Abstract
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, 51–73. 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, 167–175. 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, 237–249. 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, 284–292. 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, 69–83. 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, 303–314. 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. 86–98). 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, 401–418. Google Scholar | ISI | |
|
Canobi, K. C., Reeve, R. A., Pattison, P. E. (2002). Young children’s understanding of addition concepts. Educational Psychology, 22, 513–532. Google Scholar | Crossref | |
|
Canobi, K., Reeve, R., Pattison, P. E. (2003). Patterns of knowledge in children’s addition. Developmental Psychology, 39, 521–534. 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. 227–248). 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, 84–101. 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, 671–684. Google Scholar | Crossref | ISI | |
|
Delazer, M., Benke, T. (1997). Arithmetic facts without meaning. Cortex, 33, 697–710. 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, 495–510. 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, 635–647. 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, 1343–1359. 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, 293–304. 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, 615–626. 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. 23–40). 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. 1–27). 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, 514–528. 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, 23–38. 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, 392–400. 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. 281–326). 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, 285–303. 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, 741–749. 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, 294–208. 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. 3–21). 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. 75–110). 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, 346–362. Google Scholar | Crossref | ISI | |
|
Rivera, D. M., Bryant, B. R. (1992). Mathematics instruction for students with special needs. Intervention in School & Clinic, 28, 71–86. Google Scholar | SAGE Journals | |
|
Saffer, N. (1999). Core subjects and your career. Occupational Outlook Quarterly, 43(2), 26–40. Google Scholar | |
|
Scruggs, T. E., Mastropieri, M. A. (1998). Summarizing single-subject research: Issues and applications. Behavior Modification, 22, 221–242. 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, 324–328. 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, 181–192. 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. 245–264). 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. 219–244). Baltimore, MD: Paul Brookes. Google Scholar |

