This pilot study examined the feasibility of using worked examples as a mechanism to improve verbal explanations of fractions concepts among Grade 5 students with mathematics difficulties in a small-scale randomized controlled trial (RCT) before scaling up to a large-scale RCT. Students (N = 49) were randomly assigned to a business-as-usual (BAU) comparison group and to two variants of intervention. One intervention condition received both correct and incorrect worked example solutions, the other received correct solutions only. On a measure of verbal explanations, intervention students significantly outperformed students in BAU. Furthermore, students who received both correct and incorrect solutions significantly outperformed students who received correct solutions only on verbal explanations. On fractions proficiency outcomes, results were not significant between intervention and BAU or between the two intervention groups; however, positive findings demonstrate promise for using worked examples to elicit and develop students’ verbal explanations of fractions concepts.
BarbieriC.BoothJ. L. (2016). Support for struggling students in algebra: contributions of incorrect worked examples. Learning and Individual Differences, 48, 36–44. https://doi.org/10.1016/j.lindif.2016.04.001
2.
BentleyB.YatesG. C. (2017). Facilitating proportional reasoning through worked examples: two classroom-based experiments. Cogent Education, 4(1), 1297213. https://doi.org/10.1080/2331186X.2017.1297213
3.
BoothJ. L.CooperL. A.DonovanM. S.HuygheA.KoedingerK. R.Paré-BlagoevE. J. (2015). Design-based research within the constraints of practice: AlgebraByExample. Journal of Education for Students Placed at Risk (JESPAR), 20(1–2), 79–100. https://doi.org/10.1080/10824669.2014.986674
4.
BoothJ. L.LangeK. E.KoedingerK. R.NewtonK. J. (2013). Using example problems to improve student learning in algebra: differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34. https://doi.org/10.1016/j.learninstruc.2012.11.002
5.
BoothJ. L.NewtonK. J. (2012). Fractions: could they really be the gatekeeper’s doorman?Contemporary Education Psychology, 37(4), 247–253. https://doi.org/10.1016/j.cedpsych.2012.07.001
ChiM. T. H. (2000). Self-explaining: the dual processes of generating inference and repairing mental models. In GlaserR. (Ed.), Advances in instructional psychology: educational design and cognitive science (Vol. 5, pp. 161–238). Routledge.
8.
ChiM. T. H.BassockM.LewisM.ReimannP.GlaserR. (1989). Self-explanations: how students study and use examples in learning to solve problems. Cognitive Science, 13(2), 145–182. https://doi.org/10.1207/s15516709cog1302_1
CrooksN. M.AlibaliM. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34(4), 344–377. https://doi.org/10.1016/j.dr.2014.10.001
11.
DurkinK.Rittle-JohnsonB. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206–214. https://doi.org/10.1016/j.learninstruc.2011.11.001
12.
DurkinK.StarJ. R.Rittle-JohnsonB. (2017). Using comparison of multiple strategies in the mathematics classroom: lessons learned and next steps. ZDM Mathematics Education, 49(4), 585–597. https://doi.org/10.1007/s11858-017-0853-9
13.
Foreman-MurrayL.FuchsL. S. (2019). Quality of explanation as an indicator of fraction magnitude understanding. Journal of Learning Disabilities, 52(2), 181–191. https://doi.org/10.1177/0022219418775120
14.
FuchsL. S.MaloneA. S.PreacherK. J.FuchsD.WangA. Y.PachmayrR. (2019). Effects of fourth- and fifth-grade super solvers intervention on fraction magnitude understanding and calculation skill: a research report. Vanderbilt University.
15.
FuchsL. S.MaloneA. S.SchumacherR. F.NamkungJ.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
16.
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/
Hassinger-DasB.JordanN. C.GluttingJ.IrwinC.DysonN. (2014). Domain-general mediators of the relation between kindergarten number sense and first-grade mathematics achievement. Journal of Experimental Child Psychology, 118, 78–92. https://doi.org/https://https://doi.org/10.1016/j.jecp.2013.09.008
20.
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, 88(1), 81–100. https://doi.org/10.1177/00144029211008851
21.
JordanN. C.HansenN.FuchsL. S.SieglerR. S.GerstenR.MicklosD. (2013). Developmental predictors of fraction concepts and procedures. Journal of Experimental Child Psychology, 116(1), 45–58. https://doi.org/10.1016/j.jecp.2013.02.001
22.
KiuharaS. A.Gillespie RouseA.DaiT.WitzelB. S.MorphyP.UnkerB. (2020). Constructing written arguments to develop fraction knowledge. Journal of Educational Psychology, 112(3), 584–607. https://doi.org/10.1037/edu0000391
23.
MaloneA. S.FuchsL. S. (2019). Vanderbilt Fractions Battery. Available from L. Fuchs (lynn.fuchs@vanderbilt.edu).
NamkungJ.FuchsL. (2019). Remediating difficulty with fractions for students with mathematics learning difficulties. Learning Disabilities: A Multidisciplinary Journal, 24(2), 36–48. https://doi.org/10.18666/LDMJ-2019-V24-I2-9902
26.
NamkungJ. M.FuchsL. S.KoziolN. (2018). Does initial learning about the meaning of fractions present similar challenges 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
27.
National Mathematics Advisory Panel. (2008). Foundations for success: the final report of the national mathematics advisory panel. U.S. Department of Education.
28.
PowellS. R.FuchsL. S.CirinoP. T.FuchsD.ComptonD. L.ChangasP. C. (2015). Effects of a multitier support system on calculation, word problem, and prealgebraic performance among at-risk learners. Exceptional Children, 81(4), 443–470. https://doi.org/10.1177/0014402914563702
RenklA. (2017). Learning from worked-examples in mathematics: students relate procedures to principles. ZDM Mathematics Education, 49, 571–584. https://doi.org/10.1007/s11858-017-0859-3
31.
ResnickI.JordanN. C.HansenN.RajanV.RodriguesJ.SieglerR. S.FuchsL. S. (2016). Developmental growth trajectories in understanding of fraction magnitude from fourth through sixth grade. Developmental Psychology, 52(5), 746–757. https://doi.org/10.1037/dev0000102
32.
Rittle-JohnsonB.LoehrA. M.DurkinK. (2017). Promoting self-explanation to improve mathematics learning: a meta-analysis and instructional design principles. ZDM Mathematics Education, 49(4), 599–611. https://doi.org/10.1007/s11858-017-0834-z
33.
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/10.1037/0022-0663.99.3.561
34.
Rittle-JohnsonB.StarJ. R.DurkinK. (2012). Developing procedural flexibility: are novices prepared to learn from comparing procedures?British Journal of Educational Psychology, 82(3), 436–455. https://doi.org/10.1111/j.2044-8279.2011.02037.x
35.
RodriguesJ.JordanN. C.HansenN. (2019). Identifying fraction measures as screeners of mathematics risk status. Journal of Learning Disabilities, 52(6), 480–497. https://doi.org/10.1177/0022219419879684
36.
SchumacherR. S.WangA.DoughertyB. (2022). Verbal Assessment. R. Schumacher (robin.schumacher@inresg.org).
37.
SieglerR. S.DuncanG. J.Davis-KeanP. E.DuckworthK.ClaessensA.EngelM.SusperreguyM. I.ChenM. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. https://doi.org/10.1177/0956797612440101
38.
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
39.
StarJ. R.Rittle-JohnsonB. (2009). It pays to compare: an experimental study on computational estimation. Journal of Experimental Child Psychology, 102(4), 408–426. https://doi.org/10.1016/j.jecp.2008.11.004
40.
StarJ. R.Rittle-JohnsonB.DurkinK. (2016). Comparison and explanation of multiple strategies: one example of a small step forward for improving mathematics education. Policy Insights from the Behavioral and Brain Sciences, 3(2), 151–159. https://doi.org/10.1177/2372732216655543
41.
StataCorp (2019). Stata statistical software: Release 16.
42.
U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (1990–2009). Mathematics assessment. http://nces.ed.gov/nationsreportcard.
43.
Van GogT.PaasF.Van MerriënboerJ. J. (2004). Process-oriented worked examples: improving transfer performance through enhanced understanding. Instructional Science, 32(1–2), 83–98. https://doi.org/10.1023/B:TRUC.0000021810.70784.b0
44.
What Works Clearinghouse, Institute of Education Sciences, U.S. Department of Education (2022, August). What Works Clearinghouse: Procedures and standards handbook (Version 5.1).https://whatworks.ed.gov.
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.
