Providing multiple representations of mathematical ideas is an established strategy for supporting students in making connections between high-level conceptual ideas. One form of assistive technology that provides students with such representations is mathematics manipulatives. Virtual manipulatives, in particular, are backed by a growing research base for supporting students with special needs. However, their application and prospective benefits across many curricular domains are still unknown, including critical skills such as computations with fractions. Using a single-subject multiple-probe design, this study examined an intervention featuring explicit instruction and a virtual manipulative number line to teach computations with fractions to three middle school students with mathematics difficulties. All three students’ data exhibited a functional relation between the intervention and their accuracy performance in solving computations. These findings and the implications of using virtual representations in mathematics education for struggling learners are discussed.
AgrawalJ.MorinL. L. (2016). Evidence-based practices: applications of concrete representational abstract framework across math concepts for students with disabilities. Learn Disabil Res Pract, 31(1), 34–44. https://doi.org/10.1111/ldrp.12093
2.
BaileyD. H.HoardM. K.NugentL.GearyD. C. (2012). Competence with fractions predicts gains in mathematics achievement. J Exp Child Psychol, 113(3), 447–55. https://doi.org/10.1016/j.cedpsych.2012.07.001
3.
BarbieriC. A.YoungL. K.NewtonK. J.BoothJ. L. (2021). Predicting middle school profiles of algebra performance using fraction knowledge. Child Dev, 92(5), 1984–2005. https://doi.org/10.1111/cdev.13568
4.
BillmanR. H.ReganK.EvmenovaA.SatsangiR. (2023). Using virtual manipulatives with technology-based graphic organizers to support students in solving proportion word problems. Int J Res Learn Disabil, 6(1), 28–43. https://journals.ub.uni-koeln.de/index.php/IJRLD/article/view/1827
5.
BoneE. K.BouckE. C.SatsangiR. (2023). Comparing concrete and virtual manipulatives to teach algebra to middle school students with disabilities. Exceptionality, 31(1), 1–17. https://doi.org/10.1080/09362835.2021.1938057
6.
BouckE. C.ParkJ. (2018). A systematic review of the literature on mathematics manipulatives to support students with disabilities. Educ Treatment Children, 41(1), 65–106. https://doi.org/10.1353/etc.2018.0003
7.
BouckE. C.BassetteL.ShurrJ.ParkJ.KerrJ.WhorleyA. (2017). Teaching equivalent fractions to secondary students with disabilities via the virtual-representational-abstract instructional sequence. J Spec Educ Technol, 32(4), 220–31. https://doi.org/10.1177/0162643417727291
8.
BouckE. C.LongH.ParkJ. (2021). Using a virtual number line and corrective feedback to teach addition of integers to middle school students with developmental disabilities. J Dev Phys Disabil, 33(1), 99–116. https://doi.org/10.1007/s10882-020-09735-z
9.
BouckE. C.ParkJ.CwiakalaK.WhorleyA. (2020). Learning fraction concepts through the virtual-abstract instructional sequence. J Behav Educ, 29(3), 519–42. https://doi.org/10.1007/s10864-019-09334-9
10.
BouckE. C.SatsangiR.LongH.JakubowL.O’ReillyC. (2023). A virtual number line intervention package to support addition and subtraction for students with intellectual disability. J Spec Educ Technol, 38(4), 445–57. https://doi.org/10.1177/01626434221135144
11.
BouckE. C.ShurrJ.BassetteL.ParkJ.WhorleyA. (2018). Adding it up: comparing concrete and app-based manipulatives to support students with disabilities with adding fractions. J Spec Educ Technol, 33(3), 194–206. https://doi.org/10.1177/0162643418759341
12.
BrunerJ. S. (1966). Toward a theory of instruction. Harvard University Press.
13.
CarbonneauK. J.MarleyS. C.SeligJ. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. J Educ Psychol, 105(2), 380–400. https://doi.org/10.1037/a0031084
14.
CirinoP. T.FuchsL. S.EliasJ. T.PowellS. R.SchumacherR. F. (2015). Cognitive and mathematical profiles for different forms of learning difficulties. J Learn Disabil, 48(2), 156–75. https://doi.org/10.1177/0022219413494239
15.
DoughertyB.BryantD. P.BryantB. R.ShinM. (2017). Helping students with mathematics difficulties understand ratios and proportions. Teach Except Child, 49(2), 96–105. https://doi.org/10.1177/0040059916674897
16.
EnnisR. P.LosinskiM. (2019). Interventions to improve fraction skills for students with disabilities: a meta-analysis. Except Child, 85(3), 367–86. https://doi.org/10.1177/0014402918817504
17.
FloresM. M.HintonV. M. (2022). The effects of a CRA-I intervention on students’ number sense and understanding of addition. Remedial Spec Educ, 43(3), 183–94. https://doi.org/10.1177%2F07419325211038009
18.
FloresM. M.HintonV. M.MeyerJ. M. (2020). Teaching fraction concepts using the concrete-representational-abstract sequence. Remedial Spec Educ, 41(3), 165–75. https://doi.org/10.1177/0741932518795477
19.
FloresM. M.HintonV. M.SchweckK. B. (2024). Using CRA-I to teach fraction and decimal concepts to students with learning disabilities. Learn Disabil Q, 47(1), 44–58. https://doi.org/10.1177/0731948723117654
FuchsL. S.MaloneA. S.SchumacherR. F.NamkungJ.WangA. (2017). Fraction intervention for students with mathematics difficulties: lessons learned from five randomized controlled trials. J Learn Disabil, 50(6), 631–9. https://doi.org/10.1177/0022219416677249
22.
FuchsL. S.Newman-GoncharR.Schumacher RobinF.DoughertyB.BuckaN.KarpK.WoodwardJ.ClarkeB.JordanN. C.GerstenR. M.JayanthiM.KeatingB.MorganS. T. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades (WWC 2021006). National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education. https://ies.ed.gov/ncee/wwc/Docs/PracticeGuide/WWC2021006-Math-PG.pdf
FuchsL. S.SchumacherR. F.LongJ.NamkungJ.MaloneA. S.WangA.HamlettC. L.JordanN. C.SieglerR. S.ChangasP. (2016). Effects of intervention to improve at-risk fourth graders’ understanding, calculations, and word problems with fractions. Elem Sch J, 116(4), 625–51. https://doi.org/10.1086/686303
25.
GearyD. C.HoardM. K.NugentL.BaileyD. H. (2012). Mathematical cognition deficits in children with learning disabilities and persistent low achievement: a five-year prospective study. J Educ Psychol, 104(1), 206–23. https://doi.org/10.1037/a0025398
26.
GesuelliK. A.JordanN. C. (2024). Fraction arithmetic development: an examination of students’ patterns of growth and errors across the intermediate grades. J Educ Psychol, 116(3), 377–95. https://doi.org/10.1037/edu0000828
27.
HedgesL. V.PustejovskyJ. E.ShadishW. R. (2012). A standardized mean difference effect size for single case designs. Res Synth Methods, 3(3), 224–39. https://doi.org/10.1002/jrsm.1052
28.
HedgesL. V.PustejovskyJ. E.ShadishW. R. (2013). A standardized mean difference effect size for multiple baseline designs across individuals. Res Synth Methods, 4(4), 324–41. https://doi.org/10.1002/jrsm.1086
29.
Houghton Mifflin Harcourt. (2019). Go math! Virginia – grade 4.
30.
HwangJ.RiccominiP. J.HwangS. Y.MoranoS. (2019). A systematic analysis of experimental studies targeting fractions for students with mathematics difficulties. Learn Disabil Res Pract, 34(1), 47–61. https://doi.org/10.1111/ldrp.12187
31.
JayanthiM.GerstenR.SchumacherR. F.DiminoJ.SmolkowskiK.SpalloneS. (2021). Improving struggling fifth-grade students’ understanding of fractions: a randomized controlled trial of an intervention that stresses both concepts and procedures. Except Child, 88(1), 81–100. https://doi.org/10.1177/00144029211008851
32.
JimersonS. R.BurnsM. K.VanDerHeydenA. M. (Eds.). (2015). Handbook of response to intervention: The science and practice of multi-tiered systems of support. Springer.
33.
KratochwillT. R.HitchcockJ.HornerR. H.LevinJ. R.OdomS. L.RindskopfD. M.ShadishW. R. (2010). Single-case designs technical documentation. What Works Clearinghouse. https://files.eric.ed.gov/fulltext/ED510743.pdf
34.
LedfordJ. R.GastD. L. (Eds.). (2018). Single case research methodology: Applications in special education and behavioral sciences. Routledge.
35.
MacciniP.GagnonJ. C. (2000). Best practices for teaching mathematics to secondary students with special needs. Focus Except Children, 32(5), 1–22. https://doi.org/10.17161/fec.v32i5.6919
36.
MackN. K. (2004). Connecting to develop computational fluency with fractions. Teach Child Math, 11(4), 226–32. https://doi.org/10.5951/tcm.11.4.0226
37.
MaloneA. S.FuchsL. S.SterbaS. K.FuchsD.Foreman-MurrayL. (2019). Does an integrated focus on fractions and decimals improve at-risk students’ rational number magnitude performance?Contemp Educ Psychol, 59(2019), 1–12. https://doi.org/10.1016/j.cedpsych.2019.101782
MazzoccoM. M. M.DevlinK. T. (2008). Parts and “holes”: gaps in rational number sense in children with vs. without mathematical learning disability. Dev Sci, 11(5), 681–91. https://doi.org/10.1111/j.1467-7687.2008.00717.x
40.
MazzoccoM. M. M.MyersG. F.LewisK. E.HanichL. B.MurphyM. M. (2013). Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) versus low mathematics achievement. J Exp Child Psychol, 115(2), 371–87. https://doi.org/10.1016/j.jecp.2013.01.005
41.
MoranoS.FloresM. M.HintonV.MeyerJ. (2020). A comparison of concrete-representational-abstract and concrete-representational-abstract-integrated fraction interventions for students with disabilities. Exceptionality, 28(2), 77–91. https://doi.org/10.1080/09362835.2020.1727328
NelsonG.PowellS. R. (2018). A systematic review of longitudinal studies of mathematics difficulty. J Learn Disabil, 51(6), 523–39. https://doi.org/10.1177/0022219417714773
44.
ParkJ.BryantD. P.ShinM. (2021). Effects of interventions using virtual manipulatives for students with learning disabilities: a synthesis of single-case research. J Learn Disabil, 55(4), 325–37. https://doi.org/10.1177/00222194211006336
45.
ParkJ.BryantD. P.ShinM. (2024). Effects of a synchronous online fraction intervention using virtual manipulatives for students with learning disabilities. J Spec Educ Technol, 39(2), 234–46. https://doi.org/10.1177/0162643423119531
46.
PeltierC.MorinK. L.BouckE. C.LingoM. E.PulosJ. M.SchefflerF. A.SukA.MathewsL. A.SinclairT. E.DeardorffM. E. (2020). A meta-analysis of single-case research using mathematics manipulatives with students at risk or identified with a disability. J Spec Educ, 54(1), 3–15. https://doi.org/10.1177/0022466919844516
47.
PustejovskyJ. E.ChenM.HamiltonB.GrekovP. (2023). scdhlm: A web-based calculator for between-case standardized mean differences (Version 0.7.1)[Web application]. https://jepusto.shinyapps.io/scdhlm
48.
RauM. A. (2017). Conditions for the effectiveness of multiple visual representations in enhancing stem learning. Educ Psychol Rev, 29(4), 717–61. https://doi.org/10.1007/s10648-016-9365-3
RoessleinR. I.CoddingR. S. (2019). Fraction interventions for struggling elementary math learners: a review of the literature. Psychol Sch, 56(3), 413–32. https://doi.org/10.1002/pits.22196
51.
SatsangiR.RainesA. R. (2023). Examining virtual manipulatives for teaching computations with fractions to children with mathematics difficulty. J Learn Disabil, 56(4), 295–309. https://doi.org/10.1177/00222194221097710
52.
SatsangiR.BouckE. C.DoughtyT. T.BofferdingL.RobertsC. A. (2016). Comparing the effectiveness of virtual and concrete manipulatives to teach algebra to secondary students with learning disabilities. Learn Disabil Q, 39(4), 240–53. https://doi.org/10.1177/0731948716649754
53.
SchnotzW. (2014). An integrated model of text and picture comprehension. In MayerR. E. (Ed.), The Cambridge handbook of multimedia learning (2nd ed, pp. 72–103). Cambridge University Press.
54.
ShinM.BryantD. P. (2015). Fraction interventions for students struggling to learn mathematics: a research synthesis. Remedial Spec Educ, 36(6), 374–87. https://doi.org/10.1177/0741932515572910
55.
SieglerR. S.ThompsonC. A.SchneiderM. (2011). An integrated theory of whole number and fractions development. Cognit Psychol, 62(4), 273–96. https://doi.org/10.1016/j.cogpsych.2011.03.001
TarlowK. R. (2017). An improved rank correlation effect size statistic for single-case designs: baseline corrected tau. Behav Modif, 41(4), 427–67. https://doi.org/10.1177/0145445516676750
58.
TianJ.SieglerR. S. (2017). Fraction learning in children with mathematics difficulties. J Learn Disabil, 50(6), 614–20. https://doi.org/10.1177/0022219416662032
59.
ValentineJ. C.Tanner-SmithE.PustejovskyJ. E.LauT. S. (2016). Between-case standardized mean difference effect sizes for single-case designs: a primer and tutorial using the scdhlm web application. Campbell System Rev, 12(1), 1–31. https://doi.org/10.4073/cmdp.2016.1
WechslerD. (2014). Wechsler intelligence scale for children – fifth edition (WISC-V). Pearson.
62.
What Works Clearinghouse. (2022). What works clearinghouse procedures and standards handbook, version 5.0. U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance. https://ies.ed.gov/ncee/wwc/Handbooks
63.
ZhangD.SteckerP.BeqiriK. (2017). Strategies students with and without mathematics disabilities use when estimating fractions on number lines. Learn Disabil Q, 40(4), 225–36. https://doi.org/10.1177/0731948717704966
64.
ZhangX.RäsänenP.KoponenT.AunolaK.LerkkanenM. K.NurmiJ. E. (2020). Early cognitive precursors of children’s mathematics learning disability and persistent low achievement: a 5-year longitudinal study. Child Dev, 91(1), 7–27. https://doi.org/10.1111/cdev.13123
