Building Prealgebra Fluency Through a Self-Managed Practice Intervention: Order of Operations


Behavioral fluency refers to a combination of accuracy and speed that enables students to function proficiently in the learning environment. The present study investigated the effects of a self-managed frequency-building intervention on the behavioral fluency of a critical prealgebra skill in four 6th-grade students. The intervention involved students having access to the PEMDAS (parentheses, exponents, multiplication, division, addition, and subtraction) mnemonic during frequency building. Using an alternating-treatments design, the first experimental condition presented the intervention as three 1-min practice trials with 30 s of feedback delivered immediately after each frequency-building trial ended. The second condition offered one 3-min practice trial with 90 s of feedback once the trial ended. A baseline condition (no practice) had the students engage in a 1-min timed trial with no feedback. The alternating-treatments design demonstrated that three of the four students produced a superior performance within the two intervention conditions when compared to baseline. However, the results did not conclusively show that one frequency-building intervention was superior to the other.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school through college. Department of Education: U.S.

    Google Scholar 

  2. Archer, A., & Hughes, C. (2011). Explicit instruction: Efficient and effective teaching. Guilford Publications.

  3. 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.

    Article  Google Scholar 

  4. Beverly, M., Hughes, J. C., & Hastings, R. P. (2009). What’s the probability of that? Using SAFMEDS to increase undergraduate success with statistical concepts. European Journal of Behavior Analysis, 10, 183–195.

    Article  Google Scholar 

  5. Beverly, M., Hughes, J. C., & Hastings, R. P. (2016). Using SAFMEDS to assist language learners to acquire second-language vocabulary. European Journal of Behavior Analysis, 17, 131–141.

    Article  Google Scholar 

  6. Binder, C. (1996). Behavioral fluency: Evolution of a new paradigm. The Behavior Analyst, 19, 163–197.

  7. Binder, C. (2003). Doesn’t everybody need fluency? Performance Improvement, 42, 14–20.

  8. Brady, K. K., & Kubina, R. M. (2010). Endurance of multiplication fact fluency for students with attention deficit hyperactivity disorder. Behavior Modification, 34, 79–93.

  9. Bullara, D. T., Kimball, J. W., & Cooper, J. O. (1993). An assessment of beginning addition skills following three months without instruction or practice. Journal of Precision Teaching, 11, 11–16.

  10. Burns, M. K., VanDerHeyden, A. M., & Boice, C. H. (2008). Best practices in delivery of intensive academic interventions. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology (5th ed., pp. 1151–1162). National Association of School Psychologists.

  11. CentralReach. (2019). PrecisionX [Computer software].

  12. Chiesa, M., & Robertson, A. (2000). Precision teaching and fluency training: Making maths easier for pupils and teachers. Educational Psychology in Practice, 16, 297–310.

    Article  Google Scholar 

  13. Clark-Carter, D. (2005). Geometric mean. In B. Everitt & D. Howell (Eds.), Encyclopedia of statistics in behavioral science (pp. 744–745). John Wiley & Sons.

  14. Codding, R. S., Burns, M. K., & Lukito, G. (2011). Meta-analysis of mathematic basic-fact fluency interventions: A component analysis. Learning Disabilities Research & Practice, 26, 36–47.

    Article  Google Scholar 

  15. Common Core State Standards Initiative. (2010). Common Core State Standards for mathematics.

  16. Common Core State Standards Initiative. (2018). Common core state standards for mathematics 2010. Common Core State Standards Initiative.

  17. Cooper, J. O., Heron, T. E., & Heward, W. L. (2020). Applied behavior analysis (3rd ed.). Pearson.

  18. Daly, E. J., III., Martens, B. K., Barnett, D., Witt, J. C., & Olson, S. C. (2007). Varying intervention delivery in response to intervention: Confronting and resolving challenges with measurement, instruction, and intensity. School Psychology Review, 36, 562–581.

  19. Datchuk, S. M. (2016). Writing simple sentences and descriptive paragraphs: Effects of an intervention on adolescents with writing difficulties. Journal of Behavioral Education, 25, 166–188.

    Article  Google Scholar 

  20. Deno, S. L., & Mirkin, P. K. (1977). Data-based program modification: A manual. Reston VA: Council for Exceptional Children.

    Google Scholar 

  21. Fitzgerald, D. L., & Garcia, H. I. (2006). Precision teaching in developmental mathematics: Accelerating basic skills. Journal of Precision Teaching and Celeration, 22, 11–28.

    Google Scholar 

  22. Foegen, A., Olson, J. R., & Impecoven-Lind, L. (2008). Developing progress monitoring measures for secondary mathematics: An illustration in algebra. Assessment for Effective Intervention, 33, 240–249.

    Article  Google Scholar 

  23. Fuchs, L. S., Seethaler, P. M., Powell, S. R., Fuchs, D., Hamlett, C. L., & Fletcher, J. M. (2008). Effects of preventative tutoring on the mathematical problem solving of third-grade students with math and reading difficulties. Exceptional Children, 74, 155–173.

    Article  PubMed  PubMed Central  Google Scholar 

  24. 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.

    Article  PubMed  PubMed Central  Google Scholar 

  25. Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77, 81–112.

    Article  Google Scholar 

  26. Hughes, C. A., Korinek, L., & Gorman, J. (1991). Self-management for students with mental retardation in public school settings: A research review. Education & Training in Mental Retardation, 26, 271–291.

    Google Scholar 

  27. Hulac, D. M., Wickerd, G., & Vining, O. (2013). Allowing students to administer their own interventions: An application of the self-administered folding-in technique. Rural Special Education Quarterly, 32, 31–36.

    Article  Google Scholar 

  28. Johnson, K., & Street, E. M. (2013). Response to intervention with precision teaching: Creating synergy in the classroom. Guilford.

  29. Johnston, J. M., & Pennypacker, H. S. (2009). Strategies and tactics of behavioral research (3rd ed.). Routledge.

  30. Kazdin, A. E. (2011). Single-case research designs: Methods for clinical and applied settings (2nd ed.). Oxford University Press.

  31. Kostewicz, D. E., King, S. A., Datchuk, S. M., Brennan, K. M., & Casey, S. D. (2016). Data collection and measurement assessment in behavioral research: 1958–2013. Behavior Analysis Research and Practice, 16, 19–33.

    Article  Google Scholar 

  32. Kubina, R. M. (2019). Precision teaching implementation manual. Greatness Achieved Publishing.

  33. Kubina, R. M., Young, A. E., & Kilwein, M. (2004). Examining an effect of fluency: Application of oral word segmentation and letters sounds for spelling. Learning Disabilities: A Multidisciplinary Journal, 13, 17–23.

    Google Scholar 

  34. Kubina, R. M., & Yurich, K. K. L. (2012). Precision teaching book. Greatness Achieved Publishing.

  35. Ledford, J. R., & Gast, D. L. (Eds.). (2009). Single subject research methodology in behavioral sciences: applications in special education and behavioral sciences. Routledge. 

  36. Ledford, J. R., Barton, E. E., Severini, K. E., & Zimmerman, K. N. (2019). A primer on single-case research design: Contemporary use and analysis. American Journal on Intellectual and Developmental Disabilities, 124, 35–56.

    Article  PubMed  Google Scholar 

  37. Maccini, P., Mulcahy, C. A., & Wilson, M. G. (2007). A follow-up of mathematics interventions for secondary students with learning disabilities. Learning Disabilities Research & Practice, 22, 58–74.

    Article  Google Scholar 

  38. MacDonald, J. E., Wilder, D. A., & Binder, C. (2006). The use of precision teaching techniques to increase mathematics skills in adults with schizophrenia. Journal of Precision Teaching and Celeration, 22, 2–10.

    Google Scholar 

  39. Mace, E C., Belfiore, P. J., & Hutchinson, J. M. (2001). Operant theory and research on self-regulation. In B. Zimmerman & D. Schunk (Eds.), Self-regulated learning and academic achievement (pp. 39–66). Lawrence Erlbaum.

  40. Manalo, E., Bunnell, J. K., & Stillman, J. A. (2000). The use of process mnemonics in teaching students with mathematics learning disabilities. Learning Disability Quarterly, 23, 137–156.

    Article  Google Scholar 

  41. Mastropieri, M. A., & Scruggs, T. E. (1989). Constructing meaningful relationships: Mnemonic instruction for special populations. Educational Psychology Review, 1, 83–111.

    Article  Google Scholar 

  42. Mastropieri, M. A., & Scruggs, T. E. (1998). Enhancing school success with mnemonic strategies. Intervention in School and Clinic, 33, 201–208.

    Article  Google Scholar 

  43. McDougall, D., & Brady, M. P. (1998). Initiating and fading self-management interventions to increase math fluency in general education classes. Exceptional Children, 64, 151–166.

    Article  Google Scholar 

  44. McFarland, J., Hussar, B., Wang, X., Zhang, J., Wang, K., Rathbun, A., Barmer, A., Forrest Cataldi, E., and Bullock Mann, F. (2018). The Condition of Education 2018 (NCES 2018-144). U.S. Department of Education. Washington, DC: National Center for Education Statistics. Retrieved [date] from

  45. McTiernan, A., Holloway, J., Healy, O., & Hogan, M. (2016). A randomized controlled trial of the Morningside math facts curriculum on fluency, stability, endurance, and application outcomes. Journal of Behavioral Education, 25, 49–68.

    Article  Google Scholar 

  46. Miller, A. D., Hall, S. W., & Heward, W. L. (1995). Effects of sequential 1-minute time trials with and without inter-trial feedback and self-correction on general and special education students’ fluency with math facts. Journal of Behavioral Education, 5, 319–345.

    Article  Google Scholar 

  47. National Academies of Sciences, Engineering, and Medicine. (2017). Building America’s skilled technical workforce. The National Academies Press.

  48. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.

  49. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education.

  50. Nelson, P. M., Burns, M. K., Kanive, R., & Ysseldyke, J. E. (2013). Comparison of a math fact rehearsal and a mnemonic strategy approach for improving math fact fluency. Journal of School Psychology, 51, 659–667.

    Article  PubMed  Google Scholar 

  51. Nevin, J. A. (1992). An integrative model for the study of behavior momentum. Journal of the Experimental Analysis of Behavior, 57, 301–316.

    Article  Google Scholar 

  52. Reid, R., Trout, A. L., & Schartz, M. (2005). Self-regulation interventions for children with attention deficit/hyperactivity disorder. Exceptional Children, 71, 361–377.

    Google Scholar 

  53. Rittle-Johnson, B., & Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. In C. Donlan (Ed.), The development of mathematical skills (pp. 75–110). Psychology Press.

  54. 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.

  55. Rivera, D. M., & Bryant, B. R. (1992). Mathematics instruction for students with special needs. Intervention in School and Clinic, 28, 71–86.

    Article  Google Scholar 

  56. Sindelar, P. T., Rosenberg, M. S., & Wilson, R. J. (1985). An adapted alternating treatments design for instructional research. Education and Treatment of Children, 8, 67–76.

    Google Scholar 

  57. Stocker Jr., J. D., Kubina, R. M., Riccomini, P. J., & Mason, A. (2018). Comparing the effects of different timings to build computational and procedural fluency with complex computation. Journal of Evidence-Based Practices for Schools, 16, 206–231.

    Google Scholar 

  58. Stromgren, B., Berg-Mortensen, C., & Tangen, L. (2014). The use of precision teaching to teach basic math facts. European Journal of Behavior Analysis, 15, 225–240.

    Article  Google Scholar 

  59. Wang, X. (2013). Why students choose STEM majors: Motivation, high school learning, and postsecondary context of support. American Educational Research Journal50, 1081–1121.

  60. Witzel, B. S., & Riccomini, P. J. (2007). Optimizing math curriculum to meet the learning needs of students. Preventing School Failure: Alternative Education for Children and Youth, 52, 13–18.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to James D. Stocker Jr.

Ethics declarations

Conflict of interest

James D. Stocker declares no conflict of interest. Richard M. Kubina owns equity in CentralReach. The financial interest has been reviewed by Pennsylvania State University’s Individual Conflict of Interest Committee and is currently being managed by the University.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stocker, J.D., Kubina, R.M. Building Prealgebra Fluency Through a Self-Managed Practice Intervention: Order of Operations. Behav Analysis Practice (2021).

Download citation


  • frequency building
  • mathematics fluency
  • pre-algebra fluency
  • complex computation
  • behavioral fluency
  • feedback
  • self-managed interventions