نوع مقاله : پژوهشی
نویسندگان
1 دانشجوی دکتری، روانشناسی تربیتی، دانشگاه خوارزمی، تهران، ایران
2 دانشیار، گروه روانشناسی تربیتی، دانشگاه خوارزمی، تهران، ایران
3 دانشیار، گروه روانشناسی، دانشگاه خوارمی، تهران، ایران
4 استادیار گروه روانشناسی تربیتی دانشگاه خوارزمی، تهران، ایران
چکیده
هدف پژوهش حاضر بررسی کارآمدی رویکرد تلفیقی به آموزش خودکارسازی حقایق ضرب در دانش‌آموزان با ناتوانی ریاضی بود. پژوهش حاضر در قالب یک طرح تک آزمودنی با چند خط پایه هم‌زمان انجام شد. تعداد پنج دانش‌آموز با ناتوانی ریاضی با استفاده از نسخة ایرانی مقیاس هوشی وکسلر - ویرایش پنجم، آزمون ریاضیات ایران کی مت و بر اساس ملاک‌های ورود و خروج از میان تعدادی از دانش‌آموز معرفی شده به مرکز اختلالات یادگیری انتخاب شدند. مداخلة آموزشی شامل آموزش مفهومی حقایق و راهبردهای ضرب با استفاده از آموزش مستقیم و سه حالت بازنمایی دانش (عینی - نمادین - انتزاعی) و خودکارسازی حقایق آموخته شده با استفاده از روش تشخیص - تمرین - تصحیح در ۱۷ جلسه (به طور میانگین ۳۰ دقیقه‌ای) ارائه شد. داده‌ها با استفاده از یک آزمون خود کاری ریاضی جمع‌آوری و با استفاده از نرم‌افزار R به‌صورت دیداری و کمی تحلیل شدند. تحلیل‌های دیداری و شاخص‌های ناهمپوشانی و نیز اندازه‌های اثر به‌دست آمده (بیشترین ۸۹/۲ و کمترین ۴۵/۲) افزایش معنادار در عملکرد را برای هر پنج شرکت‌کننده نشان دادند. همچنین یافته‌ها پایداری خودکاری به‌دست آمده در مرحلة نگه‌داری را نیز تأیید کردند. به نظر میرسد یافته‌های این پژوهش می‌‌تواند رویکردی یکپارچه و مؤثر از آموزش شامل یادگیری مفهومی و یادگیری مبتنی بر حافظه را برای خودکارسازی حقایق ضرب در اختیار معلمان و درمانگران قرار دهد.
کلیدواژه‌ها
عنوان مقاله [English]
Automatization of Basic Multiplication Facts for Children with Mathematics Learning Disability: Integrating Conceptual Learning and Memory-Based Learning
نویسندگان [English]
- Sima Aghaei 1
- Hamidreza Hassanabadi 2
- valiollah farzad 3
- Hadi Keramati 4
1 Ph.D. Student, Educational Psychology, Kharazmi University, Tehran, Iran
2 Associate Professor, Department of Educational Psychology, Kharazmi University, Tehran, Iran
3 Associate Professor, Department of Psychology, Kharazmi University, Tehran, Iran
4 Assistant Professor, Department of Educational Psychology, Kharazmi University, Tehran, Iran
چکیده [English]
The purpose of this study was to examine the efficacy of an integrated approach to automatization of multiplication facts in students with math learning disability. A single case multiple base line design was used for this study. Five students with math learning disability were selected using WISC-VIR, Iran Key-Math and according to some exclusion - inclusion Criteria among students introduced to learning disabilities clinics in Karaj. The intervention including conceptual learning of multiplications facts and strategies using explicit instruction, and concrete-representational-abstract method, and automatization of learned facts using detect-practice-repair method was presented in 17 thirty-minute sessions. Data was gathered using a one-minute mad math fluency test. Gathered data was analyzed visually and quantitatively, using R software. Compared to baseline, non-overlap indices, visual analysis and effect size indices confirmed a significant increase in automatic retrieve of multiplication facts in all 5 participants. The findings of this study can help teachers and SLD therapists to build a good foundation for math proficiency in everybody who has math difficulty.
کلیدواژه‌ها [English]
- Automaticity of Multiplication Facts؛ Direct instruc-tion؛ Concrete – Representational - Abstract Method؛ Detect - Practice-Repair Method
- Cover-Copy - Com-pare Method
اسلامی، پروانه و حسن آبادی، حمیدرضا (1396). مهارت‌های اختصاصی پردازش عدد در دانش‌آموزان پسر با ناتوانی یادگیری ریاضی. فصلنامه کودکان استثنایی، 17(2)، 85-67.
امین زاده، انوشه و حسن آبادی، حمیدرضا (1389). نارسایی‌های شناختی زیربنایی در ناتوانی ریاضی. روانشناسی تحولی، 6(23)، 200-187.
امین زاده، انوشه و حسن آبادی، حمیدرضا (1391). توانایی شاخص‌های آزمون نام بردن احتمالی در پیش‌بینی عملکرد ریاضی. روانشناسی معاصر، 8(1)، 60-47.
بیگدلی، ایمان ا..؛ محمدی فر، محمدعلی؛ رضایی، علی محمد و عبدالحسین‌زاده، عباس (1395). اثر آموزش حل مسئله ریاضی با روش بازی بر توجه، حل مسئله و خودکارآمدی دانش‌آموزان مبتلا به اختلال یادگیری ریاضی، پژوهش در یادگیری آموزشگاهی و مجازی، 4(2)، 56-45.
حسن آبادی، حمیدرضا؛ شریفی، حسن پاشا؛ ایزانلو، بلال و احمدیان نسب، مریم (1399). نسخه ایرانی مقیاس هوشی وکسلر کودکان-ویرایش 5. تهران: ستاد توسعه علوم و فناوری‌های شناختی.
روزبهانی، شهره و حسن آبادی، حمیدرضا (1394). کارکرد اجرایی مرکزی و حلقه واج‌شناختی در دانش‌آموزان ناتوان در حل مسائل کلامی ریاضی. فصلنامه کودکان استثنایی، 15(4)، 20-5.
محمداسماعیل، الهه (1381). آزمون ریاضیات ایران کی مت. تهران: سازمان آموزش و پرورش استثنایی.
یزدانی، محمدجواد؛ حسن آبادی، حمیدرضا؛ کدیور، پروین و عبدالهی، محمدحسین (1398). شناسایی زیرگونه‌های ناتوانی یادگیری ریاضی در دانش‌آموزان ایرانی: رویکرد خوشه‌بندی مدل- مبنا، پژوهش در یادگیری آموزشگاهی و مجازی، 6(4)، 30-9.
Afroditi, T., & Ioannis, A. (2016). “Comparing the Effectiveness of Four Interventions for the Support of Students with Learning Disabilities in Acquiring Arithmetic Combinations of Multiplication and Division.” American Journal of Educational Research, vol. 4, no. 4: 294-301. doi: 10.12691/education-4-4.
Agaliotis, I., (2011). Instruction of Mathematics in Special Education: Nature and Educational management of mathematical difficulties. Athens: Gregori (in Greek).
American Psychiatric Association. (2013). Diagnostic and statistical manual of mental disorders (5th ed.). Arlington, VA: Author.
Archer, A. L., & Hughes, C. A. (2010). Explicit instruction: Effective and efficient teaching. New York, NY: Guilford.
Ardoin, S. P., & Daly, E. J. (2007). Introduction to the Special Series: Close encounters of the instructional kind: How the instructional hierarchy is shaping instructional research 30 years later. Journal of Behavioral Education, 16, 1-6. DOI 10.1007/s10864-006-9027-5.
Aud, S., Hussar, W., Kena, G., Bianco, K., Frohlich, L., Kemp, J., & Tahan, K. (2011). The condition of education 2011 (NCES 2011-033). Washington, DC: National Center for Education Statistics.
Axtell, P. K., McCallum, R. S., Bell, S. M., & Poncy, B. C. (2009). Developing math automaticity using a class-wide fluency building procedure for middle-school students. Psychology in the School, 46, 526–538.
Baker, S., Santoro, L., Chard, D., Fien, H., Park, Y., & Otterstedt, J. (2013). An evaluation of an explicit read aloud intervention in whole-classroom formats in first grade. The Elementary School Journal, 113(3), 331-358.
Baroody, A. (2006). Why children have difficulties mastering the basic number combinations and how to help them. Teaching Children Mathematics, 13(1), 22-31.
Baroody, A. J., Bajwa, N. P., & Eiland, M. (2009). Why can't Johnny remember the basic facts? Developmental Disabilities Research Reviews, 15, 69-79. doi: 10.1002/-ddrr.45.
Barrouillet, P., Fayol, M., & Lathulie`re, E. (1997). Selecting between competitors in multiplication tasks: An explanation of the errors produced by adolescents with learning disabilities. International Journal of Behavioral Development, 21, 253–275.
Bay-Williams, J. M., & Kling, G. (2014). Assessing basic fact fluency. Teaching Children Mathematics, 28(8), 488-497.
Biancarosa, G., & Shanley, L. (2016). What Is Fluency? In the Fluency Construct (pp. 84 1-18). Springer New York.
Binder, C. (2003). Doesn't everybody need fluency? Performance Improvement, 42, 14-20.
Boaler, J. (2015). Fluency without fear: Research evidence on the best ways to learn math facts. You cubed at Stanford University, 1-28. Retrieved from https://www.youcubed.org-/fluencywithout-fear/.
Bulté, I., & Onghena, P. (2013). The single-case data analysis package: Analysing single-case experiments with R software. Journal of Modern Applied Statistical Methods, 12(2), 450-478.
Burns, M. K., Kanive, R., & DeGrande, M. (2012). Effect of a computer-delivered math fact intervention as a supplemental intervention for math in third and fourth grades. Remedial and Special Education, 33, 184–191. https://doi.org/10.1177/0741932510381652.
Burns, M. K., Ysseldyke, J., Nelson, P. M., & Kanive, R. (2015). Number of repetitions required to retain single-digit multiplication math facts for elementary students. School Psychology Quarterly, 30, 398–405. https://doi.-org/10.1037/spq0000097.
Chen, L., Peng, Ch., & Chen. M. (2015). Computing tools for implementing standards for single case designs. Behavior Modification, 39(6), 835-869.
Cholmsky, P. (2011). From acquisition to automaticity: The Reflex solution for math fact mastery. Retrieved from reflexmath. com/assets/doc/Reflex_White_Paper.pdf.
Clarke, B., Nelson, N., & Shanley, L. (2016). Mathematics Fluency—More than the Weekly Timed Test. In the Fluency Construct (pp. 67-89). Springer New York.
Codding, R. S., Burns, M. K., & Lukito, G. (2011). Meta-analysis of mathematics basic-fact fluency interventions. Learning Disabilities Research & Practice, 26(1), 36-47.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed.). Hillsdale, NJ: Erlbaum.
Cooper, J., Heron, T., & Heward, W. (2007). Applied behavior analysis (2nd ed.). Upper Saddle River, NJ: Prentice Hall.
Cowan, R., and Powell, D., (2013). The Contributions of DomainGeneral and Numerical Factors to Third-Grade Arithmetic Skills and Mathematical Learning Disability. Journal of Educational Psychology, 1-16.
Cumming, J.J. & Elkins, J. (1999). Lack of automaticity in the basic addition facts as a characteristic of arithmetic learning problems and instructional needs. Mathematical Cognition, 5(2),149-180. doi:10.1080/-135467999387289.
Davis, B. G. (2013). Research-based spelling: Sitton spelling and word study. Online article accessed 25 February 2013 at: http://eps.-schoolspecialty.com/ downloads /research _papers /series/SSWS_research.pd.
Denckla, M. B., & Rudel, R. G. (1976). Rapid ’automatized’ naming (R.A.N.): Dyslexia differentiated from other learning disabilities. Neuropsychologia, 14, 471–479.
Doadler, C. T., & Fien, H. (2013). Explicit Mathematics Instruction. What Teachers Can Do for Teaching Students with Mathematics Difficulties. Intervention in School and Clinic, 48(5), 276-285.
Fawcett, A. J., & Nicolson, R. I. (1994). Naming speed in children with dyslexia. Journal of Learning Disabilities, 27, 641–646.
Fletcher, J. M., Lyon, G. R., Fuchs, L. S., & Barnes, M. A. (2007). Learning disabilities: From identification to intervention. New York, NY: The Guillford Press.
Flores, M., Hinton, V., & Strozier, S. (2014). Teaching Subtraction and Multiplication with Regrouping Using the Concrete-Representational- Abstract Sequence and Strategic Instruction Model. Learning Disabilities Research & Practice, 19(2), 75-88.
Fuchs L., Fuchs D., Powell S., Seethaler P., Cirino P. & Fletcher J. (2008). Intensive intervention for students with mathematics disabilities: Seven principles of effective practice. Learning Disabilities Quarterly, 31(2): 79–92.
Galen, M. S., & Reitsma, P. (2010). Learning basic addition facts from choosing between alternative answers. Learning and Instruction, 20, 47-60.
Garnett, K., & Fleischner, J. E. (1983). Automatization and basic fact performance of normal and learning-disabled children. Learning Disabilities Quarterly, 6, 223–230.
Gast, D. L., & Ledford, J. R. (2014). Single case research methodology: Applications in special education and behavioral sciences (2nd ed.). New York, NY: Routledge.
Gathercole, S. E., & Adams, A.-M. (1994). Children’s phonological working memory: Contributions of long-term knowledge and rehearsal. Journal of Memory and Language, 33, 672–688.
Geary, D. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37, 4- 15. doi: 10.1177/00222194040-370010201.
Geary, D.C. (1993). Mathematical disabilities: Cognitive, neuropsychological, and genetic components. Psychological Bulletin, 114, 345–362.
Geary, D. C. (2013). Early foundations for mathematics learning and their relations to learning disabilities. Current directions in psychological science, 22, 23-27.
Geary, D. C., & Brown, S. C (1991). Cognitive addition: Strategy choice and speed-of-processing differences in gifted, normal, and mathematically disabled children. Developmental Psychology, 27, 398–406.
Geary, D. C., & Widaman, K. F. (1992). Numerical cognition: On the convergence of componential and psychometric models. Intelligence, 16, 47–80.
Geary, D. C., Hamson, C. O., & Hoard, M. K. (2000). umerical and arithmetical cognition: A longitudinal study of process and concept deficits in children with learning disability. Journal of Experimental ChildPsychology, 77, 236–263.
Geary, D. C., Hoard, M. K., & Bailey, D. H. (2012). Fact retrieval deficits in low achieving children and children with mathematical learning disabilities. Journal of Learning Disabilities, 45, 291-307. doi: 10.1177/-0022219-410392046.
Geary, D. C., Widaman, K. F., Little, T. D., & Cormier, P. (1987). Cognitive addition: Comparison of learning disabled and academically normal elementary school children. Cognitive Development, 2, 249–269.
Gersten, R., Beckmann, S., Clarke, B., Foegen, A., March, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (Practice Guide Report No. NCEE 2009-4060). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education.
Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., & Flojo, J. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79, 1202–1242.
Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics disabilities, Journal of Learning Disabilities, 38, 293–304.
Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to intervention (RtI) for elementary and middle schools (NCEE 2009-4060). National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. http://ies.ed.gov/ncee/wwc/ publications/-practiceguides
Gojak, L. M. (2012). Fluency: Simply fast and accurate? I think not! National Council of Teachers of Mathematics. Retrieved from http://www.nctm.org/News-and-Calendar/ Messages-from-the-President/Archive/ Linda-M_-Gojak/Fluency_-Simply-Fast-and Accurate_-I-Think-Not! /
Goldman, S.R., Pellegrino, J.W. & Mertz, D.L. (1988). Extended practice of basic addition facts: Strategy changes in learning disabled students. Cognition and Instruction, 5, 2 2 3 - 2 6 5.
Griffin S., (2004). Teaching Number Sense. Improving Achievement in Math and Science, 61(5), 39-42.
Hanich, L., 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(3), 615- 626.
Haring, N. G., & Eaton, M. D. (1978). Systematic instructional procedures: An instructional hierarchy. In N. G. Haring, T. C. Lovitt, M. D. Eaton, & C. L. Hansen (Eds.), The fourth R: Research in the classroom (pp. 23–40). Columbus OH: Merrill.
Hasselbring, T. S., & Goin, L. I. (1988). Use of Computers. Chapter Ten.
Hasselbring, T., Lott, A., & Zydney, J. (2005). T e c h n o l o g y - supported math instruction for students with disabilities: Two decades of research and development. Retrieved December 12, 2005, from http://www.citedu-cation.org/ mathmatrix/default.asp.
Hawkins, R. O., Collins, T., Hernan, C., & Flowers, E. (2017). Using computer-assisted instruction to build math fact fluency: An implementation guide. Intervention in School and Clinic, 52, 141–147. https://doi.-org/10.1177/1053451216644827.
Hedges, L. V., Pustejovsky, J. E., & Shadish, W. R. (2012). A standardized mean difference effect size for single case designs. Research Synthesis Methods, 3, 224-239. Hedges, L. V., Pustejovsky, J. E., & Shadish, W. R. (2013). A standardized mean difference effect size for multiple baseline designs across individuals. Research Synthesis Methods, 4, 324-341.
Hoard, M., Geary, D., & Hamson, C. (1999). Numerical and arithmetical cognition: Performance of low- and average-IQ children. Mathematical Cognition, 5, 65 – 91.
Hudson, S., Kaden, S., Lavin, K., & Vasquez, T. (2010). Improving basic math skills using technology. Online Submission (ED512698). Retrieved from http://www.eric.ed.gov/-contentdelivery/servlet/ERICServlet?-accno-=ED512698.
Isaacs, A., & Carroll, W. (1999). Strategies for basic fact instruction. Teaching Children Mathematics, 5(9), 508-515.
Jones R.S.P., & Eayrs C.B., (1992). The Use of Errorless Learning Procedures in Teaching People with a Learning Disability. A Critical Review. Mental Handicap Research, 5(2), 204-212.
Jordan, N. C., & Montani, T. O. (1997). Cognitive arithmetic and problem solving: A comparison of children with specific and general mathematics difficulties. Journal of Learning Disabilities, 30, 624–634.
Jordan, N. C., Hanich, L. B., & Kaplan, D. (2003). A longitudinal study of mathematical competencies in children with specific mathematical difficulties verse children with comorbid mathematical and reading difficulties. Child Development, 74, 834 – 850.
Kameenui, E. J., & Simmons, D. C. (1990). Designing instructional strategies: The prevention of academic learning problems. Columbus OH: Charles E. Merrill.
Keller-Margulis, M. A., Shapiro, E. S., & Hintz, J. M. (2008). Long-term diagnostic accuracy of curriculum-based measures in reading and mathematics. School Psychology Review, 37, 374-390.
Ketterlin-Geller, L.R., Chard D.J., & Fien, H., (2008). Making Connections in Mathematics. Conceptual Mathematics Intervention for Low-Performing Students. Remedial and Special Education, 29(1), 33-45.
Kinder,d., Kubina, R., & Marchand-Martella, N., Special Education and Direct Instruction: An Effective Combination. Retrieved March 28, 2016 from http://www.nifdi.-org/docman/-journal-of-direct-instruction-jodi/-volume-5-winter 2005/469-special-education-and-direct-instruction-an-effectiv-e-combination/file.
Kling, G., & Bay-Williams, J. M. (2015). Three steps to mastering multiplication facts. Teaching Children Mathematics, 21(9), 548-559
Kratochwill, T. R., Hitchcock, J., Horner, R. H., Levin, J. R., Odom, S. L., Rindskopf, D. M., & Shadish, W. R. (2010). Single-case designs technical documentation. What Works Clearinghouse.
Ledford, J. R., Barton, E. E., Severini, K. E. & Zimmerman, K. N. (2019). A Primer on Single-Case Research Designs: Contemporary Use and Analysis. American Journal on Intellectual and Developmental Disabilities, Vol. 124, No. 1, 35–56.
LeFevre, J., DeStefano, D., Coleman, B., & Shanahan, T. (2005). Mathematical cognition and working memory. In J. I. D. Campbell (Ed.), Handbook of mathematical cognition. (pp. 361-377) Psychology Press, New York, NY.
Lin, F., & Kubina, R. J. (2005). A preliminary investigation of the relationship between fluency and applicationfor multiplication. Journal of Behavioral Education, 14(2), 73-87.
Lobo, M. A., Moeyaert, M., Baraldi Cunha, A., & Babik, I. (2017) Single-case design, analysis, and quality assessment for intervention research. Journal of Neurologic Physical Therapy: JNPT, 41(3), 187-197.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates.
Manolov, R. (2017). Reporting single-case design studies: Advice in relation to the designs’ methodological and analytical peculiarities. Anuario de Psicoligia, 47(1), 45-55.
Manolov, R., Sierra, V., Solanas, A., & Botella, J. (2014). Assessing functional relations in singlecase designs: quantitative proposals in the context of the evidence-based movement. Behavior Modification, 38(6) 878-913.
Margaret, M. F. (2014). Effects of Two Methods of Teaching Multiplication to Students with Learning Disabilities. Disertasi Dr. Fal, University of Georgia State, Georgia.
Mauro, D., LeFevre, J., & Morris, J. (2003). Effects of problem format on division and multiplication performance: Division facts are mediated via multiplication-based representations. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29(2), 163-170.
Melby-Lervag, M. & Hulme, C. (2012). Is working memory training effective? A meta-analytic review. Developmental Psychology, 49(2), 270-291.
Miller, S.P. and Hudson, P.J., (2007). Using Evidence-Based Practices to Build Mathematics Competence Related to Conceptual, Procedural, and Declarative Knowledge. Learning Disabilities Practice, 22(1), 47-57.
Mong, M. D., & Mong, K.W. (2010). Efficacy of two mathematics interventions for enhancing fluency with elementary students. Journal of Behavioral Education, 19, 273-288.
Montague, M.M., (2007). Self-Regulation and Mathematics Instruction. Learning Disabilities Research & Practice, 22(1), 75- 83
Musti-Rao, S. & Plati, E. (2015). Comparing two classwide interventions: Implications of using technology for increasing multiplication fact fluency. Journal of Behavioral Education, 24(4), 418-437.
National Research Council [NRC]. (2001). Adding it up: Helping children learn mathematics. In Kilpatrick, J. Swafford, & B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
National Council of Teachers of Mathematics [NCTM]. (2017). Procedural Fluency in Mathematics. Retrieved from http:// www.nctm-.org/Standards-and-Positions/ Position Statements/Procedural-Fluency-in Mathematics/
National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2015). Common Core State Standards for mathematics. Washington DC: Author
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: U.S. Department of Education. Retrieved from http://www2.ed.gov/-about/bdscomm/list/mathpanel/report/final- report.pdf
Nelson, P. M., Parker, D. C., & Zaslofsky, A. F. (2016). The relative value of growth in math fact skills across late elementary and middle school. Assessment for Effective Intervention, 41, 184–192. https://doi.org/10.1177/-1534508416634613.
Nelson, P., Burns, M., Kanive, R., & Ysseldyke, J. (2013). Comparison of a mathematics fact rehearsal and a mnemonic strategy approach for improving mathematics fact fluency. Journal of School Psychology, 51(6), 659-667.
O’Connell, S., & SanGiovanni, J., (2013). Putting the practices into actions: Implementing the common core state standards for mathematical practice k-6. Portsmouth, NH: Heinemann.
O'Connell, S., & San Giovanni, J. (2011). Mastering the basic math facts in addition and subtraction. Portsmouth, NH: Heinemann.
Ostad, S. A. (1997). Developmental differences in addition strategies: A comparison of mathematically disabled and mathematically normal children. British Journal of Educational Psychology, 67, 345–357.
Parker, R. I., & Vannest, K. (2009). An improved effect size for single-case research: Nonoverlap of all pairs. Behavior Therapy, 40(4), 357-367.
Parker, R. I., Vannest, K. J., Davis, J. L., & Sauber, S. B. (2011). Combining nonoverlap and trend for single-case research: Tau-U. Behavior Therapy, 42(2), 284-299.
Parker, R. I., Hagan-Burke, S., & Vannest, K. J. (2007). Percentage of all non-overlapping data (PAND): An alternative to PND. The Journal of Special Education, 40(4), 194–204. doi: 10.1177/00224669070400040101.
Parkhurst, J., Skinner, C. H., Yaw, J., Poncy, B., Adcock, W., & Luna, E. (2010). Efficient Class-Wide Remediation: Using Technology to Identify Idiosyncratic Math Facts for Additional Automaticity Drills. International Journal of Behavioral Consultation and Therapy, 6(2), 111-123.
Pellegrino, J.W. & Goldman, S.R. (1987). Information Processing and Elementary Mathematics. Journal of Learning Disabilities, 20(1), 23-32.
Poncy, B.C., Fontenelle, S.F. & Skinner, C.H. (2013). Using detect, practice, and repair (DPR) to differentiate and individualize math fact instruction in a class wide setting. Journal of Behavioral Education, 22(3), 211 228. doi: http://dx.doi.org/10.1007/s10864 013 9171 7.
Poncy, B.C., McCallum, E. & Schmit, A. (2010a). Cover, copy, and compare versus facts that last: Evaluating and comparing a behavioral and a constructivist intervention targeting computational fluency. Psychology in the Schools, 47, 917–930.
Poncy, B.C., McCallum, E. & Schmitt, A.J. (2010). A comparison of behavioral and constructivist interventions for increasing mathematics-fact fluency in a secondgrade classroom. Psychology In the Schools,47(9),917-930.
Poncy, B.C., Skinner, C.H. & Axtell, P.K. (2010b). An investigation of detect, practice, and repair (DPR) to remedy math fact deficits in third-grade students. Psychology in the Schools, 47, 342–353.
Poncy, B.C., Skinner, C.H. & Jaspers, K.E. (2006). Evaluating and comparing interventions designed to enhance math fact accuracy and fluency: Cover, copy, and compare versus taped problems. Journal of Behavioral Education, 16(1), 2737. doi:10.1007/s10864-006-9025-7
Poncy, B.C., Skinner, C.H. & Jaspers, K.E. (2007). Evaluating and comparing interventions designed to enhance math fact accuracy and fluency: Cover, copy, and compare versus taped problems. Journal of Behavioral Education, 16, 27–37
Price, G. R, Mazzocco, M. M. M., & Ansari, D. (2013). Why mental arithmetic counts: Brain activation during single digit arithmetic predicts high school math scores. The Journal of Neuroscience, 33, 156-163. doi: 10.1523/-JNEUROSCI.2936-12.2013
Raghubar, K. P., Barnes, M., & Hecht, S. A. (2010). Working memory and mathematics: A review of developmental, individual difference, and cognitive approaches. Learning and Individual Differences
Rathmell, E. (1978). Using thinking strategies to teach the basic facts. In M. Suydam & R. Reys (Eds.), Developing computational skills: 1978 yearbook of the National Council of Teachers of Mathematics (pp. 13-38). Reston, VA: NCTM.
Riccomini, P. J., Stoker, J. D., & Morano, S. (2017). Implementing an effective mathematics fact fluency practice activity. Teaching Exceptional Children, 49, 318–327.
Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346-362. doi: /10.1037//0022-0663.93.2.346
Rosenshine, B. (2008). Five meanings of direct instruction. Center on Innovation & Improvement. Retrieved from http://www.centerii.-org/search/Resources%5CFiveDirectInstruct.pdf April 15th, 2014.
Rupley, W.H., (2009). Introduction to direct/explicit instruction in reading for the struggling reader: Phonemic awareness, phonics, fluency, vocabulary, and comprehension. Reading & Writing Quarterly, 25, 119-124.
Shapiro, E. S., Keller, M. A., Lutz, J. G., Santoro, L. E., & Hintz, J. M. (2006). Curriculum based measures and performance on state assessment and standardized tests. Journal of Psychoeducational Assessment, 24, 19-35. doi: 10.1177/0734282905285237.
Shin, M., & Bryant, D. P. (2015). A synthesis of mathematical and cognitive performances of students with mathematics learning disabilities. Journal of Learning Disabilities, 48, 96- 112. doi: 10.1177/0022219413508324.
Smith, C. R., Marchand Martella, N., & Martella, R. C. (2011). Assessing the effects of the rocket math program with a primary elementary school student at risk for school failure: A case study. Education & Treatment of Children, 34(2), 247 258. Retrieved from http://pearl.-stkate.edu/login?url=http://search.proquest.com.pearl.stkate.edu/docview/868 177144?-accountid=26879.
Stickney, E. M., Sharp, L. B., & Kenyon, A. S. (2012). Technology-enhanced assessment of math fact automaticity: Patterns of performance for low- and typically achieving students. Assessment for Effective Intervention, 37(2), 84-94. doi:10.1177/1534508411430321.
Tate, R. L., Perdices, M., Rosenkoetter, U., Wakim, D., Godbee, K., Togher, L., & McDonald, S. (2013). Revision of a method quality rating scale for singlecase experimental designs and n of-1 trials: The 15- item Risk of Bias in N-of-1 Trials (RoBiNT) Scale. Neuropsychological Rehabilitation, 23(5), 619-638.
Thornton, C. A. (1978). Emphasizing thinking strategies in basic fact instruction. Journal for Research in Mathematics Education, 9(3), 214-227. doi: 10.2307/748999.
Van de Walle, J., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (8th ed.). Boston, MA: Pearson.
Van Luit J.H. and Schopman E.A.M., (2000). Improving Early Numeracy of Young Children with Special Educational Needs, Remedial and Special Education, 21(1), 27-40.
VanDerHayden, A. M., & Burns, M. K. (2008). Examination of the utility of various measures of mathematics proficiency. Assessment for Intervention, 33, 215-224.
Vaughn, S., Linan-Thompson, S., & Hickman, P., (2003). Instruction as a Means of Identifying Students with Reading/ Learning Disabilities. Exceptional Children, 69(4), 391-409.
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychologyof Education, 24, 335-359. Retrieved from jstor.org/-stable/23421697.
Whitehurst, G. J. (2003). Research on mathematics education. Washington, DC: US Department of Education. Retrieved from ed.gov/-rschstat/research/progs/mathscience/whitehurst.html.
Woodward, J. (2006). Developing automaticity in multiplication facts: Integrating strategy instruction with timed practice drills. Learning Disability Quarterly, 29(4), 269-289.
)