This study investigated the role of strategy instruction and working memory capacity (WMC) on word problem solving accuracy in children with (n = 100) and without (n = 92) math difficulties (MD). Within classrooms, children in Grades 2 and 3 were randomly assigned to one of four treatment conditions: verbal-only strategies (e.g., underlining question sentence), verbal + visual strategies, visual-only strategies (e.g., correctly placing numbers in diagrams), or untreated control. Strategy interventions included 20 sessions in both Year 1 and Year 2. The intent-to-treat as well as the “as-treated” analyses showed that treatment effects were significantly moderated by WMC. In general, treatment outcomes were higher when WMC was set to a high rather than low level. When set to a relatively high WMC level, children with MD performed significantly better under visual-only strategy conditions and children without MD performed better under verbal + visual conditions when compared to control conditions.
AckermanP. L. (1988). Determinants of individual differences during skill acquisition: Cognitive abilities and information processing. Journal of Experimental Psychology: General, 117, 288–318. doi:10.1037//0096-3445.117.3.288
2.
AllowayT. P.GathercoleS. E.KirkwoodH.ElliottJ. (2009). The cognitive and behavioral characteristics of children with low working memory. Child Development, 80, 606–621. doi:10.1111/j.1467-8624.2009.01282.x
3.
AndersonJ. R. (1987). Skill acquisition: Compilation of weak-method problem solving. Psychological Review, 94, 192–210. doi:10.1037//0033-295X.94.2.192
4.
AnderssonU. (2010). Skill development in different components of arithmetic and basic cognitive functions: Findings from a 3-year longitudinal study of children with different types of learning difficulties. Journal of Educational Psychology, 102, 115–134. doi:10.1037/a0016838
5.
BaddeleyA. D.LogieR. H. (1999). The multiple-component model. In MiyakeA.ShahP. (Eds.), Models of working memory: Mechanisms of active maintenance and executive control (pp. 28–61). Cambridge, UK: Cambridge University Press.
6.
BakerS.GerstenR.LeeD. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. Elementary School Journal, 103, 51–73. doi:10.1086/499715
7.
BarrouilletP.LépineR. (2005). Working memory and children’s use of retrieval to solve addition problems. Journal of Experimental Child Psychology, 91, 183–204. doi:10.1016/j.jecp.2005.03.002
8.
BrownV. L.CroninM. E.McEntireE. (1994). Test of Mathematical Abilities. Austin, TX: PRO-ED.
9.
BrownV. L.HammillD.WeiderholtL. (1995). Test of Reading Comprehension. Austin, TX: PRO-ED.
10.
BrykA. S.RaudenbushS. W. (2002). Hierarchical linear models: Applications and data analysis methods. London, UK: Sage.
11.
BullR.EspyK. A.WiebeS. A. (2008). Short-term memory, working memory, and executive functioning in preschoolers: Longitudinal predictors of mathematical achievement at age 7 years. Developmental Neuropsychology, 33, 205–228. doi:10.1080/87565640801982312
12.
BullR.JohnstonR. S.RoyJ. A. (1999). Exploring the roles of the visual-spatial sketch pad and central executive in children’s arithmetical skills: Views from cognition and developmental neuropsychology. Developmental Neuropsychology, 15, 421–442. doi:10.1080/87565649909540759
13.
CaseL. P.HarrisK. R.GrahamS. (1992). Improving the mathematical problem-solving skills of students with learning disabilities: Self-regulated strategy development. Journal of Special Education, 26, 1–19. doi:10.1177/002246699202600101
14.
CicchettiD. V.FeinsteinA. R. (1990). High agreement but low Kappa: II. Resolving the paradoxes. Journal of Clinical Epidemiology, 43, 551–558.
15.
CohenJ.CohenP. (1983). Applied multiple regression/correlation analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
16.
ColomR.AbadF. J.QuirogaM. ÁShihP. C.Flores-MendozaC. (2008). Working memory and intelligence are highly related constructs, but why?Intelligence, 36, 584–606. doi:10.1016/j.intell.2008.01.002
17.
ConnollyA. J. (1998). KeyMath (revised/normative update). Circle Pines, MN: American Guidance.
18.
DanemanM.CarpenterP. A. (1980). Individual differences in working memory and reading. Journal of Verbal Learning and Verbal Behavior, 19, 450–466. doi:10.1016/S0022-5371(80)90312-6
19.
EngleR. W.TuholskiS. W.LaughlinJ. E.ConwayA. R. (1999). Working memory, short-term memory, and general fluid intelligence: A latent variable approach. Journal of Experimental Psychology: General, 128, 309–331. doi:10.1037//0096-3445.128.3.309
20.
FletcherJ. M.EspyK. A.FrancisP. J.DavidsonK. C.RourkeB. P.ShaywitzS. E. (1989). Comparison of cutoff and regression-based definitions of reading disabilities. Journal of Learning Disabilities, 22, 334–338. doi:10.1177/002221948902200603
21.
FrangakisC. E.RubinD. B. (1999). Addressing complications of intention-to-treat analysis in the presence of all-or-none treatment-noncompliance and subsequent missing outcomes. Biometrika, 86, 365–379. doi:10.1093/biomet/86.2.365
22.
FuchsL.FuchsD.PrenticeK.BurchM.HamlettC. L.OwenM.SchroeterK. (2003). Enhancing third-grade student mathematical problem solving with self-regulated learning strategies. Journal of Educational Psychology, 95, 306–315. doi:10.1037/0022-0663.95.2.306
23.
GearyD. C. (2010). Mathematical disabilities: Reflections on cognitive, neuropsychological, and genetic components. Learning and Individual Differences, 20, 130–133. doi:10.1016/j.lindif.2009.10.008
24.
GearyD. C.HoardM. K.Byrd-CravenJ.NugentL.NumteeC. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78, 1343–1359. doi:10.1111/j.1467-8624.2007.01069.x
25.
GerstenR.ChardD. J.JayanthiM.BakerS. K.MorphyP.FlojoJ. (2009). Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components. Review of Educational Research, 79, 1202–1242. doi:10.3102/0034654309334431
JuddC. M.McClellandG. H.SmithE. R. (1996). Testing treatment by covariate interactions when treatment varies within subjects. Psychological Methods, 1, 366–378. doi:10.1037//1082-989X.1.4.366
29.
JustM. A.CarpenterP. A. (1992). A capacity theory of comprehension: Individual differences in working memory. Psychological Review, 99, 122–149. doi:10.1037//0033-295X.99.1.122
30.
KillipS.MahfoudZ.PearceK. (2004). What is an intracluster correlation coefficient?Annals of Family Medicine, 2, 204–208. doi:10.1370/afm.141
31.
KolloffelB.EysinkT.de JongT.WilhelmP. (2009). The effects of representation format on learning combinatorics from an interactive computer. Instructional Science, 37, 503–515. doi:10.1007/s11251-008-9056-7
32.
LeBlancM. D.Weber-RussellS. (1996). Text integration and mathematical connections: A computer model of arithmetic word problem solving. Cognitive Science, 20, 357–407. doi:10.1016/S0364-0213(99)80010-X
33.
LeonA. C.PorteraL.LowellM. A.RheinheimerD. (1998). A strategy to evaluate covariance by group interaction in an analysis of variance. Psychopharmacology Bulletin, 34, 805–808.
34.
LittellR. C.MillikenG. A.StroupW. W.WolfingerR. D.SchabenbergerO. (2010). SAS for mixed models (2nd ed.). Cary, NC: SAS.
35.
MastropieriM.ScruggsT.ShiahR. (1997). Can computers teach problem solving strategies to children with mental retardation?Remedial and Special Education, 18, 157–165. doi:10.1177/074193259701800304
36.
MayerR. E. (2005). Cognitive theory and multimedia learning. In MayerR. (Ed.), The Cambridge handbook of multimedia learning (pp. 31–48). New York, NY: Cambridge University Press. doi:10.1017/CBO9780511816819.004
37.
MayerR. E.HegartyM. (1996). The process of understanding mathematical problem solving. In SternbergR. J.Ben-ZeevT. (Eds.), The nature of mathematical thinking (pp. 29–54). Mahwah, NJ: Lawrence Erlbaum.
38.
McLeanJ. F.HitchG. J. (1999). Working memory impairments in children with specific arithmetic learning difficulties. Journal of Experimental Child Psychology, 74, 240–260. doi:10.1006/jecp.1999.2516
39.
MeyerM. L.SalimpoorV. N.WuS. S.GearyD. C.MenonV. (2010). Differential contribution of specific working memory components to mathematics achievement in 2nd and 3rd graders. Learning and Individual Differences, 20, 101–109. doi:10.1016/j.lindif.2009.08.004
40.
MontagueM. (2008). Self-regulation strategies to improve mathematical problem solving for students with learning disabilities. Learning Disability Quarterly, 31, 37–44.
41.
MontagueM.WargerC.MorganT. H. (2000). Solve it! Strategy instruction to improve mathematical problem solving. Learning Disabilities Research & Practice, 15, 110–116. doi:10.1207/SLDRP1502_7
42.
MorrisN.JonesD. M. (1990). Memory updating in working memory: The role of central executive. British Journal of Psychology, 91, 111–121. doi:10.1111/j.2044-8295.1990.tb02349.x
43.
NgS. F.LeeK. (2009). The model method: Singapore children’s tool for representing and solving algebraic word problems. Journal for Research in Mathematics Education, 40, 282–313.
44.
PassolunghiM. C.CornoldiC.De LibertoS. (2001). Working memory and intrusions of irrelevant information in a group of specific poor problem solvers. Memory & Cognition, 27, 779–790. doi:10.3758/BF03198531
45.
PassolunghiM. C.SiegelL. S. (2001). Short-term memory, working memory, and inhibitory control in children with difficulties in arithmetic problem solving. Journal of Experimental Child Psychology, 80, 44–57. doi:10.1006/jecp.2000.2626
46.
Pearson. (2009). Scott Foresman-Addison Wesley enVisionMATH. New York, NY: Author.
47.
PeughJ. L.EndersC. K. (2004). Missing data in educational research: A review of reporting practices and suggestions for improvement. Review of Educational Research, 74, 525–556. doi:10.3102/00346543074004525
48.
PodockS.AbdallaM. (1998). The hope and hazards of using compliance data in randomized controlled trials. Statistics in Medicine, 17, 303–317.
49.
PowellS. R. (2011). Solving word problems using schemas: A review of literature. Learning Disabilities Research & Practice, 26, 94–108. doi:10.1111/j.1540-5826.2011.00329.x
50.
PreacherK. J.RuckerD. D.MacCallumR. C.NicewanderW. A. (2005). Use of the extreme groups approach: A critical examination and new recommendations. Psychological Methods, 10, 178–192. doi:10.1037/1082-989X.10.2.178
51.
Psychological Corporation. (1992). Wechsler Individual Achievement Test. San Antonio TX: Harcourt Brace.
52.
RavenJ. C. (1976). Colored Progressive Matrices Test. London, UK: H. K. Lewis.
SAS Institute. (2010). SAS program version 9.3. Cary, NC: Author.
55.
SiegelL. S.RyanE. B. (1989). The development of working memory in normally achieving and subtypes of learning disabled. Child Development, 60, 973–980. doi:10.1111/j.1467-8624.1989.tb03528.x
56.
SieglerR. S. (1988). Individual differences in strategy choices: Good students, not-so-good students, and perfectionists. Child Development, 59, 833–851. doi:10.2307/1130252
57.
SingerJ. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics, 24, 323–355. doi:10.3102/10769986023004323
58.
SingerJ. D. (2002). Fitting individual growth models using SAS PROC MIXED. In MoskowitzD. S.HershbergerS. L. (Eds.), Intraindividual variability with repeated measures data: Methods and applications. Multivariate applications book series (pp. 135–170). Mahwah, NJ: Lawrence Erlbaum.
59.
SnijdersT.BoskerR. (1999). Multilevel analysis: An introduction to basic and advanced multilevel modeling (2000 ed.). Thousand Oaks, CA: Sage.
60.
SwansonH. L. (1992). Generality and modification of working memory among skilled and less skilled readers. Journal of Educational Psychology, 84, 473–488. doi:10.1037//0022-0663.84.4.473
61.
SwansonH. L. (1995). S-Cognitive Processing Test (S-CPT): A dynamic assessment measure. Austin, TX: PRO-ED.
62.
SwansonH. L.AllowayT. P. (2012). Working memory, learning, and academic achievement. In HarrisK.UrbanT.GrahamS. (Eds.), APA handbook of educational psychology (Vol. 1, pp. 327–366). Washington, DC: American Psychological Association. doi:10.1037/13273-012
63.
SwansonH. L.AshbakerM. H.LeeC. (1996). Learning-disabled readers’ working memory as a function of processing demands. Journal of Experimental Child Psychology, 61, 242–275. doi:10.1006/jecp.1996.0016
64.
SwansonH. L.Beebe-FrankenbergerM. (2004). The relationship between working memory and mathematical problem solving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 96, 471–491. doi:10.1037/0022-0663.96.3.471
65.
SwansonH. L.JermanO. (2006). Math disabilities: A selective meta-analysis of the literature. Review of Educational Research, 76, 249–274. doi:10.3102/00346543076002249
66.
SwansonH. L.JermanO.ZhengX. (2008). Growth in working memory and mathematical problem solving in children at risk and not at risk for serious math difficulties. Journal of Educational Psychology, 100, 343–379. doi:10.1037/0022-0663.100.2.343
67.
SwellerJ. (1988). Cognitive load during problem solving: Effects of learning. Cognitive Science, 12, 257–285. doi:10.1016/0364-0213(88)90023-7
68.
SwellerJ. (2005). Implications of cognitive load theory for multimedia learning. In MayerR. (Ed.), Cambridge handbook of multimedia learning (pp. 19–30). New York, NY: Cambridge University Press. doi:10.1017/CBO9780511816819.003
69.
Turley-AmesK. J.WhitfieldM. (2003). Strategy training and working memory performance. Journal of Memory and Language, 49, 446–468. doi:10.1016/S0749-596X(03)00095-0
70.
UnsworthN.EngleR. W. (2007). On the division of short-term memory and working memory: An examination of simple and complex span and their relation to higher order abilities. Psychological Bulletin, 133, 1038–1066. doi:10.1037/0033-2909.133.6.1038
71.
U.S. Department of Education, Institute of Education Sciences. (2008). What Works Clearing House: Procedures and standards handbook (Version 2.0). Retrieved from http://ies.ed.gov/ncee/wwc/documentsum.aspx?sid=19
72.
van GarderenD. (2007). Teaching students with LD to use diagrams to solve mathematical word problems. Journal of Learning Disabilities, 40, 540–553. doi:10.1177/00222194070400060501
73.
van GarderenD.MontagueM. (2003). Visual-spatial representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research & Practice, 18, 246–254. doi:10.1111/1540-5826.00079
74.
WaescheJ. S. B.SchatschneiderC.ManerJ. K.AhmedY.WagnerR. K. (2011). Examining agreement and longitudinal stability among traditional and RTI-based definitions of reading disability using the affected-status agreement statistic. Journal of Learning Disabilities, 44, 296–307.
75.
WilkinsonG. S. (1993). The Wide Range Achievement Test. Wilmington, DE: Wide Range.
76.
XinY. P. (2008). The effect of schema-based instruction in solving mathematics word problems: An emphasis on prealgebraic conceptualization of multiplicative relations. Journal for Research in Mathematics Education, 39, 526–551.
77.
XinY. P.JitendraA. K. (1999). The effects of instruction in solving mathematical word problems for students with learning problems: A meta-analysis. Journal of Special Education, 32, 207–225. doi:10.1177/002246699903200402
