This study's first purpose was to investigate effects of a fourth- and fifth-grade “next-generation” fraction intervention, which included six enhancements over a previously validated fraction intervention, designed to address career- and college-readiness standards. The study's second purpose was to assess effects of the next-generation fraction intervention at follow-up, 1 year after intervention ended. The third purpose was to isolate the effects of one of the six intervention enhancements: interleaved fraction calculations instruction. Students with intensive intervention needs were randomized to next-generation fraction intervention (Super Solvers [SSINT]) with blocked calculations instruction (SSINT_B), SSINT with interleaved calculations instruction (SSINT_I), and control. On a mix of proximal and transfer outcomes, SSINT (across conditions) produced strong, significant effects over control at posttest. At follow-up, effect sizes were weaker but remained significant on calculations: g = 1.22. On other measures, follow-up g was 0.39 to 0.58. The effect of SSINT_I over SSINT_B, although not significant at posttest (g = 0.28), was statistically significant and large at follow-up (g = 0.65), in line with the cognitive science literature showing long-term advantages for interleaved instruction. Results suggest next-generation fraction intervention efficacy for intensive-needs students and the importance of interleaved instruction.
Get full access to this article
View all access options for this article.
References
1.
BanduraA. (1986). Social foundation of thought and action: Social cognitive theory. Prentice Hall.
2.
BarbieriC. A.YoungL. K.NewtonK. J.BoothJ. L. (2021). Predicting middle school profiles of algebra performance using fraction knowledge. Child Development, 92(5), 1984–2005. https://doi.org/10.1111/cdev.13568
3.
BauerD. J.SterbaS. K.HallforsD. D. (2008). Evaluating group-based interventions when control participants are ungrouped. Multivariate Behavioral Research, 43(2), 210–236. https://doi.org/10.1080/00273170802034810
4.
BjorkE.BjorkR. (2011). Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning. In GernsbacherM. A.PewR. W.HoughL. M.PomerantzJ. R. (Eds.), Psychology and the real world: Essays illustrating fundamental contributions to society (pp. 56–64). Worth.
5.
BlackwellL. S.TrzesniewskiK. H.DweckC. W. (2007). Implicit theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention. Child Development, 78(1), 246–263. https://doi.org/10.1111/j.1467-8624.2007.00995.x
6.
BoothJ. L.NewtonK. J.Twiss-GarrityL. K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118(1) 110–118. https://doi.org/10.1016/j.jecp.2013.09.001
7.
BraithwaiteD. W.SieglerR. S. (2018). Children learn spurious associations in their math textbooks: Examples from fraction arithmetic. Journal of Experimental Psychology: Learning, Memory, and Cognition, 44(11), 1765–1777. https://doi.org/10.1037/xlm0000546
CervoneD.WoodR. (1995). Goals, feedback, and the differential influence of self-regulatory processes on cognitively complex performance. Cognitive Therapy and Research, 19(5), 519–545. https://doi.org/10.1007/BF02230512
10.
ChaseC. C.ShemwellJ. T.SchwartzD. L. (2010). Where hypotheses come from: Learning new relations by structural alignment. Journal of Cognition and Development, 11(3), 356–373. http://doi.org/10.1080/15248371003700015
11.
ClarkeB.DoableC. T.TurturaJ.SmolkowskiK.KostyD. B.SutherlandM.Kurtz-NelsonE.FienH.BakerS. K. (2020). Examining the efficacy of a kindergarten mathematics intervention by group size and initial skill: Implications for practice and policy. Elementary School Journal, 121(1). https://doi.org/10.1177/2332858417706899
12.
ClemensN. H.FuchsD. (2021). Commercially developed tests of reading comprehension: Gold standard or fool’s gold?Reading Research Quarterly, 57(2):385–397. https://doi.org/10.1002/rrq.415
13.
De CorteE.VerschaffelL.Op’t EyndeP. (2000). Self-regulation: A characteristic and a goal of mathematics education. In BoekaertsM.PintrichP. R.ZeidnerM. (Eds.), Handbook of self-regulation (pp. 687–726). Academic Press.
EdgertonA. K.DesimoneL. M. (2018). Teacher implementation of college- and career-readiness standards: Links among policy, instruction, challenges, and resources. AERA Open, 4(4), 1–22. https://doi.org/10.1177/2332858418806863
EnnisR. P.LosinskiM. (2019). Intervention to improve fraction skills for students with disabilities: A meta-analysis. Exceptional Children, 85(3), 367–386. https://doi.org/10.1177/0014402918817504
18.
EnuJ.DansoP.AwortweP. K. (2015). Effects of group size on students mathematics achievement in small group settings. Journal of Education and Practice, 6(1), 119. https://core.ac.uk/download/pdf/234636905.pdf
19.
FuchsD.MirowitzH. S.GilbertJ. K. (2022). Exploring the truth of Michael Yudin’s claim: The more time students with disabilities spend in general classrooms, the better they do academically. Journal of Disability Policy Studies. https://doi.org/10.1177/10442973221097713
20.
FuchsL. S.FuchsD.ComptonD. L.WehbyJ.SchumacherR. F.GerstenR.JordanN. C. (2015). Inclusion versus specialized intervention for very low-performing students: What does access mean in an era of academic challenge?Exceptional Children, 81(2), 134–157. https://doi.org/10.1177/0014402914551743
21.
FuchsL. S.FuchsD.SeethalerP. M.CuttingL. E.Mancilla-MartinezJ. (2019). Connections between reading comprehension and word-problem solving via oral language comprehension: Implications for comorbid learning disabilities. New Directions for Child and Adolescent Development, 2019(165), 73–90. https://doi.org/10.1002/cad.20288
FuchsL. S.MaloneA.SchumacherR. F.NamkungJ. M.HamlettC. L.JordanN. C.SieglerR. S.GerstenR.ChangasP. (2016). Supported self-explaining during fraction intervention. Journal of Educational Psychology, 108(4), 493–508. https://doi.org/10.1037/edu0000073
24.
FuchsL. S.MaloneA. S.SchumacherR. F.NamkungJ.WangA. (2017). Fraction intervention for students with mathematics difficulties: Lessons learned from five randomized control trials. Journal of Learning Disabilities, 50(6), 631–639. https://doi.org/10.1177/0022219416677249
25.
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
26.
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. Elementary School Journal, 116(4), 625–651. https://doi.org/10.1086/686303
27.
FuchsL. S.SchumacherR. F.SterbaS. K.LongJ.NamkungJ.MaloneA.HamlettC. L.JordanN. C.GerstenR.SieglerR.ChangasP. (2014). Does working memory moderate the effects of fraction intervention? An aptitude-treatment interaction. Journal of Educational Psychology, 106(2), 499–514. https://doi.org/10.1037/a0034341
28.
FuchsL. S.WangA. Y.PreacherK. J.MaloneA. S.FuchsD.PachmayrR. (2021). Addressing challenging mathematics standards with at-risk learners: A randomized controlled trial on the effects of fractions intervention at third grade. Exceptional Children, 87(2), 163–182. https://doi.org/10.1177/00144029209248
29.
GabrielF.CocheF.SzucsD.CaretteV.ReyB.ContentA. (2013). A componential analysis of children’s difficulties in learning fractions. Frontiers in Developmental Psychology, 4, 715.https://doi.org/10.3389/fpsyg.2013.00715
30.
HedgesL. V.CitkowiczM. (2014). Estimating effect size when there is clustering in one treatment group. Behavior Research Methods, 47(4), 1295–1308. https://doi.org/10.3758/s13428-014-0538-z
HughesC. A.LeeJ.-Y. (2019). Effective approaches for scheduling and formatting practice: Distributed, cumulative, and interleaved practice. TEACHING Exceptional Children, 51(6), 411–423. https://doi.org/10.1177/0040059919847194
33.
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. Exceptional Children, 80(1), 423–437. https://doi.org/10.1177/00144029211008851
34.
JordanN. C.BarbieriC.DysonN.DevlinB. (2020). Improving learning in students with mathematics difficulties. In MartinA. J.SperlingR. A.NewtonK. J. (Eds.), Handbook of educational psychology and students with special needs (pp. 107–114). Taylor & Francis.
35.
KallaiA. Y.TzelgovJ. (2009). A generalized fraction: An entity smaller than one on the mental number line. Journal of Experimental Psychology: Human Perception and Performance, 35(6), 1845–1864. https://doi.org/10.1037/a0016892
36.
LemonsC. J.FuchsD.GilbertJ. K.FuchsL. S. (2014). Evidence-based practices in a changing world: Reconsidering the counterfactual in educational research. Educational Researcher, 43(5), 242–252. https://doi.org/10.3102/0013189X14539189
MuthénL. K.MuthénB. O. (2002). How to use a Monte Carlo study to decide on sample size and determine power. Structural Equation Modeling, 9(4), 599–620. https://doi.org/10.1207/S15328007SEM0904_8
40.
MuthénL. K.MuthénB. O. (2018). Mplus statistical analysis with latent variables: User’s guide (Version 8.2).
41.
NamkungJ. M.FuchsL. S.KoziolN. (2018). Does initial learning about the meaning of fractions present similar challenging for students with and without adequate whole-number skill?Learning and Individual Differences, 61, 151–157. https://doi.org/10.1016/j.lindif.2017.11.018
42.
NozariM.BryantD. P.DoablerC. T.FalcomataT. S.LiuzM.GrimesR. (2021). Intensifying instruction to address treatment resistance to early mathematics interventions. Current Psychology. https://doi.org/10.1007/s12144-021-02320-3
43.
ObareF.BirungiH.DeaconB.BurnetR. (2013). Effectiveness of using comic books to communicate HUB and AID messages to in-school youth: Insights from a pilot intervention study in Nairobi, Kenya. African Population Studies, 27(2), 203–215. https://doi.org/10.11564/27-2-441
PowellS. R.BerryK. A.FallA.-M.RobertsG.BarnesM. A.FuchsL. S.LinX. (2021). Does word-problem performance maintain? Follow-up one year after implementation of a word-problem intervention. Journal of Research on Educational Effectiveness, 15(1), 52–77. https://doi.org/10.1080/19345747.2021.1961332
46.
ReysR. E.YangD.-C. (1998). Relationship between computational performance and number sense among sixth- and eighth-grade students in Taiwan. Journal for Research in Mathematics Education, 29(2), 225–237. https://www.jstor.org/stable/749900https://doi.org/10.2307/749900
47.
Rittle-JohnsonB.StarJ. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574. https://doi.org/10.1037/0022-0663.99.561
48.
RobertsC.RobertsS. A. (2005). Design and analysis of clinical trials with clustering effects due to treatment. Clinical Trials, 2(2), 152–162. https://doi.org/10.1191/1740774505cn076oa
49.
RohrerD. (2012). Interleaving helps students distinguish among similar concepts. Educational Psychology Review, 24(3), 355–367. https://doi.org/10.1007/s10648-012-9201-3
50.
SchumacherR. F.MaloneA. S. (2017). Error patterns with fraction calculations at fourth grade as a function of students’ mathematics achievement status. Elementary School Journal, 118(1), 105–127. https://doi.org/10.1086/692914
51.
SieglerR. S.ThompsonC. A.SchneiderM. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. https://doi.org/10.1016/j.cogpsych.2011.03.001
52.
TianJ.LeibE. R.GrigerC.OppenzatoC. O.SieglerR. S. (2022). Biased problem distributions in assignments parallel those in textbooks: Evidence from fraction and decimal arithmetic. Journal of Numerical Cognition, 8(1), 73–88. https://doi.org/10.5964/jnc.6365
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
YeagerD. S.HanselmanP.WaltonG. M., et al. (2019). A national experiment reveals where a growth mindset improves achievement. Nature, 573(7774), 364–369. https://doi.org/10.1038/s41586-019-1466-y
59.
ZiglerE.SternE. (2014). Delayed benefits of learning elementary algebraic transformations through contrasted comparisons. Learning and Instruction, 33, 131–146. https://doi.org/10.1016/j.learninstruc.2014.04.006
60.
ZiglerE.SternE. (2016). Consistent advantages of contrasted comparisons: Algebra learning under direct instruction. Learning and Instruction, 41(2), 41–51. https://doi.org/10.1016/j.learninstruc.2015.09.006
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
