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Learning Disabilities Research & Practice,28(4), 170183C2013 The Division for Learning Disabilities of the Council for Exceptional Children

Effects of Cognitive Strategy Interventions and Cognitive Moderators on WordProblem Solving in Children at Risk for Problem Solving Difficulties

H. Lee Swanson, Cathy Lussier, and Michael OroscoUniversity of CaliforniaRiverside

This study investigated the role of strategy instruction and cognitive abilities on word prob-lem solving accuracy in children with math difficulties (MD). Elementary school children(N= 120) with and without MD were randomly assigned to 1 of 4 conditions: general-heuristic(e.g., underline question sentence),visual-schematicpresentation (diagrams),general-heuristic+ visual-schematic, and an untreated control. When compared to the control condition thatincluded children with MD, an advantage at posttest was found for children with MD forthe visual-schematic-alone condition on measures of problem solving and calculation accu-racy, whereas all strategy conditions facilitated posttest performance in correctly identifyingproblem solving components. The results also suggested that strategy conditions drew upondifferent cognitive resources. The General-heuristic condition drew primarily upon the execu-tive component of working memory (WM), Visual-schematic condition drew upon the visualcomponent of WM and the combined strategies condition drew upon number processing skills.

Word problems are an important part of mathematics pro-grams in elementary schools. Recent intervention studiesdirected toward improving problem solving accuracy in chil-dren with math difficulties (MD) have found support forteaching cognitive strategies. Several studies have foundthat general-heuristic strategy instruction (e.g., Montague,2008; Montague, Warger, & Morgan, 2000), as well asvisual-spatial strategies (e.g., Kolloffel, Eysink, de Jong, &Wilhelm, 2009; van Garderen, 2007) enhance childrensmath performance relative to control conditions (see Baker,Gersten, & Lee, 2002; Gersten et al., 2009; for reviews).Despite the overall benefits of strategy instruction in remedi-ating word problem solving difficulties, the use of strategiesfor some children with MD may not always be advantageous.From an aptitude-treatment perspective, not all children withMD may be expected to benefit from strategy training be-cause of the excessive demands on their cognitive processingskills.

The purpose of this intervention study was to exploresome of the cognitive skills that come into play whenattempting to improve problem solving accuracy in chil-dren with MD. Several recent syntheses of the experi-mental literature have identified effective problem solvingstrategies for children with MD (e.g., Fuchs, Fuchs, Schu-macher, & Seethaler, 2103; Gersten et al., 2009; Powell,2011; Xin & Jitendra, 1999; Zheng, Flynn, & Swanson,2013). Two general approaches have been found effective inboosting word problem solving performance. One approachemphasizes visual-schematic representations (e.g., Jitendra,Star, Rodriguez, Lindell, & Someki, 2011; Van Garden &

Requests for reprints should be sent to H. Lee Swanson, Uni-versity of CaliforniaRiverside, Electronic inquiries should be sent [email protected].

Montague, 2003). This approach teaches children to integratesolution relevant text elements into a coherent visualizationof the word problem (e.g., Powell, 2011; Van Garderen &Montague, 2003). Such procedures draw upon retrieval, re-tention, and transformation of visual information within aspatial context (Hegarty & Kozhevnikov, 1999). Anotherapproach emphasizes the execution of action sequences tosolve a problem (e.g., Hutchinson, 1993; Montague, 2008;Rosenzweig, Krawec, & Montague, 2011). That is, whenconfronted with a problem, children are taught a general-heuristic to identify within text what information is needed(e.g., Brissiaud & Sander, 2010; Hayes, Waterman, & Robin-son, 1977), as well as what information is irrelevant to solvea problem (Low & Over, 1989). This cuing childrens at-tention to text information, such as cues to underline thequestion sentence, is assumed to facilitate integrating infor-mation into a coherent problem representation (referred toin the literature as a situation model). These steps are notnecessarily tied to specific types of problems, and thereforeare assumed to have some generalizability. The advantagesof these procedures, when applied to children with MD, havebeen attributed to facilitating meta-cognitiveskills or abilities(e.g., Butler, Beckingham, & Lauscher, 2005; Rosenzweig,Krawec, & Montague, 2011).

In this study, we compare outcomes for each of thesetwo general approaches (referred to as visual-schematic andgeneral-heuristic) as well as assess some of the cognitiveprocesses related to performance outcomes using either ap-proach. We also compared these two strategies with a treat-ment that combines the elements of each approach. Thevisual-schematic intervention drew from some of the workof Van Garderen and Montague (2007) as well as stud-ies using diagrams from the Singapore curriculum (e.g.,Kolloffell et al., 2009). The general-heuristic treatment was

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LEARNING DISABILITIES RESEARCH 171

designed to cue attention to various types of text (sentence)information and drew from some of the earlier work of Mon-tague (1992; 2008) and others (Hutchinson, 1993; Shiah,Mastropieri, Scruggs, & Fulk, 19941995) that involvesteaching a series of procedural steps to identify informationwithin text.

We also designed the strategy conditions to include anumber of activities found effective in the intervention lit-

erature. For example, because warm-up activities related tocalculation have been found to be effective in problem solv-ing interventions, this component was also included in allstrategy training sessions (Fuchs et al., 2006; Fuchs et al.,2009). In addition, consistent withliteraturereviews that haveidentified key components related to treatment effectiveness(Gersten et al., 2009; Xin & Jitendra, 1999; Zheng et al.,2013), each strategy training session involved explicit prac-tice and feedback related to strategy use and performance.The strategy conditions also directed childrens attention torelevant relational propositions across sentences, although atthe same time increasing the number of irrelevant proposi-tions embedded within the word problem (Mayer & Hegarty,1996). To this end, this study addresses three questions:

Question 1. Do cognitive strategies enhance mathematicalproblem solving accuracy when compared to controlconditions for children with MD?

Based on existing studies, strategy conditions were pre-dicted to facilitate problem solving performance when com-pared to control condition. The elements of each proce-dure have been shown to yield moderate to high effect sizes(ESs) relative to control conditions (e.g., Zheng et al., 2013).However, some studies have suggested advantages related tovisual-schematic representations over more general-heuristicapproaches (e.g., Jitendra & Hoff, 1996; Jitendra et al., 2011).Others have suggested that focusing schema representations

is domain specific, and therefore suggest heuristic training isnecessary to enhance transfer (Koichu, Berman, & Moore,2007).

The issuesrelated to the preferred strategy approach, how-ever, may be partially related to cognitive strengths foundin children with MD. A recent meta-analysis of the cogni-tive literature on MD, for example, has suggested that suchchildren experience greater processing difficulties on verbal(e.g., identifying various aspects of text information) ratherthan visual-spatial information (Swanson & Jerman, 2006),and that visual-spatial strategies (use of diagrams) havebeen found to yield large ESs relative to control conditions(Gersten et al., 2009; median ES estimates of 0.67 relative tocontrol conditions). Based on the assumption that visual spa-

tial skills in children with MD is relatively intact when com-pared to verbal memory (Swanson & Beebe-Frankenberger,2004; however see, Andersson, 2010), we predict that visual-schematic conditions will yield higher accuracy scores whencompared to the general-heuristic procedures that focus pri-marily on highlighting text information.

Question 2. Are some cognitive strategies more effectivethan others in reducing the performance differences be-tween children with and without MD?

Although several strategy conditions may be effective rel-ative to the control condition within the MD sample, somestrategies may play a more important role for reducing theperformance gap between the two groups. This may occurfor several reasons, but one possibility explored is that somestrategies may make fewer demands on the cognitive limita-tions of children with MD.

Question 3. Do cognitive strategies place different de-mands on cognitive processes in children with MD?

This study also explores whether strategy conditions com-pensate for, or place excessive demands on, cognitive pro-cesses in children with MD. Two hypotheses are consid-ered. One hypothesis tested, referred to as the compensatorymodel, suggests that cognitive training reduces processingdemands, and therefore frees additional resources to solveproblems. The compensatory model predicts that childrenwithlower cognitive abilities aremore likely to place a greaterreliance on strategy conditions. Thus, a negative correlationbetween cognitive processes at pretest and problem solving

accuracy at posttest occurs within strategy conditions. A sec-ond hypothesis, referred to as the high cognitive skills model,suggests that children with MD who are most likely to ben-efit most from strategy conditions are those with relativelyhigher cognitive skills. This hypothesis predicts that signifi-cant positive correlations occurbetween cognitive processingat pretest and problem solving at posttest.

METHOD

Participants

One hundred and twenty (120; 55 females and 65 males)

children from grade 3 participated in this study. The finalselection was based on parent approval for participation andachievement scores. Ethnic representation of the sample was:63 Anglo, 17 Hispanic, 8 African American, 8 Asian, and 24mixed/other (e.g. Anglo and Hispanic, Native American).The mean SES of the sample was primarily low to middleSES based on free lunch participation, parent education, orparent occupation.

Definition of Risk for MD

Because the majority of children in the sample were not di-agnosed with specific learning disabilities in math problem

solving, we utilized the term at risk for MD in problem solv-ing to operationally define our sample. Children were definedas at risk if they performed at or below the 25th percentile ontwo norm-referenced word problem solving math tests; thestory problem subtest from the Test of Math Ability (TOMA;Brown, Cronin, & McIntire, 1994) and the KeyMath assess-ment (Connolly, 1998). The 25th percentile cut-off scoreon standardized achievement measures has been commonlyused to identify children at risk (e.g., Fletcher et al., 1989;Siegel & Ryan, 1989).

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172 SWANSONL LUSSIER, AND OROSCO: EFFECTS OF COGNITIVE STRATEGY INTERVENTIONS

TABLE 1

Comparison of Children with and without Math Difficulties on Classification, Moderator and Pretest Variables

Children with MD Children without MD

Mean SD Mean SD KR20 ES F-Ratio

Measures

Age (Mos.) 103.24 6.63 104.41 6.23 0.18 0.96

Word problem solving

TOMA/KeyM_S 7.76 1.28 11.24 1.16 0.82 2.85 235.89**Fluid intelligence

Raven_P 51.89 23.74 65.73 23.34 0.97 0.59 10.02**

Reading

TORC_S 10.08 1.62 11.39 2.04 0.79 0.72 14.86**

WRAT_S 105.3 11.93 114.73 14.48 0.87 0.71 14.70**

Calculation

WIAT_S 101.51 14.72 105.80 10.11 0.88 0.35 3.46*

WRAT_S 101.66 9.13 102.82 9.40 0.80 0.13 0.45

Number processes

RNG_R 8.54 3.93 9.35 5.88 0.76 0.17 0.79

NUMERACY 16.49 5.52 19.06 6.98 0.8 0.41 4.87*

ESTIMATION 18.89 13.48 15.49 14.41 0.95 0.24 1.71

RDN_R 37.87 10.35 38.49 10.57 0.83 0.06 0.10

RDN_S 11.71 2.26 11.34 2.06 0.83 0.17 0.76

Compositea 0.07 0.47 0.11 0.42 0.40 4.74*

WM-Executive

Compositeb 0.24 0.47 0.34 0.79 0.84 0.92 24.43**

WM-visual-spatial

Compositec 0.15 0.64 0.21 0.85 0.89 0.48 6.77**

WM-phonological-STM

Composited 0.14 0.71 0.22 0.75 0.73 0.49 7.01**

Pretest scores

CMAT_R 5.60 2.51 8.57 2.42 0.90 1.20 41.84**

Calculation_C 0.16 0.89 0.23 0.99 0.80 0.41 4.99**

Components_C 0.11 0.7 0.15 0.65 0.81 0.39 4.30*

Note. _C= composite z-score, _S= standard or scale score, _R= raw score, _P =percentile score, TOMA= Test of Math Ability, CMAT = Comprehensive

Test of Math Abilities,KEYM=KeyMath, TOMA=Test of Math Ability, WRAT=Wide Range Achievement Test, TORC=Test of Reading Comprehension,

RNG= random generation of numbers, RDN = rapid digit naming. *p < .05, **p < .01, ***p < .001.aComposite= mean z-score of math processes (random number generation, estimation, numeracy, and rapid digit naming).

bComposite= mean z-score of executive WM (conceptual span, digit/sentence span, and updating).cComposite= mean z-score of visual-spatial WM (visual matrix and mapping/direction task).dComposite= mean z-score of STM (forward digit span, word span, and nonword span tasks).

It is also important to note that the focus of this studywas on word problem solving deficits and not on calculation;therefore, we identified those children who performed in thelower 25thpercentile on norm-referenced word problem solv-ing math tests and above the 25th percentile in calculationas being at risk for MD. We also sought to identify chil-dren whose skills in other areas (fluid intelligence, compu-tation, and reading comprehension) were within the normalrange. This allowed us to separate and retain the current sam-ple into 71 children with MD (32 females) and 49 children(23 females) without MD. No significant differences emergedbetween children with and without MD as a function ofgrade, 2 (df= 1, N= 120) = .64, p > .10; ethnicity, 2

(df= 4,N= 120) = .42,p > .05; gender, 2 (df= 1,N=120)= .04,p> .10; or chronological age,F(1,118)= . 96,p> .05. Table 1 shows the means and standard deviations forchildren with and without MD before random assignment totreatment.

Design and Treatment Conditions

Random Assignment

Based on district wide tests, children with and without MDwere randomly assigned to conditions. To control for the pos-sible impact of classroom teacher, each participating class-room included students in each of thedifferent strategy condi-tions. After random assignment, children were separated fordata analysis into those at risk and not at risk for MD basedon the aforementioned story problem measures (TOMA andKeyMath). Children were randomly assigned to either a con-trol group (N= 26, MD = 16) or to one of three treat-ment conditions (general-heuristic strategies [N= 40, MD=21], general-heuristic + visual-schematic strategiesN= 34,MD = 22], and visual-schematic-alone strategies, [N= 20,MD = 11]). 2. The uneven sample size reflected the removalof children from the data analysis not meeting the operational

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criteria (e.g., low reading scores, low fluid intelligence score)for defining the sample as at risk or not at risk for MD. Nosignificant differences emerged among the 4 conditions as afunction of the proportion of children at risk status, 2 (df=3, N= 120) = 1.34,p > 0.05, ethnicity, 2 (df= 12,N=120) = 19.81,p > 0.05, or gender, 2 (df= 3,N= 120) =1.22,p > 0.10.

Common Instructional Conditions

All children in the study participated with their peers intheir homerooms on tasks and activities related to the districtwide math school curriculum. The school-wide instructionacross conditions was the enVisionMATH Learning Curricu-lum (Pearson Publishers, 2009).

Experimental Conditions

Each experimental treatment condition included 20 scriptedlessons administered over 8 weeks. Each 30-minute lesson

was administered 3 times a week in small groups of twoto four children. Lesson administration was done by one ofsix tutors (graduate students or paraprofessionals). Childrenwere presented with individual booklets at the beginning ofthe lesson, and all responses were recorded in the booklet.Each lesson consisted of four phases: warm-up, instruction,guided practice, and independent practice.

The warm-up phase included two parts which involvedcalculation of problems that required children to provide themissing numbers (9+ 2= x, x + 1= 6; x 5= 1) and a setof puzzles based on problems using geometric shapes. Thisactivity took approximately 3 to 5 minutes to complete.

The directinstruction phaselasted approximately 5 min-utes. At the beginning of each lesson, the strategies and/or

rule cards were either read to the children (e.g., to find thewhole, you need to add the parts) or reviewed. Across the 20lessons, seven rules were presented. Depending on the treat-ment condition, children were taught the instructional inter-vention: general-heuristic strategy-alone, general-heuristic+ visual-schematic strategies, or visual-schematic strategy-alone.

Steps for the General-heuristic strategy approach includeddirecting the child to find and identify within the word prob-lem: (i) the question sentence and underline it, (ii) the sen-tences with numbers and circle them, (iii) the key words andplace a square around them and (iv) the irrelevant sentencesand cross them out (see Appendix for an example). The childwas then directed to decide what operation needed to be

done (add/subtract/or both) and then solve the problem. Forthe visual-schematic only strategy condition students weretaught how to use two types of diagrams (see Appendix forexample). The first diagram represented how parts made upa whole and the second represented how quantities are com-pared. The second diagram consisted of two empty boxes,one bigger and the other smaller, which students were askedto fill in with the correct numbers representing the quantitiesfrom the word problem. An equation with a question mark,

which acted as a placeholder for the missing number, was pre-sented. Students then identified which number was neededto replace the question mark and solve the problem. Finally,for the combined general-heuristic + visual-schematic strat-egy condition, diagramming was added to the six general-heuristic strategy steps. In the diagramming step, studentsfilled in the diagram with given numbers and identified themissing numbers in the corresponding slots in the boxes (i.e.,

part/part/whole).The third phase, guided practice, lasted 10 minutes andinvolved children working on three practice problems. Tu-tor feedback was provided on the application of steps andstrategies for each of these three problems. In this phase,children also reviewed problems from the examples from theinstructional phase. The tutor assisted children with findingthe correct operation, identifying key words, and providingcorrective feedback on the solution.

The fourth phase, independent practice, lasted 10 min-utes and required students to independently answer anotherset of three word problems without feedback. If the studentfinished the independent practice tasks before the 10 minuteswere over, they were presented a puzzle to complete. Stu-

dent responses were recorded for each session to assess theapplication of the intervention strategies and problem solv-ing accuracy. For the general-heuristic strategy condition,points were recorded for identifying the correct numbers, ap-plying strategies (e.g., underlining), identifying the correctoperations, and solution accuracy. For the visual-schematic-alone strategy condition, points were recorded for correctlychoosing the correct diagram, correctly filling in the num-bers for the diagram, identifying the correct operations, andcorrectly solving the problem. For the general-heuristic +visualschematic strategy condition, points were recordedfor correctly choosing the diagram, inserting correct num-bers, applying strategies, identifying the correct operations,and correctly solving the problem. The mean points scores

averagedacross all sessions exceed 90 percent acrosstrainingconditions.

Increments in Lesson Plans

As indicated, the materials for word problems for each in-dependent practice session included three parts: questionsentences, number sentences, and irrelevant sentences. Thenumber of sentences gradually increased across the sessions.

Treatment Fidelity

Independent evaluations were carried out to determine thetreatment fidelity. During the lesson sessions, tutors wererandomly evaluated by independent observers (a postdoc-toral student, a graduate student, or the project director).The observers independently filled out evaluation forms cov-ering all segments of the lesson intervention. Points wererecorded on the accuracy with which the tutor implementedthe instructional sequence as intended. Observations of eachtutor occurred for six sessions randomly distributed across

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instructionalsessions. Interrater agreement was calculated onall observations and exceeded 90 percent across all observedcategories. Tutors following each step of strategy implemen-tation (10 observable treatment specific items were coded)and yielded individual scores above 95 percent across allobserved sequences and conditions. The percent of strategyimplementation as intended (maximum scoreof 10) exceeded95 percent across all observed sessions.

Measures

The battery of group and individually administered tasksis described below. Tasks were divided into classification,pretest/posttest, and pretest only (moderator) measures. Thesample reliabilities for each measure varied from .80 to .98and are reported in Table 1.

Classification Measures

Norm referenced measures were administered before inter-

vention to assess word problem solving accuracy using aTOMA (Brown, Cronin, & McIntire, 1994) along with Key-Math (Connolly, 1998) and measures of fluid intelligence(Raven Colored Progressive Matrices; Raven, 1976), read-ing word recognition, and comprehension (WRAT-3; Test ofReading Comprehension, Brown, Hammill, & Weiderholt,1995). It is important to note that calculation measures (WideRange Achievement Test-3; Wilkinson, 1993; Wechsler Indi-vidual Achievement Test Psychological Corporation, 1992)were administered at pretest to insure childrens arithmeticskills were in the average range, and again at posttest to assesstreatment effects.

Pretest and Posttest Measures

Alternate forms of story problems from the ComprehensiveMathematical Abilities Test (CMAT; Hresko, Schlieve, Her-ron, Swain, & Sherbenou, 2003) and WRAT-3 and WIATwere administered at pretest and posttest. The alternate formswere counterbalanced for presentation order. We investigatedcalculation (WIAT, WRAT-3) because it was a required ac-tivity for all sessions and therefore a logical measure to assesspossible transfer effects.

Mathematical Word Problem Solving Components

This experimental test assessed the childs ability to iden-tify (retrieve) propositional information within word prob-lems (Swanson & Beebe-Frankenberger, 2004). This mea-sure is closely related to the intervention instruction thatdirects childrens attention to the components of word prob-lems outlined by Mayer and Hegarty (1996). Each bookletcontained three problems that included pages assessing therecall of text from the mathematical word problems. The cat-egories of mathematical word problems were addition, sub-traction, and multiplication. Problems were four sentences

in length and contained two-assignment propositions, andone relation, one question, and an extraneous proposition(irrelevant sentence) related to the solution. To control forreading problems, the examiner orally read (a) each problemand (b) all multiple-choice response options as the studentfollowed along. The total score possible for propositions re-lated to question, number, goal, operations, algorithms, andtrue-false questions was 12.

Pretest Only (Moderator Variables)

Meta-analyses have identified processes related to calcula-tion, speed, reading, working memory (WM) and inhibitionas potential moderators of change (growth) in children withMD problem solving performance (e.g., Swanson & Jerman,2006). Thus, it was necessary to administer several of thesemeasures at pretest because of their potential to moderatetreatment outcomes.

Memory Measures

Administered were three WM measures from a normativemeasure (S-CPT; Swanson, 1995) that capturedexecutive

processing(Conceptual Span, Updating, and Digit/Sentence)and two measures that captured visual-spatial WM (Map-ping & Directions and Visual Matrix). The inter-correlationsamong the processing measures exceeded .45 and the re-ported Cronbach Alpha reliabilities are greater than .88(Swanson & Beebe-Frankenberger, 2004). In addition, threemeasures of short-term memory (STM) were administered:Forward Digit Span (WISC III; Psychological Corporation,1991), Word Span, and Pseudoword Span tasks (Swanson &Beebe-Frankenberger, 2004).

Digit Naming Speed

The administration proceduresfollowed those specified in themanual of theComprehensive Test of Phonological Process-ing(CTOPP; Wagner, Torgesen, & Rahotte, 2000). Partici-pants were required to name the digits as quickly as possiblefor each of two stimulus arrays containing 36 items, for atotal of 72 items. The dependent measure was the total timeto name both arrays of numbers.

Random Generation of Numbers

The random number generation task has been well articu-lated in theliterature as a measureof inhibition (e.g. Towse &Cheshire, 2007). Each child was asked to write as quickly aspossible numbers in sequential order to establish a baseline.Children were then asked to write numbers in a random non-systematic order. Scoring included an index for randomness,redundancy, and percentage of paired responses to assess thetendency to suppress repetitions.

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Estimation

In addition, two instruments on estimation (numeracy esti-mation and line estimation) were adapted from Siegler andOpfer (2003) and Seigler and Booth (2004). The numeracyestimation task presented 25 groups of numbers on the firstpage of a booklet with three numbers in each group (e.g., 231811). The participant was then asked to circle the largest

number in each group on the first page as fast as they couldin 30 seconds. A second page had an additional 25 groups ofnumbers with three numbers in each group. Each participantwas then asked to circle the smallest number in each group asfast as they could in 30 seconds. Scoring included combiningthe raw scores from the two parts to get a total.

The lineestimationtask was presented to participants withtwo equivalent forms. The first page of this task required eachstudent to examine five straight lines. Each line was identicalin length and was marked with a zero on one end and onehundred on the other, creating a blank number line. A numberwas placed above the center of each line. Each child was thenasked to estimate where they thought the number presentedshould be placed on the line and indicated this by marking

an X on the line. A second page of this task required eachstudent to examine another set of five straight lines that weredifferent lengths, but with each still marked with a zero onone end and one hundred on the other end. As before, eachchild was asked to estimate where the number presentedshould be placed on the line. A raw score was calculated byusing a transparency template and counting how many unitsof measure the X was from the answer.

Statistical Analysis

Children were drawn from 21 classrooms. Because the datareflected treatments of children nested within classrooms

and children with MD and without MD (NMD) nestedwithin treatments, a hierarchical linear model (HLM; Bryk &Raudenbush, 2002) or mixed regression analysis was neces-sary to analyze treatment effects. A mixed regression modelwas selected over a mixed ANCOVA because the MD ver-sus NMD contrast was a continuous variable and thereforeentry of these contrasts simultaneously into the regressionmodel allowed for the assessment of variance within groups.The fixed and random effect parameter estimates were ob-tained using PROC MIXED in SAS 9.3 (SAS Institute, Inc,2010). The criterion variables in the analysis were posttestscores for solution accuracy (CMAT), calculation accuracy(WRAT-3 + WIAT), and identification of problem solvingcomponents. It is important to note that the treatment con-

ditions were entered as binary variables (e.g., strategy-only+ 1, other conditions 0) that by default allowed comparisonsfor treatments for MD and NMD groups to the control con-dition of children with MD (control-MD). Thus, the fixedeffect intercept shown in Table 2 reflected the performanceof the control group with MD. The intraclass correlations forthe posttest criterion measures of solution accuracy, calcu-lation accuracy, and component identification accuracy were.34, .06, .18, respectively. We used a maximum-likelihoodestimation to compute the parameters in the various models

as well as robust standard errors (Huber-White) to allow forthe nonindependence of observations from children nestedwithin classrooms.

Results

Pretest Comparisons

Table 1 provides the means, standard deviations, and relia-bility for the measures. To reduce the sample to task ratio inthe subsequent analyses, composite scores were created. Thecomposite scores for calculation (WIAT and WRAT-3), read-ing (WRAT-3 and TORC), STM (forward digit span, wordspan, phonological span), executive WM (updating, concep-tual span, digit/sentence span), word problem solving com-ponents (identification of question, number, goal, algorithm,operation, and irrelevant components), and math processes(line estimation, rapid digit naming speed, numeracy, andrandom number generation) were based on the mean z-scorecomputed from the raw scores.

Ability Group Comparisons

A MANOVA compared children with and without MD onscores for fluid intelligence (Raven Colored Progressive Ma-trices Test), computation, reading, STM, executive WM,visual-spatial WM, word problem solving components, andmath processes. The groups were also compared on wordproblem solving accuracy (CMAT). The ESs and univariatesfor individual tasks are shown in Table 1. As expected, theMANOVA was significant and in favor of children withoutMD, = .57, F (9,109) = 9.03, p < .0001. Significantunivariates in favor of children without MD are reported inTable 1.

Treatment Comparisons

A MANOVA comparing treatment conditions was computedon the total sample for pretest scores of word problem accu-racy (CMAT), computation (WIAT and WRAT-III), reading(TORC and WRAT), Fluid intelligence (Raven), math pro-cesses, STM, visual-spatial WM, and executive WM. TheMANOVA was not significant, = .72,F(24,299) = 1.55,

p > .05.

Posttest Comparisons

For comparison purposes, the posttest criterion measures(problem solving accuracy on the CMAT, calculation ac-curacy on the combined WRAT and WIAT measures, andidentification accuracy on the problem solving componentmeasure) were converted to z-scores based on the mean andstandard deviation at pretest. The adjusted posttest accuracyscores, standard errors, and ESs are shown in Table 2. Thetop section of Table 2 shows the adjusted posttest accuracyscores and standard errors. The middle section shows the ESs

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TABLE 2

Adjusted z-score Posttest Means, Standard Errors and Effect Sizes as a Function of Group and Treatment

Gen eral h euristi c Visua l-schemati c-

General heuristic + visual-schematic alone Control

Mean SE Mean SE Mean SE Mean SE

Children without MD

Problem solving .80 .25 .60 .27 .60 .31 .66 .28

Calculation .76 .21 1.24 .23 .83 .28 .84 .25

Components .28 .22 .48 .25 .37 .29 .73 .25

Children with MD

Problem Solving .35 .21 .66 .28 .84 .25 .23 .17

Calculation .83 .19 1.11 .20 1.31 .23 .71 .15

Components .07 .17 .15 .20 .17 .24 .41 .17

ES between groupsa ES ES ES ES

Problem solving 0.43 0.05 0.27 .49

Calculation .08 0.14 0.59 0.17

Components 0.24 0.34 0.24 1.42

ES between conditions

1 versus 2 1 versus 3 1 versus 4b 2 versuss 3 2 versus 4b 3 versus 4b

Children without MD

Problem solving .17 .18 .12 .001 .05 .06

Calculation .14 .07 .08 .41 .38 .01

Components .18 .09 .45 .10 .23 .37Children with MD

Problem Solving .33 .59 0.11 0.16 0.36 0.57

Calculation 0.37 .63 0.12 .20 0.38 0.62

Components 0.11 .14 0.48 .02 0.51 0.59

Note. Problem Solving = CMAT, Calculation = combined WRAT & WIAT, Components = identification of problem solving

components. 1 = general-heuristic, 2 = general-heuristic+ visual-schematic, 3 = visual-schematic-alone, 4 = control. Bold= moderate effect sizes.aNegative effect size favors of children without MD,bNegative effect size f avors control condition.

comparing children with and without MD within each condi-tion. The bottom section shows the ESs comparing treatmentconditions within each ability group.

Before the mixed regression analysis, ESswere computed.The ESs in Table 2 were adjusted for the covariates of pretest,reading, fluid intelligence, and the dependence among mea-sures within the same classroom. Hedges gwas calculatedwhere = /[(SD1

2)(N1) + (SD22)(N2)/2]

1/2 where is theHLM coefficient for the adjusted posttest mean differencebetween treatment (partial for pretest and covariates), ad-justed for both level-1 and level-2 covariates, andN1 andN2are the sample sizes. The SD1 and SD2 were the standarddeviations from the posttest. For the interpretation of themagnitude of the ESs, Cohens (1988) distinction was used;an ES of 0.20 is considered small, and ES of 0.50 and 0.80are considered moderate and large, respectively.

As shown in the bottom sectionof Table 2, when compared

to the control condition across the three criterion measures,the adjusted posttest ESs were in the moderate range, andin favor of the visual-schematic-alone condition for childrenwith MD.

Question 1. Do cognitive strategies enhance mathematicalproblem solving accuracy when compared to controlconditions for children with MD?

To address question 1, a mixed regression analysis wascomputed on the posttest measures. Model 1 included predic-

tions of posttest performance without covariates and Model 2included predictions of posttest performance with covariates.For each model, comparisons included treatment conditions

of each subgroup of math ability to the control group thatincluded children with MD.

Posttest Solution Accuracy (CMAT)

As shown in Table 3, Model 1 compared treatment condi-tions for each subgroup of math ability to the control groupthat included children with MD, but without pretest enteredinto the analysis. The posttest problem solving mean accu-racy z-score for the participants in the control condition with

MDwas -.05, and was no better than chance. As expected,all treatment conditions at posttest for children without MDwere significantly (all ps < .05) better than the control con-

dition for children with MD. In contrast, children with MDin the visual-schematic-alone condition yielded significantlyhigher posttest scores than children with MD in the controlcondition.

Model 2 entered pretest scores as well as reading andfluid intelligence scores into the mixed regression analysis.As shown in Table 2, adjusted posttest z-scores for childrenwith MD for the general-heuristic strategy, general-heuristic+ visual-schematic strategy, visual-schematic-alone strategyand control conditions were .35 (.23 + .12), .66 (.23 + .43),

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TABLE 3

Hierarchical Linear Model for Predicting Posttest Performance as a Function of Strategy Conditions

Problem Solving (CMAT) Components Calculation

Model 1 Model 2 Model 1 Model 2 Model 1 Model 2

Fixed Effects Estimate SE Estimate SE Estimate SE Estimate SE Estimate SE Estimate SE

MD-CRTL Intercept 0.05 .2 0.23 .17 0.47* .24 0.71** .15 0.46** .16 0.41** .17

GH versus MD-CRTL

MD 0.21 .25 0.12 .21 0.35 .30 0.12 .19 0.54** .20 0.48* .02

NMD 1.17** .28 0.57* .25 0.55 .33 0.05 .21 0.77** .22 0.69* .22

GH+Vis versus

MD-CRTL

MD 0.42 .26 0.43* .22 0.87** .30 0.40* .20 0.56** .21 0.56* .2

NMD 1.04** .29 0.37 .27 0.74* .36 0.53* .23 1.00** .23 0.89* .25

Vis versus MD-CRTL

MD 0.69* .3 0.61* .25 1.06** .36 0.60** .23 0.65** .24 0.58* .24

NMD 1.08** .34 0.37 .31 0.59 .42 0.12 .28 0.88** .28 0.78* .29

NMD-CRTL versus

MD-CRTL

1.17** .3 0.43 .28 -0.21 .42 0.13 .25 1.21** .25 1.14** .25

Covariate

Pretest 0.59** .07 0.83** .08 0.20* .1

Raven 0.01 .08 0.001 .01 0.02 .01

Reading 0.11 .08 .30** .09 0.03 .09

Random Effects Variance SE Variance SE Variance SE Variance SE Variance SE Variance SE

Intercept 0.34* .19 0.03 .04 0.06* .03 0.0004 .001 0.07 .05 0.02 .03

Residual 0.53** .07 .40** .06 0.80** .1 0.33*** .04 0.35* .05 0.35* .05

Note. CRTL = control condition, GH = general-heuristic, Vis = visual-schematic, Problem Solving = solution accuracy measured by CMAT, Calculation =

combined WRAT & WIAT, Components = correct identification of problem solving components (e.g., goal, question, numbers, etc.) at pretest and posttest,

MD = children with MD, NMD = children without MD,

*p < .05. **p < .01, ***p < .001.

.84 (.23 + .61), and .23, respectively. A series of post hocanalyses were computed comparing treatment effects withingroups. For children with MD, a Tukey test yielded signif-icant adjusted posttest score advantages (all ps < .05) forthe visual-schematic-alone condition when compared to the

other conditions (visual-schematic-alone> general-heuristic+ visual schematic> general-heuristic-alone= control). Forchildren without MD, a Tukey test yielded no significant dif-ferences (allps> .05) in adjusted posttest scoresas a functionof treatment conditions (general-heuristic-alone = visual-schematic + general-heuristic = visual-schematic-alone =control).

Posttest Calculation Accuracy

The two aforementioned models were computed on posttestcalculation scores (composite of WRAT-3 and WIAT).

Model 2 shown in Table 2 indicated that children withMD in the visual-schematic-alone and general-heuristic+visual-schematic condition significantly (ps < .05) out-performed children with MD in the control condition. Theadjusted posttest estimated z-scores for children with MDfor the general-heuristic strategy, general-heuristic + visual-schematic strategy and Visual-schematic-alone strategy con-ditions and control were .83 (.71 + .12), 1.11 (.71 +.40), 1.31 (.71 + .60), and .71, respectively. For childrenwith MD, a Tukey test indicated a significant advantage

(ps < .05) for the visual-schematic-alone and the general-heuristic +visual-schematic relative to the other condi-tions (visual-schematic-alone > general-heuristic + visual-schematic > general-heuristic-alone = control). The re-sults also showed that reading skill significantly moder-

ated the treatment outcomes. For children without MD, aTukey test indicated that a significant (ps < .05) posttesttreatment advantage for the General-heuristic + visual-schematic condition relative to the other conditions (general-heuristic + visual-schematic > general-heuristic-alone =visual-schematic-alone =xbrk control).

Component Identification

The two mixed regression models were computed on posttestscores correctly identifying components within word prob-lems. As shown in Model 1, all children in the treatment

conditions outperformed children with MD in the controlcondition. Model 2 showed treatment advantages within thetwo ability groups. Within the MD group, a Tukey test in-dicated that strategy conditions yielded significantly (ps control). For children without MD, a Tukey test found a sig-nificant advantage (ps< .05) for the adjusted posttest controlcondition scores relative to the other conditions (control >

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general-heuristic-alone = general-heuristic + visual-schematic = visual-schematic-alone).

Summary Question 1

Overall, the results related to the analysis for Question 1showed when compared to the control condition, an advan-

tage at posttest was found for children withMD for thevisual-schematic strategy only condition on measures of problemsolving accuracy, calculation accuracy and correctly identi-fying problem solving components.

Question 2. Are some cognitive strategies more effectivethan others in reducing the performance differences be-tween children with and without MD?

As expected, the pretest analyses showed a significant ad-vantage for children without MD when compared to childrenwith MD on the criterion measures before random assign-ment to treatment condition. However, it was of interest todetermine if ESs between the two groups varied substan-tially at posttest after partialing out the influence of pretest.

Because we do not have the power to test the hypothesisof no differences between the groups at posttest, we limitedour analysis to determining the magnitude posttest ESs (par-tialing for pretest and covariates) between the two groupswithin each condition. As shown in the middle of Table 2,negative ESs were largest (in favor of children without MD)for General-heuristic-alone and control condition. In con-trast, positive ESs were found in favor of children with MDfor problem solving and calculation accuracy for the Visual-schematic-alone condition. Interestingly, minimal variationin ESs occurred between strategy conditions for the correctidentification of problem solving components.

Question 3. Do cognitive strategies place different de-

mands on cognitive processes in children with MD?

We next tested the assumption that different strategy con-ditions draw upon different cognitive resources. Becausesubstantial differences emerged between ability groups onseveral moderator variables (reading, and WM) before treat-ment (see Table 1), cognitive variables were correlated withposttest performance after partialing out the influence ofpretest performance. For data reduction purposes, variablesconceptually related were combined into composite scores.A confirmatory factor analysis determined the model fit forthese combinations. The CALIS (Covariance Analysis andLinear Structural Equation) software program (SAS Insti-tute, Inc., 2010) was used to test whether the three WM tasks

(Listening Span, Conceptual Span, and Updating) were load-ing on one factor, the three STM tasks (Forward Digit Span,Pseudoword Span, and Word Span) were loading on a secondfactor, the two visual WM tasks were loadingon a thirdfactor(Visual Matrix, Mapping and Directions) and the math pro-cesses were loading on a fourth factor (numeracy, randomgeneration, number naming speed, and estimation). Thesefour factorswere labels as executive WM), STM, visual WM,and number processing, respectively. The fit statistics were.94 for the comparative fit index .90 for the Lewis Tucker In-

TABLE 4

Correlations Between Posttest Solution Accuracy with Reading and

Cognitive Measures

General-Heuristic General-Heuristic+

Alone Visual-Schematic

Partial Partial

Total MD Total MD

Related skillReading 0.21 0.32 0.02 0.08

Processes

Number 0.01 0.18 0.38* 0.50*

Visual-spatial WM 0.14 0.15 0.38* 0.33

STM 0.02 0.21 0.08 0.21

Executive-WM 0.36* 0.49* 0.02 0.03

Visual-Schematic

alone Control

Partial Partial

Total MD Total MD

Related skill

Reading 0.36 0.01 0.02 0.11

Processes

Number 0.25 0.37 0.22 0.13

Visual-spatial WM 0.51** 0.41* 0.21 .41*

STM 0.08 0.27 0.01 0.06

Executive-WM 0.30 0.04 0.12 0.22

Note. STM = short-term memory or phonological component of WM.

Total= children with and without MD, MD = sample of children with MD

only; Partial = coefficient partialed for pretest Calculation and Component

Identification performance. Bold=partialed coefficients > .30.

*p < .05, **p

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major role in the general-heuristic-alone condition, thexbrklatent measure for number processing in the general-heuristic+ visual-schematic condition, and the latent measure forvisual-spatial WM in the Visual-schematic-alone and con-trol condition. Second, except for the visual-schematic-alonecondition, the number processing latent variable (namingspeed, numeracy, random generation) was positively corre-lated with posttest problem solving accuracy. Finally, reading

proficiency was negatively related to posttest accuracy in thegeneral-heuristic-alone condition for children with MD, sug-gesting that focusing on text may have been more helpful tolower than higher readers.

Discussion

This study investigated the role of strategy instruction andcognitive abilities on word problem solving accuracy in chil-dren withMD. Word problem solving deficitsin children withMD have been found to be persistent across time, even whencalculation and reading skills are within the normal range(e.g., Swanson, Jerman, & Zheng, 2008). Thus, assessing the

role of intensive cognitive strategies was the focus of thisstudy. Overall, the results show that visual-schematic strat-egy instructions facilitated posttest performance for childrenwith MD on measures of problem solving accuracy, cal-culation, and the correct identification of problem solvingcomponents. Although a positive finding on the experimen-tal measure (identifying key components of problem solving)was expected, the results related to problem solving and cal-culation accuracy on normed referenced measures were im-portant findings. The results will now be discussed in termsof the three questions that directed the study.

Question 1. Do cognitive strategies enhance mathematicalproblem solving accuracy when compared to control

conditions for children with MD?

When compared to the control condition, an advantagewas found on posttest problem solving and calculation accu-racy for children with MD for the visual-schematic strategy,whereas all strategy conditions facilitated posttest perfor-mance in correctly identifying problem solving componentsrelative to the control condition. One rather straightforwardexplanation for the superior outcomes of the visual-schematiccondition was that the use of diagrams allowed for some ob-vious mapping of the numbers from the text which in turnallowed for a direct translation into a set of computations. Inaddition, the visual-schematic strategy for children with MDmay have provided a technique that allowed them to focus on

the relevant aspects of the task without being distracted byirrelevant information.

There are, of course, other interpretations to these find-ings. One possibility is that the Visual-schematic conditionmay have required more active processing than the General-heuristic condition.As shown in Table 4, posttest accuracyforthe visual-schematic condition yielded moderate ESs (corre-lations > .30) with both number and visual-WM process-ing. In contrast, posttest accuracy for the general-heuristic-alone strategy was related to domain general processing

(executive WM r= .49) and less so to number processing(r= .18). Thus, the Visual-schematic condition may haveactivated more intensive processing of numerical informa-tion, as well as provided more opportunity for actual problemsolving than the general-heuristic approach. That is, childrenin the general-heuristic condition may have attended more tothe reading text to underline, cross-out, or box various propo-sitions than engage in actual problem solving. No doubt, at-

tributing differences in outcomes to these subtle processingdifferences are highly inferential on our part because ourcognitive measures were administered before posttest condi-tions, and therefore are not directly tied to online processing.

Our interpretations also need to be placed in the contextof our findings for the general-heuristic +visual-schematiccondition. We assumed that this condition would involvemore intense processing because children needed to producenot only correct visual-schematic representations, but alsoaccurately identify (via underlining, crossing out, etc.) textinformation within the word problem. If one assumes thatcombining the two strategies is more processing intensive,then this condition should have superseded those conditionsthat focus on each strategy in isolation. In addition, one could

make the case that the combining the general-heuristic andthe visual-schematic condition would increase the chances todraw upon separate verbal (i.e., linguistic information fromtext) and visual-spatial (i.e., visual representation) storagecapacities. The combination of these storage systems wouldopen up the possibility for more information to be processedand retained without making excessive demands on WM.Such was not the case in this study. Although the posttestoutcomes for the combined condition were related to indi-vidual differences in number processing (r= .50), the com-bined condition did make demands on visual-spatial abilities(r= .33). In addition, it is important to note that althoughthe combined condition was an improvement over the controlcondition (ES = .36), the magnitude of the ES was small.

The results also need to be placed in the context ofour findings related to correctly identifying the proposi-tions within word problem that lead to correct solutions.For all three strategy conditions, children with MD wereable to infer the correct relations between solution-relevantelements from the problem texts relative to the control con-dition (ESs on the component criterion measure were .48,.51, and .59 for General-heuristic-alone, general-heuristic +visual schematic, visual-schematic-only, respectively). Thisfinding suggests that instruction related to combining propo-sitional information within text is a necessary condition forsuccessful word problem solving, but not a sufficient condi-tion. Thus, our best interpretation of the advantage relatedto the visual-schematic condition-alone is that it draws upon

the relatively strengths in childrens visual-spatial WM skills.We infer this because the magnitude of ESs between posttestsolution accuracy and visual WM was moderate for the MDgroup (r= .41) and high for the total sample (r= .51) withinthe visual-schematic-alone condition. Thus, we assume thatsome of the advantages for this condition were related to indi-vidual differences in visual-spatial skills and not necessarilydifferences in the depth of processing required. We would beremiss, however if we did not mention that, in our follow-up research with a different sample, a clear advantage was

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found across 2 years of training for the visual-schematiccondition relative to the control (Swanson, Lussier, &Orosco, in press), but the visual-schematic condition was noteffective relative to the other conditions on high stake prob-lem solving measures that required silent reading (Swanson,Orosco, & Lussier, in press). Thus, further research is neces-sary to isolate the processes that account for the advantagesand disadvantages related to the visual-schematic condition.

The overall results are interesting in light of the cogni-tive literature on children with MD. Several studies havesuggested that visual-spatial WM is closely linked with MD(e.g., Bull, Epsy, & Weibe, 2008) and therefore the ques-tion as to why children with MD were more likely to ben-efit from the visual-schematic-alone when compared to thegeneral-heuristic treatment conditions is an important one.Some studies have suggested that visual-spatial WM plays akey role in some processing struggles found in children whosuffer from MD (e.g., Bull, Johnston, & Roy, 1999). A re-cent meta-analysis synthesizing research on cognitive studiesof MD (Swanson & Jerman, 2006) suggests, however, thatmemory deficits are more apparent in the verbal than visual-spatial WM domain. These findings suggest that visual WM

in children with MD is relatively intact when compared to theverbal WM processes and, therefore, is an important routefor strategy training. It is possible that visual-spatial memoryfunctions as a mental blackboard to support number rep-resentation as well as specific associations between visual-spatial memory and encoding in problems presented visu-ally (Kolloffel, Eysink, de Jong, & Wilhelm, 2009; MeyerSalimpoor, Wu, Geary, & Menon, 2010). Therefore, our re-sults showing an advantage for the visual-spatial interventionmay coincide with a preference in processing visual infor-mation relative to interventions that place an emphasis onintegrating propositional information within word problems.

Question 2. Are some cognitive strategies more effective

than others in reducing the performance differences be-tween children with and without MD?

The answer to this question is clearly shown in Table 2.The ESs on the adjusted posttest scores were smaller betweenchildren withand without MD for the visual-schematic-aloneand general-heuristic+ visual-schematic strategy conditionswhen compared to the other conditions. However, an analy-sis of Table 4 shows that these treatment outcomes reflecteddifferent demands on childrens cognitive processes. For ex-ample, the general-heuristic condition made high demandson executive WM skills when compared to the other cogni-tive processes. Thus, this would not be the preferred strategyfor children with MD who also have low executive process-

ing skills. In contrast, as suggest by the negative correlations,the general-heuristic + visual-schematic conditions showedsome compensatory processing for children with relativelylow visual-spatial WM. Thus, although the gap between chil-dren with and without MD may be narrowed within certainstrategy conditions, it is important to note that outcomes maybe related to demands made on specific cognitive processes.

Question 3. Do cognitive strategies place different de-mands on cognitive processes in children with MD?

Although a response to this question was partially an-swered above, the results will be placed in the context ofthe two hypotheses provided in the introduction. The com-pensatory hypothesis suggested that strategy training com-pensates for individual differences in cognitive processing.For example, some studies have shown strategy training helpslow span participants allocate WM resources more efficientlywhen compared to high span participants (e.g., Turley-Ames

& Whitfield, 2003). Thus, weexpected that children withMD,especially those with relatively lower WM span, would bene-fit more from strategy instructions than those with higherWM skills. Support for this assumption would be foundif a negative correlation between posttest solution accu-racy and WM occurred. As shown, a negative correlationonly occurred for children in the general-heuristic+ visual-schematic strategy condition and this condition yielded ESsfor problem solving in the low range when compared to thecontrol condition.

In contrast, the high cognitive skills hypothesis suggestedthat training in problem solving strategies is more likely toimprove problem solving outcomes for children with rel-atively higher cognitive skills. Support for this hypothesis

would be found if a positive correlation between posttestsolution accuracy and the cognitive process occurred. Themajority of our outcomes (correlations) appear to supportthis hypothesis. The magnitude of positive correlations foundfor the General-heuristic-alone condition suggested a clearadvantage for children with higher executive WM skills. Incontrast, the positive correlations within the general-heuristic+ visual-schematic condition suggested a clear advantage forchildren with higher numerical processing skills. Likewise,the positive correlations within the visual-schematicand con-trol conditions indicated that posttest scores were higher forchildren with higher visual WM.

Implications

Ourfindings have three applications to current research.First,the study may account for why some children benefit fromstrategy instructions and others do not. This may be related tothecognitive processesdrawn upon. For example, as shown inTable 4, the general-heuristic condition made high demandson executive WM skills when compared to the other cognitiveprocesses. This would not be a particularly effective strategyfor children with low executive processing skills.

Second, general-heuristic + visual-schematic and visual-schematic-alone conditions facilitated calculation profi-ciency for children with MD. Improvement in calculationwas part of each lesson plan and therefore practice and feed-

back may have played a role in the performance of chil-dren with MD. As shown in the standard scores reported inTable 1, children with MD had lower calculation scores thanchildren without MD and therefore strategy instruction mayhave provided an additional boost in performance.

Finally, improvement occurred on a norm-referenced test.Themajority of intervention studies for problem solving haveshown gains on experimental measures and less so on stan-dardized measures (see Powell, 2011, for review). Thus, wewere able to improve performance substantially on materials

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related to our experimental measures (component identifica-tion) as well as a standardized test.

Acknowledgment

This article is a first year study of a four year grant fundedby the U.S. Department of Education, Cognition and Stu-

dent Learning (USDE R3234A090002), Institute of Educa-tion Sciences awarded to the first author. The purpose ofthis first year was identifying cognitive variables that under-lie strategy training outcomes. The authors are indebted toLoren Alberg, Catherine Tung, Dennis Sisco-Taylor, KenishaWilliams, Garett Briney, Kristi Bryant, Orheta Rice and OlgaJerman in the data collection and/or task/curriculum develop-ment. Special appreciation is given to school administratorsChip Kling, Sandra Briney and Jan Gustafson-Corea. Thereport does not necessarily reflect the views of the U.S. De-partment of Education or the School Districts.

REFERENCES

Andersson, U. (2010). Skill development in different components of arith-meticand basiccognitive functions: Findingsfrom a 3-year longitudinalstudy of children with different types of learning difficulties.Journalof Educational Psychology,102(1), 115134. doi:10.1037/a0016838.

Baker, S., Gersten, R., & Lee, D. (2002). A synthesis of empirical researchon teaching mathematics to low-achieving students. The ElementarySchool Journal,103(1), 5173. doi: 10.1086/499715.

Brissiaud, R., & Sander, E. (2010). Arithmetic word problem solving: Asituation strategy first framework. Developmental Science,13(1), 92107. doi: http://dx.doi.org/10.1111/j.1467-7687.2009.00866.x.

Brown, V. L., Cronin, M. E., & McEntire, E. (1994). Test of mathematicalability. Austin, TX: PRO-ED.

Brown, V. L., Hammill, D., & Weiderholt, L. (1995).Test of reading com-prehension. Austin, TX: PRO-ED.

Bryk, A. S., & Raudenbush, S. W. (2002). Hierarchical linear models:applications and data analysis methods. London: Sage.

Bull, R., Espy, K. A., & Wiebe, S. A. (2008). Short-term memory, work-ing memory, and executive functioning in preschoolers: Longitudinal

predictors of mathematical achievement at age 7 years.DevelopmentalNeuropsychology,33, 205228. doi: 10.1080/87565640801982312.

Bull, R., Johnston, R. S., & Roy, J. A. (1999). Exploring the rolesof the visual-spatial sketch pad and central executive in childrensarithmetical skills: Views from cognition and developmental neu-ropsychology. Developmental Neuropsychology, 15, 421442. doi:10.1080/87565649909540759.

Butler, D. L., Beckingham, B., & Lauscher, H. J. N. (2005). Promotingstrategic learning by eighth-grade students struggling in mathemat-ics: A report of three case studies. Learning Disabilities Research& Practice, 20(3), 156174. doi: http://dx.doi.org/10.1111/j.15405826.2005.00130.x.

Cohen, J. (1988).Statistical power analysis for the behavioral sciences (2nded.). Hillsdale, NJ: L. Erlbaum Associates.

Connolly, A. J. (1998). KeyMath revised/Normative update. Circle Pines,

MN: American Guidance.Fletcher, J. M., Epsy, K. A., Francis, P. J., Davidson, K. C., Rourke, B. P.,

& Shaywitz, S. E. (1989). Comparison of cutoff and regression-baseddefinitions of reading disabilities.Journal of Learning Disabilities,22,334338. doi: 10.1177/00222194890220.

Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R., Seethaler, P.M., Capizzi, A. M., et al. (2006). The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmeticword problems. Journal of Educational Psychology, 98 , 2943. doi:10.1037/0022-0663.98.1.29.

Fuchs, L. S., Fuchs, D., Schumacher,R. F., & Seethaler,P. M. (2103).Instruc-tional intervention for students with mathematics learning disabilities.

In H. L. Swanson, K. R. Harris, & S. Graham (Eds.), Handbook of learning disabilities(2nd.ed., pp. 388404). NY: Guilford.

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M.,Fuchs, D., Zumeta, R. O.(2009).Remediating number combination andword problem deficits among students with mathematics difficulties: Arandomized control trial. Journal of Educational Psychology,101(3),561576. doi: http://dx.doi.org/10.1037/a0014701.

Gersten, R., Chard, D. J., Jayanthi, M., Baker, S. K., Morphy, P., & Flojo, J.(2009). Mathematics instruction for students with learning disabilities:A meta-analysis of instructional components. Review of Educational

Research,79(3), 12021242. doi: 10.3102/0034654309334431.Hayes, J. R., Waterman, D. A., & Robinson, C. S. (1977). Identifying the

relevant aspects of problem text.Cognitive Science,1(3), 293313.doi:http://dx.doi.org/10.1207/s15516709cog0103_3

Hegarty, M., & Kozhevnikov, M. (1999). Types of visualspatial repre-sentations and mathematical problem solving.Journal of Educational

Psychology, 91(4), 684689. doi: http://dx.doi.org/10.1037/00220663.91.4.684.

Hresko, W., Schlieve, P. L., Herron, S.R., Sawain, C., & Sherbenou, R.(2003).Comprehensive math abilities test. Austin, TX: PRO-ED.

Hutchinson, N. L. (1993). Effects of cognitive strategy instruc-tion on algebra problem solving of adolescents with learn-ing disabilities. Learning Disability Quarterly, 16(1), 3463. doi:http://dx.doi.org/10.2307/1511158.

Jitendra, A. K., & Hoff, K. (1996). The effects of schema-based instructionon the mathematical word-problemsolving performance of studentswith learning disabilities.Journal of Learning Disabilities, 29(4),422

431. doi: http://dx.doi.org/10.1177/002221949602900410.Jitendra, A. K., Star, J. R., Rodriguez, M., Lindell, M., & Someki,

F. (2011). Improving students proportional thinking using schema-based instruction. Learning and Instruction, 21(6), 731745. doi:http://dx.doi.org/10.1016/j.learninstruc.2011.04.002.

Koichu, B., Berman, A., & Moore, M. (2007). The effect of promotingheuristic literacy on mathematical aptitude of middle school students.

International j ournal of Mathematical Educa tion in Science and Tech-nology,38, 17. doi: 10.1080/002207390600861161.

Kolloffel, B., Eysink, T., de Jong, T., & Wilhelm, P. (2009). The effects ofrepresentation format on learning combinatories from an interactivecomputer. Instructional Science, 37, 503515. doi: 10.1007/s112510089056-7.

Low, R., & Over, R. (1989). Detection of missing and irrelevant informa-tion within algebraic story problems. British Journal of Educational

Psychology, 59(3), 296305. doi: http:// dx.doi.org/10.1111/j.20448279.1989.tb03104.x.

Mayer, R. E., & Hegarty, M. (1996). The process of understanding mathe-matical problem solving. In R.J. Sternberg & T. Ben-Zeev (Eds.).Thenature of mathematical thinking(pp. 2954). Mahwah, NJ: Erlbaum.

Meyer,M. L.,Salimpoor, V. N., Wu, S. S.,Geary, D. C., & Menon, V. (2010).Differential contribution of specific working memory components tomathematics achievement in 2nd and 3rd graders.Learning and Indi-vidual Differences,20(2), 101109. doi: 10.1016/j.lindif.2009.08.004.

Montague, M. (1992). The effects of cognitive and metacognitive strategyinstruction on the mathematical problem solving of middle school stu-dentswith learning disabilities.Journal of Learning Disabilities, 25(4),230248. doi: http://dx.doi.org/10.1177/002221949202500404.

Montague, M. (2008). Self-regulation strategies to improve mathematicalproblem solving for students with lear ning disabilities. Learning Dis-ability Quarterly,31(1), 3744.

Montague, M., Warger, C., & Morgan, T. H. (2000). Solve it! Strat-egy instruction to improve mathematical problem solving. Learning

Disabilities Research & Practice, 15(2), 110116. doi: 10.1207/SL-

DRP1502_7.Pearson Publishers (2009). Scott Forsman-Addison Wesley EnVisionMath.NY: Pearson, INC.

Powell, S. R. (2011). Solving word problems using schemas: A review ofliterature. Learning Disabilities Research and Practice, 2 6, 94108.doi: 10.1111/j.15405826.2011.00329.x.

Psychological Corporation (1991).Wechsler intelligence scale for children(3rd ed.). San Antonio TX: Harcourt Brace Jovanovich, Inc.

Psychological Corporation (1992). Wechsler Individual Achievement Test.San Antonio TX: Harcourt Brace & Co.

Raven, J. C. (1976). Colored progressive matrices test. London, England:H. K. Lewis & Co. Ltd.

8/13/2019 ldrp12019

13/14


Rosenzweig, C., Krawec, J., & Montague, M. (2011). Metacognitivestrategy use of eighth-grade students with and without learningdisabilities during mathematical problem solving: A think-aloudanalysis. Journal of Learning Disabilities, 44(6), 508520. doi:http://dx.doi.org/10.1177/0022219410378445.

SAS Institute, Inc. (2010).SAS program version 9.3. Cary, NC: SAS Insti-tute, Inc.

Siegler, R. S. & Booth, J. (2004). Development of numerical estimation inyoung children. Child Development, 75, 428444.doi: 10.1111/j.1467-8624.2004.00684.x

Siegler, R. S.,& Opfer,J. (2003). Thedevelopment of numerical estimation:Evidence For multiple representation of numerical quantity.Psycholog-ical Science,14, 237243. doi: 10.1111/14679280.02438.

Siegel, L. S., & Ryan, E. B. (1989). The development of working mem-ory in normally achieving and subtypes of learning disabled. Child

Development,60, 973980. doi: 10.2307/1131037.Shiah, R. L., Mastroprieri, M., Scruggs, T., & Fulk, B. J. (19941995).

The effects of computer-assisted instruction on mathematical problemsolving in students with learning disabilities. Exceptionality,5 , 131161.

Swanson, H. L. (1995). S-Cognitive Processing Test (S-CPT): A DynamicAssessment Measure. Austin, TX: PRO-ED.

Swanson, H. L., & Beebe-Frankenberger, M. (2004). The relationship be-tween working memory and mathematical problem solving in childrenat risk and not a risk for serious math difficulties. Journal of Educa-tional Psychology,96, 471491. doi: 10.1037/00220663.96.3.471.

Swanson, H. L., & Jerman, O. (2006). Math disabilities: A selective meta-

analysis of the literature.Review of Educational Research, 76, 249274.doi: 10.3102/00346543076002249.

Swanson,H. L.,Jerman, O.,& Zheng, X. (2008).Growthin workingmemoryand mathematical problem solving in children at risk and not at risk forserious math difficulties. Journal of Educational Psychology, 100(2),343379. doi: 10.1037/00220663.100.2.343.

Swanson, H. L., Lussier, C. M., & Orosco, M. J. (in press). Cognitivestrategies, working memory and growth in problem solving.Journal of

Learning Disabiliti es.Swanson, H. L., Orosco, M. J., & Lussier, C. M. (in press). The effects of

strategy instruction for children with serious problem solving difficul-ties.Exceptional Children.

Towse, J., & Cheshire, A. (2007). Random generation and working mem-ory. European Journal of Cognitive Psychology, 19, 374394. doi:10.1080/09541440600764570.

Turley-Ames, K. J., & Whitfield, M. (2003). Strategy training and workingmemory performance.Journal of Memory and Language, 49, 446468.

doi: 10.1016/S0749596X(03)00095-0.Van Garderen, D., & Montague, M. (2003). Visual-spatial representation,

mathematicalproblemsolving,and students of varying abilities.Learn-ing Disabilities Research & Practice,18(4), 246254.

Van Garderen, D. (2007). Teaching students with LD to use diagrams tosolve mathematical word problems.Journal of Learning Disabilities,40(6), 540553. doi: 10.1177/00222194070400060501.

Wagner, R., Torgesen, J., & Rashotte, C. (2000). Comprehensive test of phonological processes. Austin, TX: Pro-ED.

Wilkinson, G. S. (1993).The wide range achievement test. Wilmington DE:Wide Range, Inc.

Xin, Y. P., & Jitendra, A. K. (1999). The effects of instruction in solvingmathematical word problems for students with learning problems: Ameta-analysis.The Journal of Special Education,32(4), 207225. doi:10.1177/002246699903200402.

Zheng, X., Flynn, L., & Swanson, H. L. (2013). Experimental interven-tion studies on word problem solving and math disabilities: A selec-

tive analysis of the literature. Learning Disability Quarterly, 36, 97111.

Appendix

This Appendix shows an example of a word problem fromthe guided instruction section of the general-heuristic strat-egy including tutor strategy instructions, keyword prompts,

and strategy mark ups of the problem . This appendix alsoshows an example of a word problem from the guided in-struction section of the visual-schematic strategy includ-ing tutor diagram instructions, diagramming prompts, anddiagramming boxes and missing number fill in of theproblem.

General heuristic Strategy

General heuristic Example Instructions:

1. Find the Question and underline it.

2. Circle the numbers.

3. Find and put a square around key words.

4. Find the information you do not need and cross it out.

5. Decide what you need to do (add/subtract/or both).

6. Solve it.

Tutor: Key words are the words that you should payattention to because they are clues to help you figure out howto solve the story problems.

Example Key Words Prompt: MORE, MORE THAN, TOGETHER, IN ALL,

ALTOGETHER, TOTAL, IN SUM, LESS, LESS THAN, FEWER,

FEWER THAN, AS MANY AS, THE SAME AMOUNT AS, EQUAL TO, BOTH

General heuristic Guided Example Given to Studentwith Strategy Markups:

1) Amanda and Lori were walking on a beach. Amanda found 7

shells. Lori found 13 shells. They saw 4 +seagulls. How many shells did they find in all ?

Answer: 13 + 7 = 20 shells in all.

Tutor guides student through question using strategiesalthough giving prompts and feedback.

Visual-schematic(Diagramming) Example Instruction:Tutor: Look at the diagram below; it has boxes and

numbers. This diagram represents the whole (or the totalamount) that consists of two parts. For example, if your totalor whole is 5, then 2 and 3 are the parts that add together tomake the whole.

Whole

5

2 32 + 3 = 5

Part 1 Part 2

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Teacher:Below you can see that you can have 2 parts,or three, or even four parts. However, all parts must sum upto the whole/total.

Whole Whole

OrPart 1 Part 2 Part 1 Part 2 Part 3

Guided Example:Tutor: Here is a word problem that we will do together.

The problem will have the new diagram that we learned onthe previous page. Lets start with the first problem and wewill try to find what is the whole, what are the parts, andwhere in the diagram the numbers will go.

1) Amanda and Lori were walking on a beach. Amandafound 7 shells. Lori found 13 shells. How many shells didthey find in all?

Whole(Amandas & Lori shells in all)

?

7 13

Part 1 Part 2

(Amandas shells) (Loris Shells)

? = 13 + 7 = 20 shells

Amanda has shells and Lori has shells and we need to find how

many shells they both found in all. We dont know the whole, so

lets put a question mark for the whole. Now, the parts in the

problem are Amandas shells and Loris shells. We know thenumbers, so we write 7 for Amandas part and 13 for Loris part.

Now, we can use our rule to find the whole. The rule says, to find

the whole you need to add the parts, so we add 7 + 13 and it

equals 20. So, the answer to the problem is: "Together they found

20 shells.

About the Authors

H. Lee Swanson, Ph.D., is a Distinguished Professor and Peloy Endowed Chair in Educational Psychology at the Universityof California-Riverside. His research interests include intelligence, memory, and problem solving in children with learningdisabilities. He received his Ph.D. at the University of New Mexico and completed his post-doctoral work at UCLA.

Cathy Lussier, Ph.D., is a lecturer and project director of two federal grants at the University of California-Riverside (UCR).Her research interests include math instruction, memory, and dynamic testing. She received her doctoral degree at UCR inEducational Psychology.

Michael Orosco, Ph.D., is an assistant professor in Special Education at the University of California-Riverside. His researchinterests include bilingual education, dynamic instruction, motivation, reading, and math. He received his doctoral degree fromthe University of Colorado-Boulder.

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