Competence with rational numbers is essential for mathematics proficiency in secondary mathematics. However, many students struggle with rational number concepts, and students with mathematics difficulties struggle even more. The purpose of this study was to examine the effects of an intervention that incorporated the use of explicit instruction and length models to teach the relationship between fractions and decimals to U.S. fourth-grade students with mathematics difficulties. To conduct this study, the researcher employed a multiple probe design across three intervention groups. The results of a visual analysis determined that a functional relation existed between a fraction and decimal intervention and increased performance on two dependent variables: base-10 fraction magnitude and fraction-to-decimal translation. The magnitude of change was calculated using a between-case standardized mean difference, and the results of a social validity questionnaire were presented.
AlonzoJ.LaiC. F.TindalG. (2009). The development of K-8 progress monitoring measures in mathematics for use with the 2% and general education populations: Grade 4 (Technical Report No. 0903). Behavioral Research and Teaching, University of Oregon.
2.
AndersonD.AlonzoJ.TindalG. (2010). easyCBM mathematics criterion related validity evidence: Washington state test (Technical Report No. 1010). Behavioral Research and Teaching, University of Oregon.
3.
AndersonD.AlonzoJ.TindalG.FarleyD.IrvinP. S.LaiC. F.SavenJ. L.WrayK. A. (2014). Technical Manual: easyCBM (Technical Report No. 1408). Behavioral Research and Teaching, University of Oregon.
4.
BaileyD. H.HansenN.JordanN. C. (2017). The codevelopment of children’s fraction arithmetic skill and fraction magnitude understanding. Journal of Educational Psychology, 109(4), 509–519. https://doi.org/10.1037/edu0000152
5.
BarbieriC. A.RodriguesJ.DysonN.JordanN. C. (2020). Improving fraction understanding in sixth graders with mathematics difficulties: Effects of a number line approach combined with cognitive learning strategies. Journal of Educational Psychology, 112(3), 628–648. https://doi.org/10.1037/edu0000384
BrawandA.King-SearsM. E.EvmenovaA. S.ReganK. (2020). Proportional reasoning word problem performance for middle school students with high-incidence disabilities (HID). Learning Disability Quarterly, 43(3), 140–153. https://doi.org/10.1177/0731948719837920
8.
BryantD. P.BryantB.PfannenstielK. H.ChardD.KarpK. (2012). Elementary students in Texas: Algebra ready intervention: Fraction-to-decimal relationships. University of Texas System/Texas Education Agency. https://www.texasgateway.org/resource/estar-ii-intervention-modules
9.
CalhoonM. B.EmersonR. W.FloresM.HoushinsD. E. (2007). Computational fluency performance profile of high school students with mathematics disabilities. Remedial and Special Education, 28(5), 292–303. https://doi.org/10.1177/07419325070280050401
10.
ChenM.PustejovskyJ. E.KlingbeilD. A.Van NormanE. R. (2023). Between-case standardized mean differences: Flexible methods for single-case designs. Journal of School Psychology, 98, 16–38. https://doi.org/10.1016/j.jsp.2023.02.002
DoablerC. T.ClarkeB.FienH.BakerS. K.KostyD. B.CaryM. S. (2015). The science behind curriculum development and evaluation: Taking a design science approach in the production of a tier 2 mathematics curriculum. Learning Disability Quarterly, 38(2), 97–111. https://doi.org/10.1177/07319487135205
13.
DoablerC. T.ClarkeB.KostyD.MaddoxS. A.SmolkowskiK.FienH.BakerS. K.KimmelG. L. (2020). Kindergarteners at risk for severe mathematics difficulties: Investigating tipping points of core mathematics instruction. Journal of Learning Disabilities, 54(2), 97–110. https://doi.org/10.1177/0022219420972185
14.
DysonN. I.JordanN. C.RodriguesJ.BarbieriC.RinneL. (2018). A fraction sense intervention for sixth graders with or at risk for mathematics difficulties. Remedial and Special Education, 41(4), 1–11. https://doi.org/10.1177/0741932518807139
15.
EnnisR. P.LosinskiM. (2019). Interventions to improve fraction skills for students with disabilities: A meta-analysis. Exceptional Children, 85(3), 367–386. https://doi.org/10.1177/0014402918817504
16.
FletcherJ. M.LyonG. R.FuchsL. S.BarnesM. A. (2019). Learning disabilities: From identification to intervention (2nd ed.). The Guilford Press.
17.
FloresM. M.HintonV. M.MeyerJ. M. (2020). Teaching fraction concepts using the concept-representational-abstract sequence. Remedial and Special Education, 41(3), 165–175. https://doi.org/10.1177/0741932518795477
18.
FuchsL. S.Newman-GoncharR.SchumacherR.DoughertyB.BuckaN.KarpK. S.WoodwardJ.ClarkeB.JordanN. C.GerstenR.JayanthiM.KeatingB.MorganS. (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. http://whatworks.ed.gov/
19.
FuchsL. S.SchumacherR. F.LongJ.NamkungJ.HamlettC. L.CirinoP. T.JordanN. C.SieglerR.GerstenR.ChangasP. (2013). Improving at-risk learners’ understanding of fractions. Journal of Educational Psychology, 105(3), 683–700. https://doi.org/10.1037/a0032446
20.
FuchsL. S.SchumacherR. F.LongJ.NamkungJ.MaloneA.WangA.HamlettC. L.JordanN. C.SieglerR.ChangasP. (2016). Effects of intervention to improve at-risk fourth graders’ understanding, calculations, and word problems with fractions. Journal of Educational Psychology, 105(3), 683–700. https://doi.org/10.1086/686303
21.
GastD. L.LloydB. P.LedfordJ. R. (2014). Multiple Baseline and Multiple Probe Designs. In Single Case Research Methodology (pp. 251-296). Routledge.
22.
GearyD. C.BerchD. B.OchsendorfR.KoepkeK. M. (Eds.). (2017). Acquisition of complex arithmetic skills and higher-order mathematics concepts. Academic Press.
23.
GerstenR.BeckmannS.ClarkeB.FoegenA.MarshL.StarJ. R.WitzelB. (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. https://ies.ed.gov/ncee/WWC/Docs/PracticeGuide/rti_math_pg_042109.pdf
24.
GerstenR.SchumacherR. F.JordanN. C. (2017). Life on the number line: Routes to understanding fraction magnitude for students with difficulties learning mathematics. Journal of Learning Disabilities, 50(6), 655–657. https://doi.org/10.1177/0022219416662625
25.
GöbelS.WalshV.RushworthM. F. (2001). The mental number line and the human angular gyrus. NeuroImage, 14(6), 1278–1289. https://doi.org/10.1006/nimg.2001.0927
26.
HeavinK. S.MaX. (2022). The effects of dual credit college algebra on postsecondary education outcomes. Asian Journal for Mathematics Education, 1(1), 19–35. https://doi.org/10.1177/27527263221087750
27.
HedgesL. V.PustejovskyJ. E.ShadishW. A. (2012). A standardized mean difference effect size for single-case designs. Journal of Research Synthesis Methods, 3, 224–239. https://doi.org/10.1002/jrsm.1052
28.
HedgesL. V.PustejovskyJ. E.ShadishW. A. (2013). A standardized mean difference effect size for multiple baseline designs. Journal of Research Synthesis Methods, 4, 324–341. https://doi.org/10.1002/jrsm.1086
29.
HornerR. H.CarrE. G.HalleJ.McGeeG.OdomS.WoleryM. (2005). The use of single-subject research to identify evidence-based practice in special education. Exceptional Children, 71(2), 165–179. https://doi.org/10.1177/001440290507100203
30.
HwangJ.RiccominiP. J. (2019). A descriptive analysis of the error patterns observed in the fraction-computation solution pathways of students with and without learning disabilities. Assessment for Effective Intervention, 46(2), 1–11. https://doi.org/10.1177/1534508419872256
31.
JitendraA. K.AlghamdiA.EdmundsR.McKevettN. M.MouanoutouaJ.RoesslinR. (2021). The effects of tier 2 mathematics interventions for students with mathematics difficulties: A meta-analysis. Exceptional Children, 87(3), 301–325. https://doi.org/10.1177/0014402920969187
32.
JordanN.ResnickI.RodriguesJ.HansenN.DysonN. (2016). Delaware longitudinal study of fraction learning: Implications for helping children with mathematics difficulties. Journal of Learning Disabilities, 50(6), 621–630. https://doi.org/10.1177/0022219416662033
33.
KennedyC. (2005). Single-case designs for educational research. Pearson Education.
34.
LedfordJ. R.GastD. L. (2018). Single case research methodology: Applications in special education and behavioral sciences (3rd ed.). Routledge.
35.
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?Contemporary Educational Psychology, 59(2019), 1–12. https://doi.org/10.1016/j.cedpsych.2019.101782
36.
MoranoS.RiccominiP. J.LeeJ. Y. (2019). Accuracy of area model and number line representations of fractions for students with learning disabilities. Learning Disabilities Research & Practice, 34(3), 133–143. https://doi.org/10.1111/ldrp.12197
National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards (Mathematics).
39.
National Mathematics Advisory Panel. (2008). The final report of the national mathematics advisory panel.
40.
NiY.ZhouY. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52. https://doi.org/10.1207/s15326985ep4001_3
41.
Pearson-Scott Foresman. (2017). enVision Math 2.0 Grade 4. Pearson.
42.
PowellS. R.GilbertJ. K.FuchsL. S. (2019). Variables influencing algebra performance: Understanding rational numbers is essential. Learning and Individual Differences, 74(2019), 1–9. https://doi.org/10.1016/j.lindif.2019.101758
43.
PustejovskyJ. E.ChenM.HamiltonB. (2020). scdhlm: A web-based calculator for between-case standardized mean differences (Version 0.5.2) [Web application]. https://jepusto.shinyapps.io/scdhlm
44.
PustejovskyJ. E.HedgesL. V.ShadishW. R. (2014). Design-comparable effect sizes in multiple baseline designs: A general modeling framework. Journal of Educational and Behavioral Statistics, 39(5), 368–393. https://doi.org/10.3102/1076998614547577
45.
RojoM.GersibJ.PowellS. R.ShenZ.KingS. G.Sharjina AktherS.ArsenaultT. L.BosS. E.LariviereD. O.LinX. (2024). A meta-analysis of mathematics interventions: Examining the impacts of intervention characteristics.Educational Psychology Review, 36(1), 9. https://doi.org/10.1007/s10648-023-09843-0
46.
RojoM.KingS. G.GersibJ.BryantD. P. (2023). Rational number interventions for students with mathematics difficulties: A meta-analysis. Remedial and Special Education, 44(3), 225–238. https://doi.org/10.1177/07419325221105520
47.
ShadishW. R.HedgesL. V.PustejovskyJ. E. (2014). Analysis and meta-analysis of single-case designs with a standardized mean difference statistic: A primer and applications. Journal of School Psychology, 52(2), 123–147. https://doi.org/10.1016/j.jsp.2013.11.005
48.
SieglerR. S. (2016). Magnitude knowledge: The common core of numerical development. Developmental Science, 19(3), 341–361. https://doi.org/10.1111/desc.12395
49.
SieglerR. S.Lortie-ForguesH. (2014). An integrative theory of numerical development. Child Development Perspectives, 8(3), 144–150. https://doi.org/10.1111/cdep.12077
50.
SieglerR. S.PykeA. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994–2004. https://doi.org/10.1037/a0031200
51.
SieglerR. S.ThompsonC. A.SchneiderM. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(2011), 273–296. https://doi.org/10.1016/j.cogpsych.2011.03.001
52.
TianJ.SieglerR. S. (2018). Which type of rational numbers should students learn first?Educational Psychology Review, 30, 351–372. https://doi.org/10.1007/s10648-017-9417-3
53.
TorbeynsJ.SchneiderM.XinZ.SieglerR. S. (2015). Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, 37, 5–13. https://doi.org/10.1016/j.learninstruc.2014.03.002
54.
VamvakoussiX.VosniadouS. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14(5), 453–467. https://doi.org/10.1016/j.learninstruc.2004.06.013
55.
VamvakoussiX.VosniadouS. (2010). How many decimals are between two fractions? Aspects of secondary school students’ understanding of rational numbers and their notation. Cognition and Instruction, 28(2), 181–209. https://doi.org/10.1080/07370001003676603
56.
Van de WalleJ. A.KarpK.Bay-WilliamsJ. M.WrayJ. A.BrownE. T. (2019). Elementary and middle school mathematics: Teaching developmentally / John A. Van de Walle, Karen S. Karp, Jennifer M. Bay-Williams; with contributions by Jonathan Wray, Elizabeth Todd Brown (10th ed.). Pearson.
57.
Van HoofJ.DegrandeT.CeulemansE.VerschaffelL.Van DoorenW. (2018). Towards a mathematically more correct understanding of rational numbers: A longitudinal study with upper elementary school learners. Learning and Individual Differences, 61, 99–108. https://doi.org/10.1016/j.lindif.2017.11.010
58.
Van HoofJ.VanvakoussiX.Van DoorenV. W.VershaffelL. (2017). The transition from natural to rational number knowledge. In GearyD. C.BerchD. B.OchosendorfR.KoepkeK. M. (Eds.), Acquisition of complex arithmetic skills and higher-order mathematics concepts: Vol. 3. Mathematical cognition and learning (1st ed., pp. 102–126). Academic Press.
59.
WangA. Y.FuchsL. S.FuchsD.GilbertJ. K.KrowkaS.AbramsonR. (2019). Embedding self-regulation instruction within fractions intervention for third graders with mathematics difficulties. Journal of Learning Disabilities, 52(4), 337–348. https://doi.org/10.1177/0022219419851750
60.
WangY.SieglerR. S. (2013). Representations of and translation between common fractions and decimal fractions. Chinese Science Bulletin, 58, 4630–4640. https://doi.org/10.1007/s11434-013-6035-4
61.
What Works Clearinghouse. (2017). What Works Clearinghouse procedures handbook, version 4.1. U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance. https://ies.ed.gov/ncee/wwc/handbooks
62.
What Works Clearinghouse. (2020). What Works Clearinghouse procedures handbook, version 4.1. U.S. Department of Education, Institute of Education Sciences, National Center for Education Evaluation and Regional Assistance. https://ies.ed.gov/ncee/wwc/handbooks
