mathematical problem-solving strategy instruction for third-grade students with learning...

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2002 23: 268Remedial and Special EducationRhoda L. Owen and Lynn S. Fuchs

Mathematical Problem-Solving Strategy Instruction for Third-Grade Students with Learning Disabilities

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268 R E M E D I A L A N D S P E C I A L E D U C A T I O N

Volume 23, Number 5 September/October 2002, Pages 268–278

Mathematical Problem-Solving StrategyInstruction for Third-Grade Students with Learning DisabilitiesR H O D A L . O W E N A N D L Y N N S . F U C H S

A B S T R A C T

The purpose of this study was to examine the effects ofstrategy instruction on the mathematical problem solving of 3rd-grade students with learning disabilities. Participants were 24 stu-dents whose teachers were randomly assigned to 4 conditions: (a) control, (b) acquisition, (c) low-dose acquisition plus transfer, or(d) full-dose acquisition plus transfer. During the 3-week study, stu-dents in each experimental group received instruction on a 6-stepprocedure for solving word problems that required finding half of anumber. Across Groups A, B, and C, treatment comprised explicitinstruction with heavy use of worked examples and practice witha higher achieving classmate. Analyses of variances were con-ducted on improvement between pre- and posttreatment mea-sures in terms of number of problems solved correctly and amountof work showing the steps taught in the treatment. For problemssolved correctly, statistically significant improvement favored thefull-dose acquisition plus transfer group over the control and overthe low-dose acquisition plus transfer groups. For amount of work,significant differences favored the low-dose acquisition plus trans-fer and full-dose acquisition plus transfer groups over the controlgroup. Student and teacher attitudes about the instructional strat-egy and working with a partner were positive. Mathematical strat-egy instruction and pairing students for instruction is discussed withrespect to directions for practice and future research.

IN EVERYDAY ADULT LIFE, WE ARE FACED WITH

numerous decisions that involve mathematical problem solv-ing. Children with learning disabilities (LD) grow up to liveindependently and become participating members of society

(Wilson & Sindelar, 1991). It is crucial that these studentspossess the skills to solve the wide range of mathematicalproblems they will encounter (Patton, Cronin, Bassett, &Koppel, 1997).

Mathematical problem solving can be characterized asthe ability to draw upon previous knowledge to solve a novelproblem (Carnine, 1997). Although math problem-solvingdifficulties are noted for students with and without disabili-ties (National Council of Teachers of Mathematics, 1989),research has shown that students with LD who have difficul-ties in language or mathematics are at particular risk forfailure in mathematical problem solving (Bryant, Bryant, &Hammill, 2000; Carnine, 1997; Englert, Culatta, & Horn,1987; Miller & Mercer, 1997; Montague, 1997; Parmar, Caw-ley, & Frazita, 1996; Scruggs & Mastropieri, 1986).

Although many factors may contribute to the poor mathperformance of students with disabilities, one promisingclass of interventions is cognitive methods for teaching stu-dents to attack word problems in a systematic and strategicmanner (Harris & Pressley, 1991; Jitendra & Xin, 1997; Mas-tropieri, Scruggs, & Shiah, 1991; Montague, 1997; Montague& Bos, 1986). These cognitive interventions frequently havebeen referred to as “strategy instruction.” With strategyinstruction, students follow a series of steps to facilitateunderstanding and problem solving. A strategy may bestraightforward with rote application of specific steps to solvea particular type of problem or a broad set of guidelines thatprovide a general direction to solve a problem (Wong, 1994).

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269R E M E D I A L A N D S P E C I A L E D U C A T I O N

Volume 23, Number 5, September/October 2002

application of basic skills to become mathematical problemsolvers, encouraging analytical thinking and success inpostschool life. However, current policies require studentswith disabilities to be educated in the general education class-room to the maximum extent possible. As children with LDreceive more of their math instruction in the general educa-tion classroom, teachers must accommodate the needs ofthese students, as well as those of all the students, while cov-ering the prescribed curriculum. Due to large class size ingeneral education classrooms, teachers are unable to providemore than a few minutes of individual attention to each stu-dent throughout the school day. Putting students together inpairs or small groups (peer-mediated instruction) has shownpromise as an effective way to integrate students with dis-abilities into the general education classroom. With peer-mediated instruction, students serve as instructional agentsfor classmates or other children. Research has shown that theacademic performance of students with disabilities canimprove when they work on structured learning activities incollaborative groups (D. Fuchs, Fuchs, Mathes, & Simmons,1997; Greenwood, Delquadri, & Hall, 1989).

Although peer-mediated instruction has different fea-tures (e.g., cooperative learning vs. peer tutoring), one com-monality is that students in the collaborative group seem tobenefit from observational learning (Jackson, Fletcher, &Messer, 1992). Students working alone are often at a disad-vantage because the teacher may be too busy to assist them orthe student may be reluctant to ask for help. In collaborativegroups, students can draw upon the abilities of other studentsin their group by remembering how others solved a previousproblem. Although there is evidence that cooperative studentgroups for instruction is beneficial for elementary studentswith disabilities (L. S. Fuchs et al., 1997; Greenwood et al.,1989; Maheady, Harper, & Mallette, 1991), previous researchon mathematical problem solving with collaborative groupsdoes not exist.

The purpose of this study was to examine the effects ofa strategy designed to enable students to solve mathematicalword problems. This study replicates previous research byteaching students with disabilities a specific math strategyand extends prior research by comparing three types of strat-egy instruction, each in combination with peer-mediatedinstruction, on young students with disabilities in generaleducation settings. We used three conditions to teach a mathstrategy to see which strategy had the greatest effect on stu-dents’ mathematical problem solving. Each treatment condi-tion incorporated similar instructional components (strategyinstruction, explicit instruction, worked examples, and peermediation), but variations in instruction occurred with regardto whether transfer was explicitly addressed. In addition,because treatments paired students with disabilities withhigher achieving math partners for practice, we could explorestudent perceptions about working with a partner. We wereinterested in (a) differences in performance levels on mathe-matical word problems among the four groups, (b) student

Studies that have evaluated the effectiveness of mathe-matical problem-solving strategy instruction for studentswith LD have revealed useful information. However, themajority of research on mathematical problem-solving strat-egy instruction for students with LD has focused on middle-and high-school–age students. We located only seven studiesthat focused on systematic strategy instruction for mathe-matical problem solving of elementary students with LD (i.e., Case, Harris, & Graham, 1992; Cassel & Ried, 1996;Jitendra et al., 1998; Jitendra & Hoff, 1996; Mercer & Miller,1992; Miller & Mercer, 1993; Wilson & Sindelar, 1991).

Mercer and Miller (1992) and Miller and Mercer (1993)used a sequence with a graduated level of difficulty to teachone-step subtraction and multiplication word problems. Thetreatment began with simple words and concrete materials.As students progressed, objects were replaced with tally marksand then tally marks were replaced with objects or drawings.Once mastery was achieved, students solved word problemsin which numbers were written in complete sentences, fol-lowed by the traditional paragraph format. In the final phaseof the study, students created their own word problems thatrelated to real-world math experiences. As in the previoustwo studies, Jitendra and Hoff (1996) and Jitendra et al. (1998)taught students to draw diagrams to discriminate betweendifferent problem types. Then students designed a solutionstrategy by performing the appropriate arithmetic operationto solve the problem.

Cassel and Reid (1996) taught students a nine-stepproblem-solving strategy and self-regulation procedure. Stu-dents also generated self-instruction statements to help themmonitor their problem-solving behavior (e.g., “What is it Ihave to do?” “Does what I am doing make sense?”). In astudy similar to Cassel and Reid’s, Case et al. (1992) exam-ined a self-regulated cognitive strategy (self-assessment, self-recording, self-instruction) to solve one-step addition andsubtraction word problems. Students were taught to identifycue words and phrases found in addition and subtractionproblems and to draw pictures to tell what was happening.Students generated their own words to describe each step ofthe self-instruction strategy needed to solve the problem.Finally, Wilson and Sindelar (1991) compared the effective-ness of strategy instruction (subtract when the big numberwas given and add when the big number was not given) andsequencing problems (simple action problems followed byclassification problems, complex action problems, and com-parison problems). The strategy instruction component wasconsidered more effective than sequence instruction.

These seven studies reported overall improvement ofstudents’ math word-problem–solving performance as a re-sult of strategy instruction. Thus, a common theme emergingfrom this research is that elementary students with learningdisabilities need explicit instruction in mathematical problem-solving strategies. As demonstrated in this body of work,strategy instruction holds promise to improve mathematicalunderstanding and help students with LD move beyond rote

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opinions about the math treatment and about working with apartner, and (c) teacher judgments of the extent to which themath treatment met the instructional needs of their studentswith disabilities and the extent to which working with part-ners enhanced student achievement. This study was part of alarger investigation (L. S. Fuchs et al., in press) that examinedthe effects of explicit instruction about transfer on the mathe-matical problem-solving abilities of third-grade students. Thecurrent study looked specifically at effects of treatment onone problem-solving unit of students who were identifiedunder state criteria as having a disability and were included ina general education third-grade classroom for math instruc-tion. There is no overlap in the data reported in the currentstudy and the larger study.

METHOD

Participants

Participants were 24 third-grade students: 20 with a learn-ing disability, 1 with mild/moderate mental retardation, 2with speech/language disorders, and 1 with attention-deficit/hyperactivity disorder. All children spoke English as their

primary language. None dropped out of the study. Table 1provides demographic data. These students were from 14third-grade classrooms in six elementary schools from thelarger study. Their teachers were surveyed to find out howmany minutes they typically allocated to math instruction perweek and to what extent they paired students together forinstructional activities. Analyses of variance (ANOVAs) indi-cated no significant differences among groups for the numberof minutes per week typically spent on math instruction or forthe extent to which teachers partnered students for the in-structional activities.

Measures

Half Assessment. Two forms of a word-problem–solving test were written for pre- and posttesting. Pre- andposttests were administered by research assistants (RAs) toeach class in a large group during regularly scheduled mathtime. Four problems were on each test, with one problem ona page. The questions did not contain extraneous linguistic orirrelevant numerical information that might distract the stu-dents. Two problems used the word half, whereas the othertwo problems used the numerical representation (1⁄2). (Tominimize differences due to reading ability, test items were



TABLE 1. Demographics of Students with Disabilities by Treatment

Low-dose Full-doseacquisition acquisition

Controla Acquisitionb + transferc + transferd

Variable n % n % n % n % χ2

Gender .36Girls 3 37.50 1 25.00 2 40.00 2 28.57Boys 5 62.50 3 75.00 3 60.00 5 71.43

Received free lunch 3 37.50 4 100.00 2 40.00 3 42.85 .31

Race 4.60*African American 4 50.00 2 50.00 5 100.00 3 42.85White 4 50.00 2 50.00 0 00.00 4 57.14

Math IEP 4 50.00 2 50.00 2 40.00 4 57.14 .34

Reading IEP 7 87.50 4 100.00 5 100.00 5 71.43 2.94

Class behavior 1.72Acceptable 2 25.00 2 50.00 2 40.00 4 57.14Problematic 6 75.00 2 50.00 3 60.00 3 42.86

Reading level 2.94At grade level 1 12.50 0 00.00 0 00.00 2 28.57Below grade level 7 87.50 4 100.00 5 100.00 5 71.43

Note. Classroom teachers were asked to rate each student’s class behavior as acceptable, occasional problem, or frequent problem. IEP = Individualized EducationProgram.an = 8. bn = 4. cn = 5. dn = 7. *p < .05.

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read aloud to the class while the students followed along.)Students were given time to solve each problem after it wasread. The RA reread questions upon request but did not pro-vide other assistance. Alpha coefficient was .89.

Tests were scored in two ways. The difference betweenthese two types of procedures, product scoring and processscoring, is that product points were awarded for correctresponses only, whereas process points were awarded forapplying steps taught in the treatment, regardless of whetherthe numbers used were correct. The product score (maximumscore = 22), which reflected correct answers, was awarded forwriting the correct number in the answer, providing the cor-rect answer label, and showing any steps taught in the treat-ment correctly (e.g., making the same number of marks as thenumber they were to find half of, drawing a box, putting thesame number of marks in each box). The process score (max-imum score = 28) reflected students’ ability to show steps thathad been taught during the treatment. Students earned pointsfor correct work as well as incorrect work that reflected stepstaught in the treatment. For example, one problem asked stu-dents to find half of 7. Whereas one student earned 1 point forcorrectly drawing seven symbols, another student earned halfa point simply for drawing symbols, even though seven werenot drawn.

Two special education doctoral students scored pre- andposttests according to a scoring rubric. For training, two scor-ers completed the same three tests, discussed their differ-ences, and then scored another three tests with 100%reliability. Each graduate student independently scored halfof the remaining tests. Twenty-five percent of the tests were

rescored to determine agreement, which was calculated bydividing the number of agreements by the sum of the numberof agreements and disagreements and multiplying by 100.Agreement was 99.5%.

Student Surveys. Each student with a disability and hisor her partner completed a survey. Students with disabilitieswere read five statements soliciting opinions about the mathtreatment and working with a partner. Students without dis-abilities were read similar statements as well as statementsabout their impression of the difficulty of the math treatmentfor other students in the class and how they thought theirassistance helped their partner learn. See Table 2 for studentsurvey questions. As the RA read each statement, the studentcircled his or her response (“I agree,” “I disagree,” “I’m notsure”). When a student selected “I agree” or “I disagree,” theRA asked a scripted probe to solicit additional information.For example, one question asked for a student response to thestatement, “Learning how to find half was easy.” If the stu-dent selected “I agree,” the RA asked the student, “How waslearning to find half easy?” If the student selected “I dis-agree,” the RA asked the student, “How was learning to findhalf not easy?” The RA recorded the student’s response onthe probe script.

Teacher Surveys. Each teacher completed a survey thatasked questions about the extent to which the treatment metthe instructional needs of their students with disabilities andhow well working with partners enhanced student achieve-ment. Teachers provided responses on a Likert-type scale.See Table 3 for teacher survey questions.



TABLE 2. Student Survey Question Frequencies and Chi-Square Values

Question Agree Disagree Not sure χ2

Students with disabilitiesLearning how to find half was easy. 12 2 2 12.50**The worksheets on finding half were hard. 1 12 3 12.88**I liked working with my partner on finding half. 16 0 0 —I did better finding half on the worksheets because my partner helped me. 12 1 3 12.88**I will do better finding half on my own because my partner helped me learn how to

find half. 12 1 3 12.88**

Students without disabilitiesLearning how to find half was easy for me. 13 1 2 16.63***The worksheets on finding half were easy for all of the students in my class. 1 1 14 21.13***Some of the worksheets on finding half were hard for some students in my class. 7 2 7 3.13 I liked working with my partner on finding half. 15 0 1 12.25***My partner did better on the finding half worksheets because I helped him. 9 1 6 6.13 I think my partner will do better finding half next time because I helped him learn

how to find half. 11 1 4 9.88**Helping my partner helped me learn how to find half. 5 8 3 2.38 I would have liked to work with a different partner. 4 6 6 .50

Note. Dashes indicate the variable was a constant so a chi-square test could not be performed. **p < .01. ***p < .001.

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Commonalities Across Treatment Groups

Across the three experimental conditions, treatment was im-plemented during the students’ general education math timeto control for instructional time across conditions. Lessonswere conducted in the students’ general education third-gradeclassroom with the entire class (18–20 students). Explicitexplanation of the math strategy was broken down into com-ponent steps and presented to the students along with workedexamples and modeling of the procedure. One-step wordproblems (acquisition problems) were developed for instruc-tion. Problem types were selected specifically because theywere a formal component of the district’s third-grade corecurriculum (e.g., “Every day Tony spends 8 hours at school.Yesterday he got sick and had to go home after 1⁄2 of theschool day. How many hours was he at school?”). Studentswere taught to solve this type of problem using six steps:

1. Read the problem.

2. Draw individual circles to show the number forwhich you will find half.

3. Draw a rectangle divided in half, creating twoboxes.

4. Cross out the first circle and draw it in the leftbox; cross out the second circle and draw it inthe right box; continue crossing out circles anddrawing them in alternating boxes until yourun out of circles. If you end up with one extracircle, draw a vertical line down the middle ofthe remaining circle and draw half a circle inthe left box and half a circle in the right box.

5. Count the circles in each box to make sure thesame numbers of circles are in each box.

6. Count the circles in one box and write thisanswer along with the appropriate word labelfor the problem.

After large-group instruction, students worked with apartner on practice problems. At the beginning of the study,teachers had been asked to rank students according to mathability. Partners were assigned so that each pair comprised



TABLE 3. Means, Standard Deviations, and F Values for Questions on Experimental Teacher Survey


Acquisition + transfer + transfer

No. Question M SD M SD M SD F(2, 14)

1 To what extent has the math problem solving project’s math skill of finding half matched the instructional needs of your students with disabilities? 4.25 .96 4.25 .96 3.33 .52 2.32

2 To what extent has the instruction on the skill of finding half matched the way you usually teach your students with disabilities? 1.67 .58 2.75 1.71 2.17 1.17 .63

3 To what extent do you think the math strategy taught in the half unit was confusing to your students with disabilities? 1.75 .50 3.25 1.26 2.33 1.03 2.33

4 To what extent was the level of difficulty in the half unit appropriate for your students with disabilities? 4.00 .82 3.25 1.50 3.33 1.21 .48

5 To what extent did your students with disabilities master the half skill? 3.25 .50 3.00 1.41 3.33 1.21 .11

6 To what extent did your students with disabilities actively participate in the half unit lessons? 4.25 .50 4.00 1.41 4.00 .89 .09

7 To what extent has the use of “partners” helped your students with disabilities learn the skill of finding half? 4.33 .58 4.25 .96 3.67 1.37 .49

8 To what extent has the use of “partners” with finding half encouraged the interaction of your students with disabilities with other children with whom they might not otherwise have interacted? 4.75 .50 3.75 .96 4.33 .82 1.63

9 To what extent did you feel your students with disabilities liked working with a partner during the half unit? 4.75 .50 4.75 .50 4.50 1.22 .13

Note. Teachers provided responses on a Likert-type scale ranging from 1 = not at all to 5 = to a great extent.

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one higher achieving and one lower achieving student, ac-cording to the class rank. In every pairing, the student withthe disability was the lower achieving partner.

Differences Among Treatment Groups

Classrooms were randomly assigned to four conditions. Threeexperimental conditions were acquisition (four students withdisabilities), low-dose acquisition plus transfer (five studentswith disabilities), and full-dose acquisition plus transfer(seven students with disabilities). A control group of eightstudents with disabilities was also included for comparisononly. They received instruction from the same basal mathe-matics program and on a curriculum that included word prob-lems involving halves.

The acquisition condition consisted of four scriptedlessons that taught a six-step method of solving word prob-lems involving halves. Two terms representing one half wereused: 1⁄2 and half (e.g., “Each day Brad goes for a run. Heruns 9 miles. Today he only ran 1⁄2 of that distance and walkedthe other 1⁄2. How many miles did he walk?”). During the firstlesson, half was defined through visual examples. Next, stu-dents were shown how to divide the circles into two equalparts. Then, the RA displayed an acquisition-type word prob-lem and asked students what they were to do. Studentsresponded, “Find half of eight.” The RA then called on indi-vidual students to tell the class the steps for finding half of anumber. Students had a copy of the problem at their desksand wrote the steps along with the RA, who demonstrated theprocess on a transparency. Students were then given two ac-quisition problems to solve with their partners. When stu-dents finished, the RA asked them what their answers wereand explained the worked problems on the overhead projector.

Lessons 2 through 4 began with a review of the steps forfinding half and two to four practice problems that the classcompleted together. As the RA wrote the steps on the trans-parency, the students wrote the steps on copies of the prob-lems they had at their desks. Students then were given four toeight acquisition problems to work with their partners. Whenboth students and their partners were finished, the RA orclassroom teacher gave the pairs answer keys. Students weretold that the pairs must agree on an answer. If they did not,they were to talk and explain how they got their answers. Ifthe two students could not figure out why they got a wronganswer, they raised their hands and the RA helped them withthe problem.

In the low-dose acquisition plus transfer condition, stu-dents received the same lessons on Days 1 and 2 as studentsin the acquisition condition. On Days 3 and 4, students trans-ferred the skill of finding half to other related problems.These transfer problems looked different, used differentwords, were small pieces of larger problems, or asked differ-ent final questions. Problems that looked different includedcharts or tables. Problems that used different words used one

half, half price, or divide equally. Problems that were a smallpiece of a larger problem contained extraneous information.Problems that asked a different question had students findhalf of a number and compute a simple subtraction problem(e.g., “Tim went to the mall with his sister. His mom gavethem $12 to spend. His mom told Tim to keep half and givehalf the money to his sister. At the mall, Tim bought a drinkfor $1. When Tim left the mall, how much money did Timhave left?”).

On Day 3, the RA presented four transfer problems onthe overhead. Students were given copies of the problems tocomplete while the teacher modeled the steps of finding halfon the overhead. On Day 4, a worksheet with six problems(one acquisition problem, two transfer problems, and threeproblems on a skill taught earlier in the larger study) wasgiven to the students to work on with their partner. Again,when a pair of students were both finished, they were givenan answer key to check and discuss their answers.

In the full-dose acquisition plus transfer condition, thestudents received the same 4 days of instruction and work-sheets as the acquisition condition. After the four acquisitionlessons were completed, the students in the full-dose acquisi-tion plus transfer condition received the two transfer lessonsand worksheets that were presented to the students in the low-dose acquisition plus transfer condition for a total of sixlessons.

Fidelity of Treatment

Lessons in the experimental conditions were audiotaped toassess fidelity of treatment. One graduate student in specialeducation listened to 10% of the audiotapes: two tapes eachfrom three RAs and one tape each from two RAs. As the RAlistened, she used a checklist to identify which key elementsof the treatment were and were not conducted. For acqui-sition, low-dose acquisition plus transfer, and full-dose ac-quisition plus transfer, 95.9%, 91.9%, and 97.0% of keyelements were correctly implemented, respectively. Interraterreliability was 100%.

Data Collection

Pretesting occurred 1 week prior to the start of the treatment.Posttesting was conducted 1 week after the last treatmentsession. Administration of the two test forms was counter-balanced across pre- and posttesting and treatment groups.Trained university RAs administered both tests. Although allstudents in each condition took the pre- and posttest, only thescores of the students with disabilities are reported for thisstudy. Following posttesting, RAs interviewed students withdisabilities and their partners separately. At the end of treat-ment, the experimental students’ classroom teachers com-pleted a survey.



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Design/Data Analysis

One-way ANOVAs (treatment: four levels) conducted onpretest product scores and on process scores yielded no sig-nificant main effect for treatment on process scores, F(3, 23) = .44, ns, or on product scores, F(3, 23) = .27, ns, indicatingthat the four groups’ performances on the pretest was compa-rable. To test for differential improvement as a function oftreatment, a two-way ANOVA (treatment and time, withrepeated measures on the second variable) was conducted.Tukey tests were run post hoc on significant interactionsbetween time and treatment. On the student surveys, frequen-cies and chi-square tests were calculated for each questionacross treatment groups using disability status as the factor.On the experimental teacher survey, one-way ANOVAs wererun on each question with treatment as the factor.

RESULTS

Half Assessment

The two-way ANOVA on the product scores indicated a sig-nificant effect for improvement from the pre- to posttreatment[F(1, 20) = 75.32, p < .001]; a significant effect for treatment[F(1, 20) = 10.30, p < .001], indicating differences amonggroups existed across pre- and posttreatment scores; and ofgreatest interest to this study, a significant two-way interac-tion between time and treatment [F(3, 20) = 9.57, p < .001],indicating differences in growth as a function of treatment.With respect to differences among groups, Tukey post hoccomparisons indicated that students in the full-dose acquisi-tion plus transfer group improved significantly more thanthose in the control and the low-dose acquisition plus transfergroups. (See Table 4 for means and standard deviations forthe product scores by condition.)

The two-way ANOVA on the process scores indicated asignificant effect for improvement from the pre- to posttreat-

ment [F(1, 20) = 47.62, p < .001]; a significant effect fortreatment [F(1, 20) = 8.43, p < .001], indicating a differencein the gains among groups existed across pre- and posttreat-ment scores; and of greatest interest to this study, a significanttwo-way interaction between time and treatment [F(1, 20) =8.54, p < .001], indicating differences in growth as a functionof treatment. Tukey post hoc comparisons indicated that stu-dents in the low-dose acquisition plus transfer and full-doseacquisition plus transfer groups scored significantly betterthan those in the control group. (See Table 4 for means andstandard deviations for the process scores by condition.)

Student Surveys

Data from the student surveys were analyzed by combiningtreatment groups. Frequencies and chi-square values are pre-sented in Table 2 for students with and without disabilities.Students with and without disabilities reported positive atti-tudes about the math strategy and working with a partner.Both groups reported that learning how to find half was easy(75% of students with a disability; 81% of students without adisability). It is interesting to note that the majority of stu-dents without a disability were not sure if the worksheets onfinding half were easy for all of the students in the class(88%). About half felt that some of the worksheets were hardfor some of the students (44%), and about half were not sureif the worksheets were hard for students in their classes (44%).

Students without disabilities were more aware of the dif-ficulties their partners had with the problems (e.g., “Some-times he understood part of the problem but not all of it. Itaught him the things he didn’t understand”). Fifty-six per-cent of the students without a disability reported that theirpartners did better on finding half because of their help; 69%reported that their partners would do a better job finding halfnext time because of the help they had provided. Some stu-dents without disabilities (32%) reported that helping part-ners facilitated their own learning (e.g., “When you help



TABLE 4. Means and Standard Deviations for the Product and Process Scores by Condition


Control Acquisition + transfer + transfer

Variable M SD M SD M SD M SD

ProductPre 1.38 1.38 2.13 3.07 1.50 2.40 2.29 2.36Post 3.13 1.36 7.00 3.63 7.20 2.93 12.50 2.50Improvement 1.75 2.17 4.88 3.28 5.70 1.92 10.21 4.24

ProcessPre .50 0.27 .75 0.96 .30 0.45 .64 0.85Post .63 0.44 9.50 9.39 14.20 7.93 17.43 8.17Improvement .13 0.52 8.75 9.68 13.90 7.73 16.79 8.10

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other people, you learn from it too”) and that working withpartners was “fun”; however, 50% did not feel that helpingtheir partners facilitated their own learning. Students with adisability agreed that they did better on the worksheetsbecause of their partners’ help (75%) and that they would dobetter finding half on their own because of their partners’assistance (75%).

Overall, both groups of students liked working with apartner (100% for students with a disability; 94% for studentswithout disabilities). When students with a disability wereasked how their partner helped them do better, responseswere positive about their partners’ assistance (e.g., “At first Ididn’t know how to do it and she helped me get the hang ofit,” “When I forgot things, I asked her and she helped me”).

Although the majority of responses to the students’interview questions were favorable, when students withoutdisabilities were asked if they would have liked to work witha different partner, responses were mixed (agree 25%, dis-agree 38%, and not sure 38%). Students without disabilitieswho would have liked to work with a different partner citedreasons such as, “My partner didn’t understand” and “Henever helped me.” Two female students said they would havepreferred working with “girl partners.” Students without dis-abilities who did not want to work with different partnerscited social reasons (e.g., “She’s fun,” “I liked how he acted

and didn’t hurt my feelings”) or that their partners helpedthem. One student’s reason for not wanting a different partnerwas, “Steven needed my help.” (See Table 5 for demographicinformation for students without disabilities by condition.)

Teacher Surveys

Table 3 displays means and standard deviations of teacherresponses. One-way ANOVAs on the experimental teachersurvey responses revealed no significant differences amongtreatment groups. On a 5-point Likert-type scale (1 = not atall, 5 = to a great extent), ratings of 3.3 to 4.3 indicated thatthe teachers felt that the math skill matched the instructionalneeds of the students and that the level of difficulty wasappropriate. Mean ratings of 1.7 to 2.8 indicated that the skilldid not match the way the teachers usually taught the skill offinding half. When asked if the math strategy taught was con-fusing, teachers in the low-dose acquisition plus transfergroup gave a rating of 3.3, whereas the teachers in the othertwo conditions gave ratings of 1.8 and 2.3. Teachers per-ceived a high level of active participation by the students (rat-ings of 4.0– 4.3) and some level of skill mastery (ratings of3.0–3.3).

Across treatments, teachers had positive perceptions thatthe use of partners helped the students learn the skill of find-



TABLE 5. Descriptive and Inferential Statistics on Students Without Disabilities Demographics by Treatment


Acquisitiona + transferb + transferc

Variable n % n % n % χ2

Gender 1.98Girl 4 100 3 60 5 71Boy 0 0 2 40 2 29

Received free lunch 0 0 1 20 1 14 1.48

Race 3.07African American 2 50 2 40 2 29White 2 50 3 60 3 43Other 2 28

Class behavior 2.35Acceptable 4 100 4 80 7 100Problematic 0 0 1 20 0 0

Reading level 2.96Above grade level 2 50 2 40 6 86At grade level 2 50 3 60 1 14

Math level —Above grade level 4 100 5 100 7 100

Note. Classroom teachers were asked to rate each student’s class behavior as acceptable, occasional problem, or frequent problem. Dashes indicate the variable was aconstant so a chi-square test could not be performed.an = 4. bn = 5. cn = 7.

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ing half, as indicated in ratings that ranged from 3.7 to 4.3.High ratings of 3.8 to 4.8 indicated that having partnersencouraged interaction between students with disabilitieswith other students with whom they might not otherwise haveinteracted. In addition, teachers felt that their students withdisabilities liked working with a partner during the treatment(ratings of 4.5–4.8).

DISCUSSION

Results clearly indicate that when elementary-age studentswith mild disabilities were taught to use a strategy to solvemathematical word problems, their performance on measuresof process (using steps taught in the treatment, regardless ofwhether the answers were correct) and product (finding cor-rect answers) was better than that of students with disabilitieswho received more conventional instruction. Results werecomparable for students with and without math IEPs.Although the intervention was effective in improving perfor-mance levels of all three experimental treatments, differencesin growth were also evident among the three experimentalgroups. Overall, the full-dose acquisition plus transfer groupachieved the greatest gains on both posttest measures. How-ever, to understand the effect of the strategy instruction oneach treatment group, the process scores and product scoresmust be inspected.

On the process scores (applying steps taught in the treat-ment), significant differences with large effect sizes emerged,favoring the growth of the two conditions that incorporatedtransfer instruction over the control group (ES = 1.54 and2.11). This indicates that transfer instruction, designed to pro-mote awareness of strategy use with problems with novel fea-tures, enhanced students’ capacity to apply the basic steps ofthe treatment. Although not statistically significant (due tosmall sample size), the acquisition group’s process scoresimproved substantially more than those of the control groups(ES = .96). Consequently, findings support continued studyon explicit strategy instruction, with and without explicitinstruction to transfer, with a larger number of students in asearly as third grade with disabilities.

With respect to differences among the experimentaltreatment groups on the process scores, a moderate effect size(.59) was found between the two conditions that each incor-porated 4 days of instruction. The difference between thesetwo groups was that the acquisition group received 4 days ofacquisition practice, whereas the low-dose acquisition plustransfer group received 2 days of acquisition and 2 days oftransfer instruction. The effect size favored the growth of thelow-dose acquisition plus transfer group, suggesting thattransfer instruction may have contributed an added value tostudents’ ability to solve mathematical word problems. Inaddition, a large effect size (.96) favored the full-dose acqui-sition plus transfer group over the acquisition group. Both ofthese groups received 4 days of acquisition instruction, withthe full-dose acquisition plus transfer group receiving 2 addi-

tional days of transfer instruction. Also, between the twoexperimental groups with the added transfer component,greater gains in applying the steps of the strategy wereobserved for the students who received more instructionaltime (ES = .46). Consequently, results on the process measuresuggest the usefulness of more instructional time, withexplicit instruction to transfer.

In terms of students’ product scores (finding correctanswers), results were similar but even more clear: The full-dose acquisition plus transfer treatment was necessary toeffect a statistically significant difference over the perfor-mance of the control group (ES = 1.63). Moreover, althoughnot statistically significant (due to small sample size), largeeffect sizes were noted for the full-dose acquisition plustransfer group over the other two experimental conditions (ES = .93 and 1.29). This again indicates the importance ofmore instructional time with explicit instruction to transfer.Of course, large effect sizes also favored the performance ofthe other two experimental treatments over the control group(ESs = .69 and 1.17), but only a small effect size (.27) favoredthe low-dose acquisition plus transfer group over the acquisi-tion group. The large effect sizes of the experimental treat-ments over the control group suggest that using explicitstrategy instruction was more effective than conventionalinstruction in improving students’ ability to use the basicsteps of the treatment to find correct answers. However, thesmall difference between the two treatment conditions thateach received 4 days’ instruction and practice indicates littledifference between these two groups.

It is not surprising that on both dependent measures,time appears to be an important instructional dimension.Results provide evidence that the 4 days of acquisition in-struction and practice gave students greater opportunity tomaster the basic steps of the strategy. At the same time, thetransfer component called students’ attention to details of the math problems that could change without altering therequired solution. This instruction may have encouraged thestudents to read problems carefully, as they recognized famil-iar problem structures presented in novel ways. This focus onproblem comprehension may have prompted students to readproblems carefully rather than react to the problem at a sur-face level of analysis.

With respect to academic aspects via the use of partners,75% of the students with a disability reported that they did abetter job of finding half because of their partners’ assistance,whereas only 31% of the students without disabilitiesresponded that working with partners helped them do a betterjob of finding half. This is logical because in each pair thestudent with a disability was the lower achieving partner whounderstandably required more assistance. However, eventhough only a few students without disabilities reported doingbetter because of working with a partner, prior research inmath has demonstrated that providing more skilled studentsthe opportunity to construct explanations for less skilledlearners also facilitates higher learning for the more skilledstudents (L. S. Fuchs et al., 1997) because constructing expla-



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nations requires them to engage in deeper reflective thoughtprocesses (Hiebert & Wearne, 1993).

A primary theme in previous research on social interac-tions between students with LD and peers has indicated thatpeers without disabilities identify students with LD as amongthe most unpopular students in the classroom (e.g., Fox,1989). In the present study, the few students without disabil-ities who expressed the desire to work with a different part-ner did not seem to base their preference on popularity, asmuch as a lack of social acceptance and tolerance for the lessable student. Social acceptance of students with disabilities isimportant for inclusion to be successful (D. Fuchs & Fuchs,1994; Gartner & Lipsky, 1987), but inclusion does not ensuresocial acceptance and peer support (Roberts & Mather,1995). Future research should continue to explore methods ofimproving social acceptance.

Overall, student and teacher survey results indicate thatthe use of partners not only provides students with disabilitiesimmediate feedback on class work but also may facilitateopportunities for social interactions between students withand without disabilities. As suggested by previous research-ers (e.g., Baker, Wang, & Walberg, 1994; D. Fuchs & Fuchs,1998), social interactions between students with and withoutdisabilities benefit both groups. These social interactions mayhelp students with disabilities develop more appropriatesocial skills, which should enable them to interact with andbe more accepted by a variety of individuals in diverse set-tings (e.g., home, school, community, work). This may alsobenefit peers without disabilities by changing stereotypicalthinking about disabilities and promoting increased self-esteem caused by helping others (Phillips, Fuchs, & Fuchs,1994).

Several factors may have contributed to the success ofthe strategy we used. First, the intervention was designed withtypical third-grade general education classrooms in mind, toaddress day-to-day classroom realities. Second, the mathproblems were written to correspond to real-life experiencesand to invoke student motivation and understanding that thestrategy could be used to solve real-world problems (Gold-man, Hasselbring, & the Cognition and Technology Group atVanderbilt, 1997). Third, the strategy was simple and explicit,using concrete materials and examples that were at an appro-priate developmental level for young students with cognitivedelays. Prior research (Jitendra et al., 1998; Jitendra & Hoff,1996; Montague, Applegate, & Marquard, 1993) shows howstudent understanding of mathematical problem solving canbe enhanced through the use of visual and concrete represen-tations. Fourth, we relied on peer mediation, which has beenassociated with positive learning outcomes for both groups ofstudents in previous research (L. S. Fuchs et al., 1997; Green-wood et al., 1989). In that vein, this study enhances previouswork by combining strategy instruction with peer mediationin a way that can be used with relatively young students withdisabilities in typical general education settings.

Although the results of this study provide support forstrategy instruction in combination with peer mediation, lim-

itations must be recognized. First, despite making significantgains in math problem solving, as compared to the students inthe control group (who failed to perform at an acceptablelevel), the accuracy of posttest responses for students withdisabilities was 45%. Of course, it is important to note thateven with this intensive intervention, the high-achieving stu-dents in the larger study did not reach 100% mastery (L. S.Fuchs et al., in press). Future research needs to explore stu-dent errors after mathematical problem-solving strategyinstruction, which may provide direction for interventions toimprove mathematical word-problem solving. Second, thestudy did not focus on the effects of strategy instruction inisolation, without the use of peer mediation. However, signif-icant differences among the experimental groups, who allused similar partner configurations, indicate that the strategyinstruction treatment did contribute in important ways to out-comes. Also, impressive effect sizes favoring each of thethree experimental treatments over the control group providedirection for future research to explore the effects of strategyand peer-mediated instruction, in combination as well as sep-arately. Third, although the results of this study show promisefor instructional interventions in the general education class-room setting, this study focused on one skill area over a rela-tively short period of time. However, it is important to notethat this study was part of a larger investigation that alsoshowed strong effects for students with disabilities over a 16-week study on four problem-solving units with performancemeasures on acquisition, near-, and far-transfer measures.

In the final analysis, it is clear that students with dis-abilities will continue to receive instruction in general edu-cation settings. The challenge to teachers is to deliverinstruction that is effective for the full range of students. Asshown in the achievement data, our intervention corroboratesthe effectiveness of specific strategy instruction on mathe-matical word-problem solving, an instructional method thatcan easily be used in general education settings with studentsof diverse academic abilities. Results also suggest the poten-tial for explicitly teaching children about transfer (as in thefull-dose acquisition plus transfer condition) and the impor-tance of providing a strong instructional foundation (as in thefull set of four acquisition lessons). In addition, as shown inthe teacher and student self-report data, peer mediationappears to benefit students with and without disabilities, bothacademically and socially. By not singling out students withdisabilities and by providing individualized instruction, peermediation permits students with disabilities to work produc-tively, alongside their classmates who are not disabled.

In summary, effective instructional practices in mathe-matics for students with learning disabilities are emerging. It is important that these practices, as well as future re-search, continue to be explored from several perspectives inorder to benefit students with learning disabilities in inclu-sive settings. ■

RHODA L. OWEN, EdS, is assistant professor in the Department of Edu-cation at Regis University. She is a doctoral candidate in special education in



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the Department of Education and Human Development at Vanderbilt Uni-versity and is writing her dissertation on the mathematical errors third-gradestudents with and without disabilities make when solving mathematical wordproblems. LYNN S. FUCHS, PhD, is professor of special education, co-director of the Kennedy Center research program on learning accommo-dations, and co-director of the Peabody Reading Clinic, all at VanderbiltUniversity. Her research focuses on classroom-based assessment and instruc-tional methods to enhance reading and mathematics outcomes for studentswith disabilities. Address: Rhoda L. Owen, Mail Code H-12, 3333 RegisBlvd., Regis University, Denver, CO 80221; e-mail: [email protected]

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Received February 2001Revision received July 2001

Initial acceptance September 2001Final acceptance November 2001



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