychology, 22, 513532.
Google Scholar | Crossref
Canobi, K., Reeve, R., Pattison, P. E. (2003). Patterns of knowledge in children’s addition. Developmental Psychology, 39, 521534.
Google Scholar | Crossref | Medline
Chard, D. J., Ketterlin-Geller, L. R., Jitendra, A. (2008). Systems of instruction and assessment to improve mathematics achievement for students with disabilities: The potential and promise of RTI. In Grigorenko, E. L. (Ed.), Educating individuals with disabilities: IDEIA 2004 and beyond (pp. 227248). New York, NY: Springer.
Google Scholar
Cowan, R., Dowker, A., Christakis, A., Bailey, S. (1996). Even more precisely assessing children’s understanding of the order-irrelevance principle. Journal of Experimental Child Psychology, 62, 84101.
Google Scholar | Crossref | Medline
Cronbach, L., Snow, R. (1977). Aptitudes and instructional methods: A handbook for research on interactions. New York, NY: Irvington.
Google Scholar
Cronbach, L. J. (1957). The two disciplines of scientific psychology. American Psychologist, 12, 671684.
Google Scholar | Crossref | ISI
Delazer, M., Benke, T. (1997). Arithmetic facts without meaning. Cortex, 33, 697710.
Google Scholar | Crossref | Medline
Fuchs, L. S., Fuchs, D., Karns, K. (2001). Enhancing kindergarteners’ mathematical development: Effects of peer-assisted learning strategies. Elementary School Journal, 101, 495510.
Google Scholar | Crossref
Fuchs, L. S., Fuchs, D., Prentice, K., Hamlett, C. L., Finelli, R., Courey, S. J. (2004). Enhancing mathematical problem solving among third-grade students with schema-based instruction. Journal of Educational Psychology, 96, 635647.
Google Scholar | Crossref | ISI
Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78, 13431359.
Google Scholar | Crossref | Medline | ISI
Gersten, R., Jordan, N. C., Flojo, J. R. (2005). Early identification and intervention for students with mathematical difficulties. Journal of Learning Disabilities, 38, 293304.
Google Scholar | SAGE Journals
Hanich, L. B., Jordan, N. C., Kaplan, D., Dick, J. (2001). Performance across different areas of mathematical cognition in children with learning difficulties. Journal of Educational Psychology, 93, 615626.
Google Scholar | Crossref | ISI
Haring, N. G., Eaton, M. D. (1978). Systematic instructional technology: An instructional hierarchy. In Haring, N. G., Lovitt, T. C., Eaton, M. D., Hansen, C. L. (Eds.), The fourth R: Research in the classroom (pp. 2340). Columbus, OH: Merrill.
Google Scholar
Hiebert, J., Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In Hiebert, J. (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 127). Hillsdale, NJ: Lawrence Erlbaum.
Google Scholar
Hintze, J. M., Christ, T. J., Keller, L. A. (2002). The generalizability of CBM survey-level mathematics assessments: Just how many samples do we need? School Psychology Review, 31, 514528.
Google Scholar | ISI
Jitendra, A., DiPipi, C. M., Perron-Jones, N. (2002). An exploratory student of schema-based word problem-solving instruction for middle-school students with learning disabilities: An emphasis on conceptual and procedural understanding. Journal of Special Education, 36, 2338.
Google Scholar | SAGE Journals
Jolivette, K., Lingo, A. S., Houchins, D. E., Barton-Arwood, S. M., Shippen, M. E. (2006). Building math fluency for students with developmental disabilities and attentional difficulties using great leaps math. Education and Training in Developmental Disabilities, 41, 392400.
Google Scholar
Kavale, K. A., Forness, S. R. (2000). Policy decisions in special education: The role of meta-analysis. In Gersten, R., Schiller, E. P., Vaughn, S. (Eds.), Contemporary special education research: Syntheses of knowledge base on critical instructional issues (pp. 281326). Mahwah, NJ: Lawrence.
Google Scholar
Kennedy, C. H. (2005). Single-case designs for educational research. Boston, MA: Allyn & Bacon.
Google Scholar
Kilpatrick, J., Swafford, J., Finell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC.: National Academy Press.
Google Scholar
Learning Resources . (n.d.-a). Hands-on standards photo-illustrated lessons for teaching with math manipulatives grades 3–4. Vernon Hills, IL: Author.
Google Scholar
Learning Resources . (n.d.-b). Hands-on standards photo-illustrated lessons for teaching with math manipulatives grades 1–2. Vernon Hills, IL: Author.
Google Scholar
LeFevre, J. A., Smith-Chant, B. L., Fast, L., Skwarchuk, S. L., Sargla, E., Arnup, J. S., . . . Kamawar, D. (2006). What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through grade 2. Journal of Experimental Child Psychology, 93, 285303.
Google Scholar | Crossref | Medline | ISI
Messick, S. (1995). Validity of psychological assessment: Validation of inferences from persons’ responses and performances as scientific inquiry into score meaning. American Psychologist, 50, 741749.
Google Scholar | Crossref | ISI
National Council of Teachers of Mathematics . (2000). Principles and standards for school mathematics. Reston, VA: Author.
Google Scholar
National Council of Teachers of Mathematics . (2006). Curriculum focal points for prekindergarten through grade 9 mathematics: A quest for coherence. Reston, VA: Author.
Google Scholar
National Mathematics Advisory Panel . (2008). Foundations for success: Final report of the national math advisory panel. Washington, DC: U.S. Department of Education.
Google Scholar
Nist, L., Joseph, L. M. (2008). Effectiveness and efficiency of flashcard drill instructional methods on urban first-graders’ word recognition, acquisition, maintenance, and generalization. School Psychology Review, 37, 294208.
Google Scholar | ISI
Northwest Evaluation Association . (2004). Measures of academic progress. Lake Oswego, OR: Author.
Google Scholar
Reschly, D. J., Ysseldyke, J. E. (2002). Paradigm shift: The past is not the future. In Thomas, A., Grimes, J. (Eds.), Best practices in school psychology (4th ed., pp. 321). Bethesda, MD: National Association of School Psychologists.
Google Scholar
Riley-Tillman, T. C., Burns, M. K. (2009). Single case design for measuring response to educational intervention. New York, NY: Guilford.
Google Scholar
Rittle-Johnson, B., Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In Donlan, C. (Ed.), The development of mathematical skill (pp. 75110). Hove, UK: Psychology Press.
Google Scholar
Rittle-Johnson, B., Siegler, R. S., Wagner, M. A. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346362.
Google Scholar | Crossref | ISI
Rivera, D. M., Bryant, B. R. (1992). Mathematics instruction for students with special needs. Intervention in School & Clinic, 28, 7186.
Google Scholar | SAGE Journals
Saffer, N. (1999). Core subjects and your career. Occupational Outlook Quarterly, 43(2), 2640.
Google Scholar
Scruggs, T. E., Mastropieri, M. A. (1998). Summarizing single-subject research: Issues and applications. Behavior Modification, 22, 221242.
Google Scholar | SAGE Journals | ISI
Shinn, M. R. (1989). Curriculum based measurement: Assessing special children. New York, NY: Guilford.
Google Scholar
Stiggins, R. (2005). From formative assessment to assessment for learning: A path to success in standards-based schools. Phi Delta Kappan, 87, 324328.
Google Scholar | SAGE Journals | ISI
Tucker, J. A. (1989). Basic flashcard technique when vocabulary is the goal. Unpublished teaching materials, School of Education, University of Chattanooga. Chattanooga, TN: Author.
Google Scholar
VanDeWalle, J. A., Karp, K. S., Bay-Williams, J. M. (2010). Elementary and middle school mathematics: Teaching developmentally (7th ed.). Boston, MA: Allyn & Bacon.
Google Scholar
Xin, Y. P., Jitendra, A. K., Deatline-Buchman, A. (2005). Effects of mathematical word problem-solving instruction on middle school students with learning problems. Journal of Special Education, 39, 181192.
Google Scholar | SAGE Journals | ISI
Zamarian, L., Lopez-Rolon, A., Delazer, M. (2007). Neuropsychological case studies on arithmetic processing. In Berch, D. B., Mazzocco, M. M. M. (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 245264). Baltimore, MD: Paul Brookes.
Google Scholar
Zentall, S. Z. (2007). Math performance of students with ADHD: Cognitive and behavioral contributors and interventions. In Berch, D. B., Mazzocco, M. M. M. (Eds.), Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (pp. 219244). Baltimore, MD: Paul Brookes.
Google Scholar
View access options

My Account

Welcome
You do not have access to this content.



Chinese Institutions / 中国用户

Click the button below for the full-text content

请点击以下获取该全文

Institutional Access

does not have access to this content.

Purchase Content

24 hours online access to download content

Your Access Options


Purchase

AEI-article-ppv for $15.00