Abstract
In two experiments, we used multiple probe designs to first examine the correlation between the presence of transformation of stimulus function (TSF) across math operants and the presence of TSF across saying and writing for spelling, and then tested the effects of the establishment of TSF across saying and writing words on the establishment of TSF across math operants. Eight middle school students with learning disabilities participated in Experiment 1. Four participants demonstrated TSF across saying and writing for spelling. The dependent variables of Experiment 1 were the accuracy and rate of solving word problems after receiving fluency training on math facts, and the number of counting strategies used when solving word problems. Results showed that all participants who demonstrated TSF demonstrated significant increases in accuracy and rate when solving word problems whereas three participants without TSF did not. Three participants subsequently participated in Experiment 2, which tested the effects of establishment of TSF across saying and writing words on the establishment of TSF across math operants. All three participants demonstrated increased accuracy and rate for word problems and extinction of counting strategies. The results of the two experiments suggested the importance of teaching math as a verbal behavior, in particular, as a speaker-as-own-listener behavior.
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Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.
References
Albers, A. E., & Greer, R. D. (1991). Is the three-term contingency trial a predictor of effective instruction? Journal of Behavioral Education, 1(3), 337–354. https://doi.org/10.1007/BF00947188
Alexander, R. (2006). Towards dialogic teaching: Rethinking classroom talk. Dialogos.
Al-Makahleh, A. A. A. (2011). The effect of direct instruction strategy on math achievement of primary 4th and 5th grade students with learning difficulties. International Education Studies, 4(4), 199–205. https://doi.org/10.5539/ies.v4n4p199
Altman, D. G. (1999). Some common problems in medical research. Practical Statistics for Medical Research. https://doi.org/10.1002/sim.4780101015
Arntzen, E. (2004). Probability of equivalence formation: Familiar stimuli and training sequence. The Psychological Record, 54(2), 275–291. https://doi.org/10.1007/BF03395474
Barnes-Holmes, D., Barnes-Holmes, Y., & Cullinan, V. (2000). Relational frame theory and Skinner’s Verbal behavior: A possible synthesis. The Behavior Analyst, 23(1), 69–84. https://doi.org/10.1007/BF03392000
Barron, B. F., Verkuylen, L., Belisle, J., Paliliunas, D., & Dixon, M. R. (2019). Teaching “then-later” and “here-there” relations to children with autism: An evaluation of single reversals and transformation of stimulus function. Behavior Analysis in Practice, 12(1), 167–175. https://doi.org/10.1007/s40617-018-0216-1
Belanich, J., & Fields, L. (1999). Tactual equivalence class formation and tactual-to-visual cross-modal transfer. The Psychological Record, 49(1), 75–91. https://doi.org/10.1007/BF03395308
Belisle, J., Stanley, C. R., Schmick, A., Dixon, M. R., Alholail, A., Galliford, M. E., & Ellenberger, L. (2020). Establishing arbitrary comparative relations and referential transformations of stimulus function in individuals with autism. Journal of Applied Behavior Analysis, 53(2), 938–955. https://doi.org/10.1002/jaba.655
Blackledge, J. T. (2003). An introduction to relational frame theory: Basics and applications. The Behavior Analyst Today, 3(4), 421–433. https://doi.org/10.1037/h0099997
Boaler, J. (2015). Fluency without fear: Research evidence on the best ways to learn math facts. Reflections, 40(2), 7–12.
Broto, J. A., & Greer, R. D. (2014). The effects of functional writing contingencies on second graders' writing and responding accurately to mathematical algorithms. Behavioral Development Bulletin, 19(1), 7. https://doi.org/10.1037/h0100568
Browder, D. M., Spooner, F., Ahlgrim-Delzell, L., Harris, A. A., & Wakemanxya, S. (2008). A meta-analysis on teaching mathematics to students with significant cognitive disabilities. Exceptional Children, 74(4), 407–432. https://doi.org/10.1177/001440290807400401
Cai, J., & Lester, F. (2010). Why is teaching with problem solving important to student learning. National Council of Teachers of Mathematics, 13(12), 1–6.
Chiesa, M., & Robertson, A. (2000). Precision teaching and fluency training: Making maths easier for pupils and teachers. Educational Psychology in Practice, 16(3), 297–310. https://doi.org/10.1080/713666088
Curriculum Associates. (2021). iReady diagnostic: Linking study with New Jersey Student Learning Assessments (NJSLA). https://www.curriculumassociates.com/-/media/mainsite/files/i-ready/iready-diagnostic-assessments-linking-study-overview-new-jersey-2020.pdf
Delaney, J. M., & Devereux, P. J. (2020). Math matters! The importance of mathematical and verbal skills for degree performance. Economics Letters, 186(108850)
Din, F. S. (1998, March 25–28). Direct Instruction in remedial math instructions. [Conference presentation]. National Conference on Creating the High Quality School, Arlington, VA, United States. https://files.eric.ed.gov/fulltext/ED417955.pdf.
Dougher, M. J., & Markham, M. R. (1996). Stimulus classes and the untrained acquisition of stimulus functions. In T. R. Zentall & P. M. Smeets (Eds.), Stimulus class formation in humans and animals (pp. 137–152). Elsevier Science. https://doi.org/10.1016/S0166-4115(06)80107-X.
Dye, M. W., Hauser, P. C., & Bavelier, D. (2008). Visual skills and cross-modal plasticity in deaf readers: Possible implications for acquiring meaning from print. Annals of the New York Academy of Sciences, 1145, 71–82. https://doi.org/10.1196/annals.1416.013
Dymond, S., & Rehfeldt, R. A. (2000). Understanding complex behavior: The transformation of stimulus functions. The Behavior Analyst, 23, 239–254. https://doi.org/10.1007/BF03392013
Eby, C. M., Greer, R. D., Tullo, L. D., Baker, K. A., & Pauly, R. (2010). Effects of multiple exemplar instruction on transformation of stimulus function across written and vocal spelling responses by students with autism. Journal of Speech & Language Pathology, 5(1), 20–31. https://doi.org/10.1037/h0100262
Fiorile, C. A., & Greer, R. D. (2007). The induction of naming in children with no prior tact responses as a function of multiple exemplar histories of instruction. Analysis of Verbal Behavior, 23(1), 71–87. https://doi.org/10.1007/BF03393048
Firdaus, F. M. (2017). Improving primary students' mathematical literacy through problem based learning and direct instruction. Educational Research Review, 12(4), 212–219. https://doi.org/10.5897/ERR2016.3072
Gilic, L., & Greer, R. D. (2011). Establishing naming in typically develo** two-year-old children as a function of multiple exemplar speaker and listener experiences. Analysis of Verbal Behavior, 27(1), 157–177. https://doi.org/10.1007/BF03393099
Greer, R. D. (1994). A systems analysis of the behaviors of schooling. Journal of Behavioral Education, 4(3), 255–264. https://doi.org/10.1007/BF01531981
Greer, R. D., & McDonough, S. H. (1999). Is the learn unit a fundamental measure of pedagogy? The Behavior Analyst, 22(1), 5–16. https://doi.org/10.1007/BF03391973
Greer, R. D., & Ross, D. E. (2007) Verbal behavior analysis: Inducing and expanding new verbal capabilities in children with language delays. Allyn and Bacon, New York.
Greer, R. D., & Speckman, J. M. (2009). The integration of speaker and listener responses: A theory of verbal development. The Psychological Record, 59(3), 449–488. https://doi.org/10.1007/BF03395674
Greer, R. D., Keohane, D. D., & Healy, O. (2002). Quality and comprehensive applications of behavior analysis to schooling. The Behavior Analyst Today, 3(2), 120. https://doi.org/10.1037/h0099977
Greer, R. D., Stolfi, L., Chavez-Brown, M., & Rivera-Valdez, C. (2005a). The emergence of the listener to speaker component of naming in children as a function of multiple exemplar instruction. Analysis of Verbal Behavior, 21(1), 123–134. https://doi.org/10.1007/BF03393014
Greer, R. D., Yuan, L., & Gautreaux, G. (2005b). Novel dictation and intraverbal responses as a function of a multiple exemplar history. Analysis of Verbal Behavior, 21(1), 99–116. https://doi.org/10.1007/BF03393012
Griffin, C. C., & Jitendra, A. K. (2009). Word problem-solving instruction in inclusive third-grade mathematics classrooms. Journal of Educational Research, 102(3), 187–202. https://doi.org/10.3200/JOER.102.3.187-202
Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684. https://doi.org/10.1037/0022-0663.91.4.684
Horner, R. D., & Baer, D. M. (1978). Multiple-probe technique: A variation on the multiple baseline. Journal of Applied Behavior Analysis, 11(1), 189–196. https://doi.org/10.1901/jaba.1978.11-189
Ingham, P., & Greer, R. D. (1992). Changes in student and teacher responses in observed and generalized settings as a function of supervisor observations. Journal of Applied Behavior Analysis, 25(1), 153–164. https://doi.org/10.1901/jaba.1992.25-153
Invernizzi, M., Johnston, F. R., Bear, D. R., & Templeton, S. (2017). Words their way: Word sorts for within word pattern spellers (3rd ed.). Pearson.
Johnson, K. (2008). Morningside mathematics fluency: Math facts (Vols. 1–6; curriculum program). Seattle, WA: Morningside Press.
Johnston, J. M., & Pennypacker, H. S. (2010). Strategies and tactics of behavioral research. Routledge. https://doi.org/10.4324/9780203837900
Kinder, D., & Carnine, D. (1991). Direct instruction: What it is and what it is becoming. Journal of Behavioral Education, 1(2), 193–213. https://doi.org/10.1007/BF00957004
Kleinert, K. L. (2018). A comparison of bidirectional naming for familiar and non-familiar stimuli and the effects of a repeated probe procedure for first grade students (Doctoral dissertation, Columbia University). https://doi.org/10.7916/D8ZG8871
Kobari-Wright, V. V., & Miguel, C. F. (2014). The effects of listener training on the emergence of categorization and speaker behavior in children with autism. Journal of Applied Behavior Analysis, 47, 1–6. https://doi.org/10.1002/jaba.115
Krawec, J. (2014). Problem representation and mathematical problem solving of students of varying math ability. Journal of Learning Disabilities, 47(2), 103–115. https://doi.org/10.1177/0022219412436976
Lamarre, J., & Holland, J. G. (1985). The functional independence of mands and tacts. Journal of the Experimental Analysis of Behavior, 43(1), 5–19. https://doi.org/10.1901/jeab.1985.43-5
Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29–63. https://doi.org/10.3102/00028312027001029
Lang, P. J. (1985). The cognitive psychophysiology of emotion: Fear and anxiety. In A. H. Tuma & J. D. Maser (Eds.), Anxiety and the anxiety disorders (pp. 131–170). Lawrence Erlbaum Associates, Inc.
Lee, G. P., Miguel, C. F., Darcey, E. K., & Jennings, A. M. (2015). A further evaluation of the effects of listener training on the emergence of speaker behavior and categorization in children with autism. Research in Autism Spectrum Disorders, 30, 286–293. https://doi.org/10.1016/j.rasd.2015.04.007
Lo, C. (2016). How the listener half of naming leads to multiple stimulus control (Doctoral dissertation, Columbia University). https://doi.org/10.7916/D8C82988
Lodhi, S., & Greer, R. D. (1989). The speaker as listener. Journal of the Experimental Analysis of Behavior, 51(3), 353–559. https://doi.org/10.1901/jeab.1989.51-353
Marshall, S. P. (1995). Schemas in problem-solving. Cambridge University Press. https://doi.org/10.1017/CBO9780511527890
Mayfield, K. H., & Vollmer, T. R. (2007). Teaching math skills to at-risk students using home-based peer tutoring. Journal of Applied Behavior Analysis, 40(2), 223–237. https://doi.org/10.1901/jaba.2007.108-05
Miguel, C. F., Petursdottir, A. L., Carr, J. E., & Micheal, J. (2008). The role of naming in stimulus categorization by preschool children. Journal of the Experimental Analysis of Behavior, 89, 383–405. https://doi.org/10.1901/jeab.2008-89-383
Mill, J. S. (1846). A system of logic, ratiocinative and inductive, being a connected view of the principles of evidence and the methods of scientific investigation. Harper & Brothers. https://doi.org/10.1037/11967-000
Morgan, C., Craig, T., Schuette, M., & Wagner, D. (2014). Language and communication in mathematics education: An overview of research in the field. Zdm, 46(6), 843–853. https://doi.org/10.1007/s11858-014-0624-9
Morgan, G., Greer, R. D., & Fienup, D. M. (2020). Descriptive analyses of the relations between bidirectional naming, arbitrary, and non-arbitrary relations. The Psychological Record. https://doi.org/10.1007/s49732-020-00408-z
National Mathematics Advisory Panel. (2008) Foundations for success: Final Report of the National Mathematics Advisory Panel. Washington, DC: United States Department of Education. Retrieved from http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf.
Neef, N. A., Nelles, D. E., Iwata, B. A., & Page, T. J. (2003). Analysis of precurrent skills in solving mathematics story problems. Journal of Applied Behavior Analysis, 36(1), 21–33. https://doi.org/10.1901/jaba.2003.36-21
Neu, J. A., & Greer, R. D. (2019). Fifth graders learn math by observation faster when they observe peers receive corrections. European Journal of Behavior Analysis. https://doi.org/10.1080/15021149.2019.1620044
Ninness, C., Barnes-Holmes, D., Rumph, R., McCuller, G., Ford, A. M., Payne, R., Ninness, S. K., Smith, R. J., Ward, T. A., & Elliott, M. P. (2006). Transformations of mathematical and stimulus functions. Journal of Applied Behavior Analysis, 39(3), 299–321. https://doi.org/10.1901/jaba.2006.139-05
Obama, B. H. (2015, March 23). Remarks by the President at White House Science Fair. The White House. https://obamawhitehouse.archives.gov/the-press-office/2015/03/23/remarks-president-white-house-science-fair
Organisation for Economic Co-operation and Development. (2019). PISA 2018 mathematics framework. In: PISA 2018 assessment and analytical framework. OECD Publishing. https://doi.org/10.1787/13c8a22c-en.
Parker, R. I., Vannest, K. J., & Brown, L. (2009). The improvement rate difference for single-case research. Exceptional Children, 75(2), 135–150.
Phillips, G. W. (2007). Chance favors the prepared Mind: Mathematics and science indicators for comparing states and nations. American Institutes for Research. https://doi.org/10.1037/e538242012-001
Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. New York: Routledge & Kegan Paul.
Pohl, P., Douglas Greer, R., Du, L., & Lee Moschella, J. (2020). Verbal development, behavioral metamorphosis, and the evolution of language. Perspectives on Behavior Science, 43(1), 215–232. https://doi.org/10.1007/s40614-018-00180-0
Powell, S. R. (2011). Solving word problems using schemas: A review of the literature. Learning Disabilities Research & Practice, 26(2), 94–108. https://doi.org/10.1111/j.1540-5826.2011.00329.x
Quine, W., & Van, O. (1960). Word and object: An inquiry into the linguistic mechanisms of objective reference. John Wiley.
Rakap, S. (2015). Effect sizes as result interpretation aids in single-subject experimental research: Description and application of four nonoverlap methods. British Journal of Special Education, 42(1), 11–33. https://doi.org/10.1111/1467-8578.12091
Reys, B., Reys, R. E., & Rubenstein, R. N. P. National Council of Teachers of Mathematics. (2010). Mathematics curriculum: issues, trends, and future directions. Yearbook.
Ross, D. E., & Greer, R. D. (2003). Generalized imitation and the mand: Inducing first instances of functional speech in nonvocal children with autism. Journal of Research in Developmental Disabilities, 24(1), 58–74. https://doi.org/10.1016/S0891-4222(02)00167-1
Singer-Dudek, J., Park, H. S. L., Lee, G. T., & Lo, C. (2017). Establishing the transformation of motivating operations across mands and tacts for preschoolers with developmental delays. Behavioral Development Bulletin, 22(1), 230. https://doi.org/10.1037/bdb0000045
Skinner, B. F. (1957). Verbal behavior. Appleton-Century-Crofts. https://doi.org/10.1037/11256-000
Stafford, M. W., Sundberg, M. L., & Braam, S. J. (1988). A preliminary investigation of the consequences that define the mand and the tact. Analysis of Verbal Behavior, 6(1), 61–71. https://doi.org/10.1007/BF03392829
Strømgren, B., Berg-Mortensen, C., & Tangen, L. (2014). The use of precision teaching to teach basic math facts. European Journal of Behavior Analysis, 15(2), 225–240. https://doi.org/10.1080/15021149.2014.11434723
Tsiouri, I., & Greer, R. D. (2003). Inducing vocal verbal behavior in children with severe language delays through rapid motor imitation responding. Journal of Behavioral Education, 12(3), 185–206. https://doi.org/10.1023/A:1025508311022
U.S. Department of Education (2010). Science, technology, engineering and math: Education for global leadership. https://www.ed.gov/sites/default/files/stem-overview.pdf
Verdun, V. R., Chiasson, B. A., & Fienup, D. M. (2020). At the intersection of derived relations and observational learning: Teaching fraction–percentage relations. Journal of Behavioral Education, 29(4), 741–762. https://doi.org/10.1007/s10864-019-09343-8
Zhang, D., & **n, Y. P. (2012). A follow-up meta-analysis for word-problem-solving interventions for students with mathematics difficulties. Journal of Educational Research, 105(5), 303–318. https://doi.org/10.1080/00220671.2011.627397
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Appendix 1
Appendix 1
Sample Word Problem Worksheet
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Sun, Y., Greer, R.D. Stimulus Control for Making Math Verbal. Psychol Rec 73, 33–52 (2023). https://doi.org/10.1007/s40732-022-00531-z
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DOI: https://doi.org/10.1007/s40732-022-00531-z