Mathematical Word Problem Solving in Third-Grade Classroomsweb.fsktm.um.edu.my/~nizam/article4.pdf · pare problems are characteristic of most addition and sub-traction word problems

urrent education reforms support challenginglearning standards and school accountability. Inmathematics education, the emphasis is on the

development of conceptual understanding and reasoningover memorization and rote learning (Goldsmith & Mark,1999; Hiebert et al., 1996; National Research Council,2001). Mathematical problem solving is a central theme inthe Principles and Standards for School Mathematics (Nation-al Council of Teachers of Mathematics [NCTM], 2000).Although the current emphasis is on solving complexauthentic problems situated in everyday contexts, storyproblems that range from simple to complex represent “themost common form of problem-solving” assignment inschool mathematics curricula (Jonassen, 2003, p. 267).

Story problems pose difficulties for many elementarystudents because of the complexity of the solution process(Jonassen, 2003; Lucangeli, Tressoldi, & Cendron, 1998;Schurter, 2002). Mathematics tasks that involve story-context problems are much more challenging than are no-context problems (Cummins, Kintsch, Reusser, &

Weimer, 1988; Mayer, Lewis, & Hegarty, 1992; Nathan,Long, & Alibali, 2002). Although children may knowprocedures for solving no-context problems, solving storyproblems requires them to integrate several cognitiveprocesses that are difficult for children with an insuffi-cient knowledge base or limited working memory capaci-ty. When solving story problems, for example, childrenneed to (a) understand the language and factual informa-tion in the problem, (b) translate the problem with rele-vant information to create an adequate mental represen-tation, (c) devise and monitor a solution plan, and (d)execute adequate procedural calculations (Desoete, Roey-ers, & De Clercq, 2003; Mayer, 1999). In short, solvingword problems relates closely to comprehension of therelations and goals within the problem (e.g., Briars &Larkin, 1984; Cummins et al.; De Corte, Verschaffel, &De Win, 1985; Kintsch & Greeno, 1985; Riley, Greeno,& Heller, 1983).

Story problems are critical for helping children connectdifferent meanings, interpretations, and relationships tomathematics operations (Van de Walle, 2004). How thendo educators enhance students’ mathematical word prob-lem-solving skill? Traditional textbook problem-solvinginstruction has not effectively improved the learning of stu-dents at risk for mathematics difficulties. Many mathemat-ics textbooks are organized so that the same procedure(e.g., subtraction) is used to solve all problems on a page.As a result, students do not have the opportunity to dis-criminate among problems that require different solutionstrategies. Furthermore, traditional instruction teaches stu-dents to use keywords (e.g., in all suggests addition, left sug-gests subtraction, share suggests division; Lester, Garofalo,& Kroll, 1989, p. 84) that are limiting. According to Vande Walle, “Key words are misleading,” “Many problems donot have key words,” and “Key words send a terribly wrongmessage about doing math” (p. 152). Thus, this approach

Address correspondence to Asha K. Jitendra, Lehigh University,College of Education, 315 A Iacocca Hall, 111 Research Drive, Beth-lehem PA 18015. (E-mail: [email protected])

Copyright © 2007 Heldref Publications

283

Mathematical Word Problem Solving inThird-Grade Classrooms

ASHA K. JITENDRA CYNTHIA C. GRIFFIN ANDRIA DEATLINE-BUCHMANLehigh University University of Florida Easton School District

EDWARD SCZESNIAKEaston School District

ABSTRACT The authors conducted design or classroomexperiments (R. Gersten, S. Baker, & J. W. Lloyd, 2000) at2 sites (Pennsylvania and Florida) to test the effectiveness ofschema-based instruction (SBI) prior to conducting formalexperimental studies. Results of Study 1 conducted in 2 3rd-grade, low-ability classrooms and 1 special education class-room indicated mean score improvements from pretest toposttest on word problem solving and computation fluencymeasures. In addition, student perceptions of SBI according toa strategy satisfaction questionnaire revealed SBI as effectivein helping solve word problems. Results of Study 2, whichincluded a heterogeneous (high-, average-, and low-achieving)sample of 3rd graders, also revealed student improvement onthe word problem solving and computation fluency measures.However, the outcomes were not as positive in Study 2 as inStudy 1. Lessons learned from the 2 studies are discussedwith regard to teaching and learning mathematical word prob-lem solving for different groups of students.

Keywords: elementary grade students, mathematics instruc-tion, schema-based instruction, word problem solving

C

283-302 Jitendra M_J 07 5/8/07 10:57 AM Page 283

ignores the meaning and structure of the problem and failsto develop reasoning and making sense of problem situa-tions (Van de Walle).

In contrast, current mathematics reform efforts advocatethat teachers act as facilitators to help students constructtheir own understandings of mathematical concepts andrelationships (e.g., developing the abilities of inquiry, prob-lem solving, and mathematics connections). However, theliterature is inconclusive about the benefits of that approachfor all learners (Baxter, Woodward, Voorhies, & Wong,2002). As the range of skills that students bring to the class-room increases, the challenge for teachers is synthesizingknowledge regarding mathematics content and processes,student learning, effective instruction models, and appropri-ate classroom opportunities and experiences (Ma, 1999).Learners who lack sufficient prior knowledge may need moresupportive instructional strategies in which the teacher scaf-folds information processing by supplying a greater degree ofinstructional facilitation during the learning process.

To promote elementary students’ mathematical problem-solving skills, we relied on models for understanding andassessing children’s solutions for addition and subtractionword problems derived from schema theories of cognitivepsychology (Briars & Larkin, 1984; Carpenter & Moser,1984; Kintsch & Greeno, 1985; Riley, Greeno, & Heller,1983). On the basis of those models, we designed an inter-vention (i.e., schema-based instruction [SBI]) for studentswith high-incidence disabilities and for those at risk forfailure in mathematics (e.g., Jitendra et al., 1998; Jitendra,Hoff, & Beck, 1999). Contrary to earlier investigations ofSBI that provided individual or small-group instruction byresearchers in rooms adjacent to the students’ classrooms,we deemed it critical to broaden the learning environmentto a typical classroom context wherein the classroomteacher provided all instruction. In addition, we added self-monitoring to strategy use because teaching students toself-regulate their learning has an added positive effect ontheir mathematics problem-solving performance (L. S.Fuchs et al., 2003; Schunk, 1998; Verschaffel et al., 1999).

Theoretical Framework

Results of a number of research studies on solving addi-tion and subtraction word problems have shown that

semantic or mathematical structure of problems (i.e., specificcharacteristics of the problem and the semantic relation-ships among the various problem features) is much morerelevant than is syntax (i.e., how a problem is worded; Car-penter, Hiebert & Moser, 1983; Carpenter & Moser, 1984).Notwithstanding the ease or difficulty of the syntacticstructure of problems, students who lack a well-developedsemantic structure use a “bottom-up or text-drivenapproach to comprehend a problem statement” (Kameenui& Griffin, 1989, p. 581). Given the critical role of seman-tic structure in problem solution, Carpenter and Moser pos-tulated a classification of addition and subtraction word-problem types that include change, combine, compare, andequalize. Of those problem types, change, combine, and com-pare problems are characteristic of most addition and sub-traction word problems presented in elementary mathe-matics textbooks, indicating the need for research focusingon these problem types. Table 1 shows the three differentproblem types and their characteristics. Change problemsusually begin with an initial quantity and a direct orimplied action that causes either an increase or decrease inthat quantity. The three sets of information in a changeproblem are the beginning, change, and ending. In thechange situation, the object identities (e.g., video games)for beginning, change, and ending are the same (see Figure1). In contrast, combine or group problems involve two dis-tinct groups or subsets that combine to form a new group orset. Group problems require an understanding of part-part-whole relations. The relation between a particular set andits two distinct subsets is static (i.e., no action is implied).Compare problems involve the comparison of two disjointsets (compared and referent); the emphasis is on the staticrelation between the two sets (see Table 1). The three setsof information in a compare problem are the compared, ref-erent, and difference. For each problem type, there arethree items of information; the position of the unknown inthese problems may be any one of the three items, whichcan be found if the other two items are given.

Also critical to successful mathematics problem solvingis domain-specific knowledge (conceptual and procedural;e.g., Hegarty, Mayer, & Monk, 1995). An important aspectof domain-specific concept knowledge is problem compre-hension/representation, which involves translating thetext of the problem into a semantic representation on the

284 The Journal of Educational Research

TABLE 1. Semantic Structure of Addition and Subtraction Problems

Problem characteristicsRelation

Problem type Component Object identity between sets

Change Beginning, change, ending Same DynamicCombine or group Part, part, whole Different StaticCompare Compared, referent, difference Different Static


basis of an understanding of the problem structure.Although procedural knowledge (e.g., knowing the series ofsequential steps used to solve routine mathematical tasks,such as adding 27 + 19 as well as knowing mathematical

symbolism, such as +, =, and >) is also important, it “isextremely limited unless it is connected to a conceptualknowledge base” (Prawat, 1989, p. 10). In fact, a lack ofunderstanding of the problem situation may result in a solu-

May/June 2007 [Vol. 100 (No. 5)] 285

FIGURE 1. Schematic diagrams for change, group, and compare problem situations. Change diagram adapted from “Schemas inproblem solving,” by S. P. Marshall, 1995, p. 133. Copyright 1995 by Cambridge University Press. Adapted with permission ofthe author.

Change: Jane had 4 video games. Then her mother gave her 3 more video games for her birthday. Jane now has 7 video games.

Group: Sixty-eight students at Hillcrest Elementary took part in the school play. There were 22 third graders, 19 fourth graders, and27 fifth graders in the school play.

Compare: Joe is 8 years older than Jill. Jill is age 7 years and Joe is age 15 years.

+ 3video games

7video games

4video games

Change

Beginning

Bigger Smaller Difference

Small Group or Parts

Large Group or Whole (Sum)

Ending

All students (3rd,4th & 5th graders)

68

5thgraders

27

4thgraders

19

3rdgraders

22

Jill7

years

8years

Joe15

years


tion plan that is immaturely developed. Successful problemsolvers can translate and integrate information in the prob-lem into a coherent mental representation that mediatesproblem solution (Mayer, 1999; Mayer & Hegarty, 1996).However, many students have difficulty with problem com-prehension and solution and would benefit from instruc-tion in constructing a model to represent the situation inthe text, followed by solution planning based on the model(Hegarty et al., 1995). Evidently, schema-based instruction(SBI), with its emphasis on semantic structure and problemrepresentation, is one solution to advancing students’mathematical problem-solving skills.

The goal of SBI is to help students establish and expandon the domain knowledge in which schemata are the cen-tral focus. A schema is a general description of a group ofproblems that share a common underlying structure requir-ing similar solutions (Chen, 1999; Gick & Holyoak, 1983).According to Marshall (1995), schemata “capture both thepatterns of relationships as well as their linkages to opera-tions” (Marshall, 1995, p. 67). SBI analyzes explicitly theproblem schema (e.g., the part-part whole) and the linkspertaining to how different elements of the schema arerelated (e.g., parts make up the whole). Understandingthese links is crucial in selecting appropriate operationsneeded for problem solution. For example, if the whole inthe problem is unknown, adding the parts is necessary tosolve for the whole; if one of the parts is unknown, sub-tracting the part(s) from the whole is needed to solve forthe unknown part. An important difference between SBIand other instructional approaches is that only SBI empha-sizes integrating the various pieces of factual informationessential for problem solving. Although factual details areimportant, they should not be the central focus of instruc-tion and learning. In short, SBI allows students to approachthe problem by focusing on the underlying problem struc-ture, thus facilitating conceptual understanding and ade-quate word-problem-solving skills (Marshall).

Purpose

We conducted design or classroom experiments concur-rently at two different geographical sites. Our research teamcollaborated with teachers to conduct a series of word prob-lem solving teaching sessions using SBI with a small sampleof students to understand what works, prior to conducting aformal experiment (Cobb, Confrey, diSessa, Lehrer, &Schauble, 2003). Design experiments involve in-depthstudy of an innovation in a small scale to understand notonly how responsive an intervention is to the dynamics ofthe classroom but also to understand “the nature of effectiveadaptations for students with disabilities” (Gersten et al.,2000, p. 29). Specifically, a design experiment is aimed atimproving the connections between teaching and learningthrough the process of continuous cycles of “design, enact-ment, analysis, and redesign” of lessons in authentic class-rooms (Cobb et al., 2003; The Design-Based Research Col-

lective, 2003). Researchers use the processes of iterationand feedback loops to empirically refine the innovation(e.g., curriculum product, tasks, scaffolds) and design aneffective and efficient intervention for use in experimentalor quasi-experimental studies (Gersten et al., 2000).

The learning environment in our design experimentsincluded a typical classroom context rather than an excep-tional instructional setting (i.e., special rooms outside theregular classroom), wherein the teacher conducted allinstruction. The purposes of the design experiments weretwofold. First, we investigated the effects of SBI on theacquisition of skills for solving mathematical word prob-lems by third graders and explored the differential effects ofSBI for different groups of children (e.g., children withlearning disabilities; low, average, and high achievers). Sec-ond, we examined the influence of word problem-solvinginstruction on the acquisition of computational skills giventhe role that word problems play in the development ofnumber operations (Van de Walle, 2004).

The basic methods were similar across both studies. InStudy 1, we addressed the issue of the efficacy of SBI for ahomogeneous sample of students from low-ability mathe-matics classrooms and potential differences between stu-dents with learning disabilities and students without dis-abilities who were low achievers. Study 2 focused on theefficacy of SBI for a heterogeneous sample of third graders,including students differing in initial mathematics grade-level achievement.

DESIGN STUDY 1

Method

Participants

Participants included 40 third-grade students whoattended an elementary school in a suburban school districtin Pennsylvania serving 472 students in Grades K–5.Approximately 15% of the student population was AfricanAmerican, Hispanic, or Asian; 17% were economically dis-advantaged; and 5% were English language learners (ELLs).Third-grade students were grouped according to ability lev-els (two low- and two high-ability classrooms). Teachers inthe two low-ability third-grade classrooms (Classrooms 1and 2), as well as the special education teacher (Classroom3), participated to learn about innovative approaches toimproving students’ mathematical problem-solving perfor-mance. The final sample, however, comprised 38 students(20 boys and 18 girls) because 2 students moved out of theschool district before completing the study. The meanchronological age of students was 102.60 months (range =91 to 119 months; SD = 5.54). Twenty-eight students (74%)were Caucasian, 3 (8%) were African American, 6 (16%)were Hispanic, and 1 was Middle Eastern (3%).

The sample included 9 students (6 boys and 3 girls) withlearning disabilities (LD). While 3 of the students with LDreceived instruction in the general education classrooms,



the remaining 6 students were instructed in the learningsupport classroom. In addition, we designated 9 of the 38students as low achievers (LA) on the basis of their score(below the 35th percentile) on the Computation and Con-cepts and Applications subtests of the TerraNova mathe-matics achievement test (CTB/McGraw-Hill, 2001).Tables 2 and 3 show a summary of participant demograph-ic information by classroom and group status.

Separate one-way analyses of variance (ANOVAs) indi-cated no significant differences between classrooms on themathematics subtests of the TerraNova: Concepts andApplication, F(2, 35) = 0.38, ns, and Computation, F(2,35) = 0.80, ns; Total Reading subtest, F(2, 35) = 1.03, ns.However, there was a significant difference between class-rooms on chronological age, F(2, 35) = 15.99, p < .000.The mean age of special education students in Classroom 3was greater than that of students in Classrooms 1 and 2,which we expected given that the special education stu-dents were in the third and fourth grades. Chi-squareanalyses revealed no significant between-classroom differ-ences on gender, χ2(2, N = 38) = 2.57, ns, and race, χ2(6,N = 38) = 8.68, ns.

Similarly, separate one-way ANOVAs indicated no signif-icant between-group (LD and LA) differences on the Con-cepts and Application subtest of the TerraNova, F(1, 16) =2.84, ns, and on the Total Reading subtest, F(1, 16) = 1.96,ns. However, there was a significant between-group differ-ence on the Computation subtest, F(1, 16) = 5.53, p < .05.The mean score for students with LD was greater than thatfor LA students. In addition, there was a significant between-group difference on chronological age, F(1, 16) = 10.73, p <.01. The mean age of LD students was greater than was thatof LA students. Chi-square analyses revealed no significantbetween-group differences on gender, χ2(1, N = 18) = .90,ns, and race, χ2(2, N = 18) = 1.62, ns.

Three female teachers participated in Study 1, eachteaching in one classroom. The two general educationteachers were certified in elementary education and eachhad more than 25 years of teaching experience; the specialeducation teacher was certified in special education andhad 19 years of teaching experience. Teachers attended 1hr of inservice training on ways to implement the inter-vention; they received ongoing support from two researchassistants (doctoral students in special education) through-out the study.

Procedures

In all classrooms, teachers taught mathematics five timesa week for 50 min with the district-adopted basal text,Heath Mathematics Connections (Manfre, Moser, Lobato, &Morrow, 1994). Teachers taught the word problem-solvingunit during regularly scheduled mathematics instruction for15 weeks. They taught students how to solve one-step addi-tion and subtraction problems involving change, group,and compare problems for 30 min daily, 3 days per week.

Instruction was scripted to ensure consistency of informa-tion and included an instructional paradigm of teachermodeling with think-alouds, followed by guided practice,paired partner work, independent practice, and homework.In addition, instruction during guided practice emphasizedfrequent teacher–student exchanges to facilitate problemsolving. Word problem-solving instruction involved twophases: (a) problem schema and (b) problem solution. Wedeveloped a strategy checklist to help scaffold studentlearning. The checklist included the following six steps: (a)read and retell the problem to discover the problem type:(b) underline and map important information in the wordproblem onto the schematic diagram, (c) decide whether toadd or subtract to solve the problem, (d) write the mathe-matics sentence and solve it, (e) write the complete answer,and (f) check the answer.

Schema for Problem-Solving Instruction

During the problem-solving phase, students receivedstory situations that did not contain any unknown infor-mation. Instruction focused on identifying the problemschema for each of the three problem types (change, group,compare) and on representing the features of the story sit-uation with schematic diagrams (see Figure 1). That is, stu-dents learned to interpret and elaborate on the main fea-tures of the story situation and map the details of the storyonto the schema diagram. Teachers introduced each prob-lem type successively and cumulatively reviewed them tohelp students discern the three problem types.

Change story situation. Students learned to identify theproblem type by using story situations such as, “Jane hadfour video games. Then her mother gave her three morevideo games for her birthday. Jane now has seven videogames.” Using Step 1 of the strategy checklist, studentsidentified the story situation as change because it initiallyinvolved four video games, then an action occurred thatincreased this quantity by three, which resulted in a totalof seven video games. Also, the object identity of thebeginning, change, and ending is video games. As such,this story situation is considered a change problemschema. For Step 2, teachers prompted students to use thecorresponding diagram (see Figure 1) to organize or repre-sent the information. That step involved identifying andwriting the object identity or label (e.g., video games) forthe three items of information (i.e., beginning, change,and ending) in the change diagram. Students then readthe story to find the quantities associated with the begin-ning, change, and ending and wrote them in the diagram.Next, students summarized the information in the storywith the completed diagram and checked the accuracy ofthe representation.

Group story situation. Students identified the group storyproblem type by using story situations such as, “Sixty-eightstudents at Hillcrest Elementary took part in the schoolplay. There were 22 third graders, 19 fourth graders, and 27

May/June 2007 [Vol. 100 (No. 5)] 287



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fifth graders in the school play.” Using Step 1 of the strate-gy checklist, students learned that because the storydescribed a situation in which three small groups (third-,fourth-, and fifth-grade students) combine to form a largegroup (all students in the play), it is a story situation of thegroup problem type. For Step 2, teachers prompted studentsto use the group diagram (see Figure 1) to represent theinformation. Instruction involved identifying the threesmall groups and the large group and writing the groupnames in the diagram. Students then read the story to findthe quantities associated with each group and wrote themin the diagram. Next, students summarized the information

in the story with the completed diagram and checked theaccuracy of the representation.

Compare story situation. Students identified the storycomparison problem type by using story situations such as,“Joe is 15 years old. He is 8 years older than Jill. Jill is 7years old.” Using Step 1 of the strategy checklist, studentsidentified the story situation as compare because it requireda comparison of Joe’s age to Jill’s age. For Step 2, studentsused the corresponding diagram (see Figure 1) to organizeor represent the information. That diagram involved read-ing the comparison sentence (He is 8 years older than Jill)to identify the two sets that were compared in the story—


TABLE 3. Student Demographic Characteristics and Mathematics Performance by Group: Site 1

Group

LD (n = 9) LA (n = 9)

Variable n % M SD n % M SD ES

Age (in months) 108.44 6.50 100.22 3.80Gender

Male 6 66.67 4 44.44Female 3 33.33 5 55.56

RaceEA 6 66.67 8 88.89AA 0 0.00 0 0.00Hispanic 2 22.22 1 11.11Other 1 11.11 0 0.00

IQFull scale 94.78 6.92Verbal 92.17 6.82Performance 95.00 7.75

TerraNovaC & A 567.00 18.46 552.29 14.81Comp. 537.50 18.24 511.71 24.18Total reading 588.63 33.40 607.14 10.61

WPS-CRT (25/50)Pretest 14.83 5.40 18.22 8.34 −0.48Posttest 22.61 5.61 25.83 7.64 −0.48Growth 3.57 4.20 4.83 7.43 0.03

WPS-F (16/32)Pretest 14.29 5.71 13.94 5.19 0.06Posttest 17.50 4.58 18.78 6.91 −0.22Growth 8.89 2.40 9.72 4.01 −0.21

Computationa (25/43)Pretest 10.44 4.67 4.00 1.73 +1.83Posttest 25.56 8.09 25.78 11.95 −0.02Growth 15.12 6.85 21.78 11.95 −0.56

Strat. Quest. (5/5)Enjoyed 4.56 0.73 3.44 1.13 +1.18Diagram helps 4.44 1.13 4.00 0.87 +0.44Help solve 4.89 0.33 3.44 1.01 +1.93Recommend 4.56 1.01 3.89 1.05 +0.65Continue 4.44 0.73 3.00 1.32 +1.35

Total 22.89 3.26 17.78 2.73 +1.70

Note. Parenthetical numbers separated by slashes are total number of test items and total possible score. LD = learning disability; LA = low achiev-ing; EA = European American; AA = African American; C & A = Concepts and Application mathematics subtest; Comp. = Computation math-ematics subtest; WPS-CRT = word problem solving criterion test; WPS-F = word problem solving fluency measure; ES = effect size.aDigits correct.


determining the identity of the large (Joe’s age) and small(Jill’s age) sets—labeling them in the diagram, and writingthe difference amount in the diagram. Students then readthe story to find the quantities associated with the two setsand wrote them in the diagram. Next, students summarizedthe information in the story with the completed diagramand checked the accuracy of the representation.

Problem-Solution Instruction

During the problem solution phase that followed prob-lem schema instruction, students solved problems withunknowns by using either addition or subtraction; SBI inthis phase included six steps.

Change problems. Teachers prompted students to identifyand represent the problem (i.e., Steps 1 and 2) with thechange schematic diagram similar to the problem schemainstruction phase. The only difference was that studentsused a question mark to represent the unknown quantity inthe diagram. Step 3 involved selecting the appropriateoperation and transforming the information in the diagraminto a number sentence. That is, students learned that theyneeded to add the parts if the whole was unknown and sub-tract for the part when the whole was known. Instructionemphasized that when the change action caused anincrease, the ending quantity represented the whole; whenthe change action involved a decrease, the beginningquantity was the whole. For Step 4, students had to writethe mathematics sentence and solve for the unknown usingthe operation identified in the previous step. Step 5prompted students to write a complete answer. Finally, forStep 6, students had to check the reasonableness of theiranswer and ensure the accuracy of the representation andcomputation.

Group problems. Students learned to identify and repre-sent the problem (i.e., Steps 1 and 2) with the groupschematic diagram as in the problem schema instructionphase. For Step 3, when selecting the operation to solve forthe unknown quantity in group problems, students learnedthat the large group represents the whole and the smallgroups are the parts that make up the whole. Steps 4through 6 were identical to those described for the changeproblem.

Compare problems. Students identified and representedthe problem (i.e., Steps 1 and 2) using the compareschematic diagram as in the problem schema instructionphase. For Step 3, when selecting the operation to solve forthe unknown quantity in compare problems, studentslearned that the larger set is the big number or whole,whereas the smaller set and difference are the parts thatmake up the larger set. Steps 4 through 6 were identical tothose described for the change and group problems.

When instructors taught students to solve word prob-lems using SBI, only one type of story situation or wordproblem with the corresponding schema diagram initiallyappeared on student worksheets following the instruction

of that problem type (e.g., change problem). After studentslearned how to map the features onto schematic diagramsor solve change and group problems, teachers presentedstory situations or word problems with both types, alongwith a discussion of the samenesses and differences betweenthe change and group problems. Later, when students com-pleted instruction for change, group, and compare problemtypes, teachers distributed worksheets with word problemsthat included all problem types.

We measured fidelity of treatment with a checklist ofcritical instructional steps. The checklist included salientinstructional features (e.g., providing clear instructions,reading word problems aloud, modeling the strategy appli-cation, providing guided practice). Two graduate studentsin special education collected fidelity data as they observedone of the instructors for approximately 30% of the teach-ing sessions. Treatment fidelity, estimated as the percentageof steps completed correctly by the instructor, was 93%(range = 85–100%) across the 3 teachers.

Measures and Data Collection

Two research assistants administered and scored themathematical word problem-solving tests and computationtests with scripted directions and answer keys. All datawere collected in a whole-class arrangement.

Word problem-solving criterion referenced test (WPS-CRT).To assess student growth on third-grade addition and sub-traction word problems, students completed the WPS-CRTprior to and at the end of the intervention. The test con-sisted of 25 one-step and two-step addition and subtractionword problems. Part I of the CRT comprised 9 one-stepword problems derived from the Test of MathematicalAchievement (Brown, Cronin, & McEntire, 1994). Of the9 problems, 5 items included distracters. Part II of the WPS-CRT included 16 addition and subtraction word problemsselected from five commonly used third-grade mathematicstextbooks. The items consisted of 12 one-step and 4 two-step problems that met the semantic criteria for change,group, and compare problem types. In addition, 2 of theproblems included distracters. The design of the measureincluded the three problem types and both operations.

Word problems on Parts I and II of the WPS-CRTrequired applying simple (e.g., single-digit numbers) tocomplex computation skills (e.g., three- and four-digitnumbers; regrouping). Students had 50 min to completethe test. Directions for administering the word problem-solving test required students to show their complete workand to write the answer and label. Scoring involved assign-ing one point for the correct number model and one pointfor correct answer and label for a possible total score of 2points for each item. Cronbach’s alpha for each of thepretest and posttest measures was 0.84. Interscorer agree-ment assessed by two research assistants independentlyscoring 30% of the protocols was .92 at pretreatment and.97 at postreatment.

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Word problem-solving fluency (WPS-F) measure. Wedeveloped six forms to represent items in Part II of theWPS-CRT but modified them to include fewer problems,less advanced computation (one- and two-digit numbersonly), and no problems with distracters to address thetimed nature of the task. The problems for each WPS-flu-ency probe differed with respect to numbers, context, andposition of problems, which were random. Given that eachprobe included only half the number of problems on theWPS-CRT, the probes were not identical and differed withrespect to the unknown quantity to be solved (e.g., result,beginning, compared). We covered all possible combina-tions of problem types and operations in two probes ratherthan one probe. To control for the difficulty of the probes,we designed odd- and even-numbered probes to be parallelforms. Teachers administered the probes every 3 weeks tomonitor students’ progress in solving word problems. Stu-dents had 10 min to complete eight problems. For the pur-pose of this study, we used the combined score of the firsttwo probes and the combined score of the last two probesin the data analysis. Cronbach’s alphas for the two aggre-gated probes were 0.83 and 0.80, respectively. Interscoreragreement was .93 at pretreatment and .98 at postreatment.

Basic mathematics computation fluency measure (Fuchs,Hamlett, & Fuchs, 1998). We monitored student progresstoward proficiency on third-grade mathematics computa-tion curriculum prior to and at the completion of the inter-vention with basic mathematics computation probes. Stu-dents had 3 min to complete 25 problems, with a maximumscore of 43 correct digits. We scored the performance on thecomputation probes as the total number of correct digits,which provided credit for correct segments of responses.That assessment system is known to have adequate reliabil-ity and validity (see Fuchs, Fuchs, Hamlett, & Allinder,1989). Interscorer agreement was .99 at pretreatment and1.00 at postreatment.

Strategy satisfaction questionnaire. We developed andadministered a strategy-satisfaction questionnaire followingthe intervention to provide information about student per-ceptions regarding the problem-solving strategy interven-tion (i.e., SBI). The questionnaire included five items thatrequired students to rate whether they (a) enjoyed thestrategy, (b) found the diagrams helpful in understandingand solving problems, (c) improved their problem-solvingskills, (d) would recommend using the strategy with otherstudents, and (e) would continue to use it to solve wordproblems in the classroom. Ratings for the Likert-typeitems on the questionnaire ranged from a high score of 5(strongly agree) to a low score of 1 (strongly disagree). Cron-bach’s alpha for the questionnaire was 70.

Data Analysis

The unit of analysis was each student’s individual score(N = 16 for the general education classrooms; n = 6 for thespecial education classroom) rather than the classroom

because of sample limitations. On the WPS-CRT, WPS-F,and computation pretest measures, we conducted separateone-way, between-subjects (classroom or group) analysis ofvariance (ANOVA) to examine initial classroom or group(LD and LA) comparability. On the WPS-CRT, WPS-F,and computation pretest and posttest scores, we conducteda one between-subjects (classroom or group), one within-subjects (time: pretest vs. posttest) ANOVA. We used theFisher least significance difference (LSD) post-hoc proce-dure to evaluate any pairwise comparisons for significanteffects for the full sample. If a lack of classroom or groupcomparability on pretreatment measures occurred, we con-ducted a one-factor ANOVA on the change scores, withclassroom or group as the between-subjects factor. (Weused change scores for analyzing the two-wave data becausetheir interpretation is straightforward.) In addition, weanalyzed scores from the strategy questionnaire with multi-variate analysis of variance (MANOVA) with teacher orgroup as the between-subjects factor. To estimate the prac-tical significance of effects for classroom and group, wecomputed effect sizes (ESs) by subtracting the differencebetween the posttest means, then dividing by the pooledstandard deviation of the posttest. On the computationmeasure, we calculated improvement effects by dividingthe difference between the improvement means by thepooled standard deviation of the improvement divided bythe square root of 2(1– rxy; Glass, McGraw, & Smith,1981).

Results

Table 2 shows the means and standard deviations for allmeasures for the full sample; Table 3 shows means and stan-dard deviations for the measures by group (LD and LA).

Pretreatment Differences Among Classrooms on Mathematics Performance

Results indicated that differences between classrooms onthe WPS-CRT, F(2), 35) = 1.95, ns, and WPS-F, F(2, 35 =0.99, ns, were not significant prior to the study.

On the computation pretest, however, there was a sig-nificant effect for classrooms, F(2, 35) = 4.87, p < .05. LSDfollow-up tests indicated that mean scores for Classroom 3were significantly different than those for Classroom 1 (p <.05) and Classroom 2 (p < .01); Classroom 3 > Classroom1 > Classroom 2. However, the difference between Class-rooms 1 and 2 was not significant. We found large effectsizes of 1.08 and 1.71 for Classroom 3 when compared withClassrooms 1 and 2, respectively.

Posttreatment Differences Among Classrooms on Mathematics Performance

Results of the repeated-measures ANOVA applied to thepretest and posttest scores demonstrated a significant main



effect for time on the WPS-CRT, F(1, 35) = 40.90, p < .000,and WPS-F, F(1, 35) = 12.56, p < .01. Effect sizes for timeat posttest when compared with pretest were moderate forthe WPS-CRT (ES = 0.77) and large for the WPS-F (ES =0.83). There was no significant main effect for classroom orinteraction between classroom and time. ANOVA valuesfor the classroom main effect on WPS-CRT and WPS-Fwere, F(2, 35) = 2.14, ns, and F(2, 35) = 0.11, ns, respec-tively. The corresponding values for the interaction effecton the two measures were, F(2, 35) = 0.11, ns, and F(2, 35)= 0.64, ns.

On the computation measure, results of an ANOVA onchange scores indicated a significant effect for classrooms,F(2, 35) = 13.19, p < .000. Follow-up tests indicated thatthe differences between Classrooms 1 and 2 and betweenClassrooms 1 and 3 only were significant (p < .01). (Meanposttest scores for Classroom 1 > Classroom 2 > Classroom3.) We found large effects of 1.49 and 1.96 for Classroom 1when compared with Classrooms 2 and 3, respectively.

Posttreatment Differences Between Classrooms on the Strategy Satisfaction Questionnaire

Results of the MANOVA applied to the Strategy Satis-faction Questionnaire posttreatment scores revealed no sig-nificant differences among classrooms, Wilks’s lambda =.69, approximate F(2, 35) = 1.29, ns.

Pretreatment Differences Between Groups (LD and LA) onMathematics Performance

Results indicated that differences between groups on theWPS-CRT, F(1, 16) = 1.05, ns, and WPS-F, F(1, 16) =0.36, ns, were not significant prior to the study.

On the computation pretest, however, there was a sig-nificant main effect for group, F(1, 16) = 15.09, p < .01.The mean computation score for LD students was higherthan that for LA students, with a large effect size of 1.83 forLD students.

Posttreatment Differences Between Groups on Mathematics Performance LD and LA Sample

Results of repeated-measures ANOVA applied to thepretest and posttest mathematics test scores demonstrated asignificant main effect for time on the WPS-CRT, F(1, 16)= 26.94, p < .000, and WPS-F, F(1, 16) = 8.56, p < .01.Effect sizes for time at posttest when compared with pretestwere large for the WPS-CRT (ES = 1.12) and moderate forthe WPS-F (ES = 0.76). There were no significant maineffects for group or interaction between group and time.ANOVA values for the main effect of group on WPS-CRTand WPS-F were F(1, 16) = 1.32, ns, and F(1, 16) 0.96, ns,respectively. The corresponding values for the interactioneffect on the two measures were, F(1, 16) = 0.00, ns, andF(1, 16) = 0.89, ns. On the computation measure, results of

an ANOVA on change scores indicated the lack of a sig-nificant main effect for group, F(1, 16) = 2.11, ns.

Posttreatment Differences Between Groups on the Strategy Satisfaction Questionnaire

On the strategy satisfaction posttreatment scores, wefound significant differences between groups, Wilks’s lamb-da = .43, approximate F(1, 16) = 3.16, p < .05. Results ofseparate univariate ANOVAs indicated significant differ-ences between groups regarding total strategy satisfactionscores, F(1, 16) = 13.02, p < .01, enjoying the strategy, F(1,16) = 6.15, p < .05, usefulness of the strategy in solvingword problems, F(1, 16) = 16.49, p < .01, and continuingto use the strategy in solving word problems, F(1, 16) =8.24, p < .01. Specifically, LD students rated the strategyhigher than did LA students.

Discussion

Results must be interpreted in light of two serious limi-tations. First, the full sample (N = 38) and group samplesizes (n = 9 each for LD and LA students) were small. Sec-ond, the design was unbalanced; classroom sizes rangedfrom 16 to 6 students. As such, the findings indicate onlypreliminary evidence regarding the effectiveness of SBI.Within the constraints of the limitations, results fromStudy 1 provide evidence that SBI led to improvements inword problem-solving performance for the three class-rooms. Although the overall treatment fidelity was high,teaching styles varied concerning teachers’ adherence tothe scripted curriculum (i.e., read verbatim or used theirown explanations). The high level of treatment fidelityfinding suggests that SBI accounted for improved studentlearning, which is encouraging.

Furthermore, SBI was effective in enhancing the wordproblem-solving performance of students with LD, whetherthey received instruction in general education mathemat-ics classrooms or in a special education classroom. Whenwe separated outcomes on the word problem-solving mea-sures for students with LD and their LA peers, the effects ofthe word problem-solving curriculum on students’ perfor-mance was comparable for both groups. Those results arenotable because teachers implemented the treatment in awhole-class format and taught students to solve only one-step problems, although students were tested on one-stepand two-step problems. The findings support and extendprevious research regarding the effectiveness of SBI in solv-ing arithmetic word problems (e.g., Fuchs, Fuchs, Finelli,Courey, & Hamlett, 2004; Jitendra et al., 1998; Zawaiza &Gerber, 1993).

Our results indicated that classrooms and groups werenot comparable on their computational skills prior to thestudy and that computational skill improvement varied asa function of classroom and group. In general, students withLD at pretreatment demonstrated better computational

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skills than did the other students at pretreatment. Weexpected that result given that many of those students wereolder students receiving instruction in a special educationclassroom in which the focus of instruction was on theacquisition of basic skills. However, the general findingthat computation improvement was evident for all studentswas encouraging because opportunities for word problemsolving facilitate computation skills. The effect size forposttest over pretest was large for the entire sample (ES =2.98) and for the LD and LA students (ES = 2.16). Evi-dently, having students solve story problems enhanced thedevelopment of number operations or computational skills.The use of schematic diagrams, which provided meaning tonumber sentences, had an added value in promoting com-putation (Van De Walle, 2004).

Finally, the positive evaluation of SBI by students in thestudy seemed to play a role in enhancing mathematics per-formance as in several previous investigations (e.g., Case,Harris, & Graham, 1992; Jitendra et al., 1999). Studentswith learning disabilities especially were more enthusedabout SBI than were their peers with regard to treatmentacceptability and benefits. Wood, Frank, and Wacker(1998) stated that “Student preference is an important fac-tor, because students are not as likely to exhibit effort overtime with strategies that they do not like or do not feel arehelpful” (p. 336).

Overall, the study helped us learn about effective ways toenhance the problem-solving curriculum as well as facili-tate teacher implementation and student learning. Teach-ers became conversant about modifying the curriculumonly when they had completed the problem-solving uniton teaching change and group problems. The lessons thatwe learned from our observations and teacher input (priorto instruction on compare problems) during SBI imple-mentation allowed us to make several modifications thatwe organized in the following paragraphs according to cur-riculum, teacher, and student enhancements.

Curriculum

Although Marshall (1995) discussed the importance ofpresenting problem schema instruction in concert with thethree problem types followed by problem-solution instruc-tion, teachers raised concerns that problem solution wasseemingly removed from problem schema instruction foreach problem type. Therefore, we revised the curriculumsuch that instruction for the change problem type beganwith problem schema instruction, followed by problemsolution instruction. We used that same sequence for groupand compare problem types.

We redesigned the self-monitoring strategy checklist toinclude four steps, and used an acronym, FOPS (Find theproblem type, Organize information in the problem usingthe schema diagram, Plan to solve the problem, and Solvethe Problem), to help students remember the steps. Fur-thermore, we elaborated on each step to align with

domain-specific knowledge consistent with SBI. For exam-ple, to find the problem type (Step 1), we prompted stu-dents to examine information pertaining to each set(beginning, change, and ending) in the problem (e.g.,change).

Given that the compare problem type was difficult formany students during the problem schema phase, we col-laborated with teachers to develop a compare structure thatwas coherent to third-grade students. For example, wefocused on three sets (bigger, smaller, and difference) ofinformation and eliminated reference to difficult terms(e.g., compared, referent) when discussing the problem. Inaddition, we added oral exercises prior to written work toemphasize the critical features and relations in the problemto ensure that students were familiar with the informationneeded for problem comprehension, a key aspect of SBI.

We found that teachers were spending considerableamounts of instructional time explaining unfamiliar termsto students who did not have the necessary experientialbackground, which detracted from the focus on problemsolving. Therefore, we modified the textbook problems tomeet the needs of the students. For example, unknownwords such as “alpaca” were replaced by more familiar terms(sheep).

An important issue that emerged was teacher account-ability for student performance on statewide testing. Teach-ers believed that the use of word problems presented onlyin text format would not adequately prepare students togeneralize to word problems on the state mathematics test.Therefore, revisions to the problem-solving curriculumcontent entailed inclusion of items that presented informa-tion in tables, graphs, and pictographs. Also, because thestate test required the use of mathematical vocabulary (e.g.,addend) and emphasized written communication, we mod-ified our instruction to include key mathematical terms andprovided students with practice in writing explanations forhow the problem was solved.

Furthermore, teachers noted that the time required toimplement SBI was unrealistic given the need to coverother topics in the school curriculum. In the redesign of theproblem-solving curriculum, we reduced problem schemainstruction for each problem type from three 30-minlessons to one 50-min lesson and developed four lessons foreach problem type that addressed problem solution instruc-tion. In addition, we incorporated fading of schematic dia-grams to ensure that students were able to apply learnedsolution procedures independent of diagrams. Also, weeliminated homework problems from the curriculumbecause of teacher concerns regarding students’ inconsis-tent completion of homework.

Teachers

Our observations revealed that teachers need ongoingsupport during the initial implementation of a newly devel-oped intervention. Although we provided teaching scripts



to ensure consistency in implementing the critical content,we suggested that teachers use them as a framework forinstructional implementation. However, our observationsindicated that Teacher 1 read the script verbatim and fol-lowed it in its entirety, whereas Teacher 2 followed thescript inconsistently and required reminders to adhere tothe relevant information needed to promote problem solv-ing. In contrast, Teacher 3 (special education teacher)familiarized herself with the script and used her own expla-nations and elaborations to implement the interventionwith ease. Obviously, general education teachers in thisstudy needed more support to implement the intervention.Also, we noticed that some students in one classroom werestruggling during partner work and had to wait until whole-class discussion to get corrective feedback, indicating theneed for providing teachers with explicit guidelines aboutdirect monitoring student work.

In addition, we found that the general education teach-ers seemed more at ease in communicating their concernsto the two research assistants who supported them duringthe course of the project than to the primary researcher,which raised the issue of how and with whom communica-tion should be facilitated.

Students

Our observations indicated that for many students, espe-cially those with LD, scaffolding instruction (modeling, useof schematic diagrams and checklists) was critical as theylearned to apply the strategy. Explicit modeling and expla-nations using several examples enhanced their problem-solving skills. Following teacher-led instruction, we noticedthat several students with LD, as well as low-achieving stu-dents, were enthused and participated actively, as evidencedby their raised hands in response to teacher questioning.

DESIGN STUDY 2

Method

Participants

Students in Study 2 were 56 third-grade students in twoheterogeneous classrooms in a parochial school located ina small city in Florida that served 570 students in Gradespre-K–8. Approximately 20% of the student populationwas African American, Hispanic, or Asian. The total sam-ple included 27 boys and 29 girls. The mean chronologicalage of the students was 108 months (range = 96 to 113months). Forty-four of the students (78%) were Caucasian,5 (8%) were African American, 6 (10%) were Hispanic,and 1 was Asian American (2%). Although not formallyidentified at the school, the two third-grade classroomteachers reported that 9 of the 56 third-grade students hadeither LD, attention deficit disorder, or were consideredlow achievers. On the basis of scores from the ProblemSolving and Data Interpretation (PSDI) mathematics sub-

test of the Iowa Test of Basic Skills (ITBS), we designatedeach student’s initial mathematics achievement status aslow performing (LO; below the 34th percentile), averageperforming (AV; between the 35th and 66th percentiles),or high performing (HI; above the 66th percentile). Tables4 and 5 summarize participant demographic information bythe two classrooms and by student type (LO, AV, and HIstudents).

Separate one-way analysis of variances (ANOVAs) indi-cated no significant differences between classrooms on themathematics and reading subtests of the ITBS, ProblemSolving and Data Interpretation, F(1, 54) = 0.11, ns; Com-putation, F(1, 54) = 0.04, ns; Vocabulary, F(1, 54) = 0.35,ns; and Comprehension, F(1, 54) = 0.01, ns. In addition,there was a lack of a significant difference between class-rooms on chronological age, F(1, 54) = 0.31, ns. Chi-squareanalyses revealed no significant between-classroom differ-ences on gender, χ2(1, N = 56) = .27, ns, and ethnicity,χ2(3, N = 56) = 1.96, ns.

Similarly, separate one-way ANOVAs indicated signifi-cant differences between student type (i.e., LO, AV, andHI) on the PSDI, F(2, 53) = 239.81, p < .000; computa-tion, F(2, 53) = 9.498, p < .000; vocabulary, F(2, 53) =17.80, p < .000; and comprehension, F(2, 53) = 25.54, p <.000. For all analyses, LSD post-hoc tests revealed that thedifferences between the three groups were significant.(Performance of HI students > AV students > LO stu-dents.) In addition, there was no significant between-stu-dent type difference on chronological age, F(2, 53) = 0.95,ns. Chi-square analyses revealed no significant between-student type differences on gender, χ2(2, N = 56) = 0.27,ns, or race χ2(6, N = 56) = 1.96, ns.

The two general education teachers who participated inStudy 2 were women; one teacher held a master’s degreewith 10 years’ teaching experience, and the other teacher,a bachelor’s degree with 30 years’ experience. The teachersattended 1 hr of inservice training on ways to implementthe intervention; they received support from one of theresearchers periodically throughout the study.

Procedures and Measures

Procedures and measures were similar to those used inStudy 1. In this section, we report only how components ofthe two studies differed. First, although students received thesame amount of instructional time as those in Study 1,teachers began Study 2 later in the school year. Second,teachers in Study 2 used the textbook series, “Mathemat-ics—The Path to Math Success! Grade 3” by Silver, Burdett,and Ginn (as cited in Fennell et al., 1998) for daily mathe-matics instruction. Third, instructional procedures differedfrom Study 1 with regard to compare instruction. On thebasis of student difficulty with the compare problem type,teachers in Study 1 used the revised compare diagram andinstruction. However, teachers in Study 2 decided to imple-ment the original compare instruction and determine how

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their students, who were predominately average and highperforming, would respond. Fourth, in Study 2, we did notuse the student strategy satisfaction questionnaire adminis-tered in Study 1 because of time constraints at the end of theschool year. Treatment fidelity was 97.5% (range =91–100%) across the two teachers.

Results

Table 4 shows the means and standard deviations for allmeasures for the full sample. Table 5 displays means andstandard deviations for the measures by group (LD and LA).

Pretreatment Differences Between Classrooms onMathematics Performance

Results indicated that differences between classrooms onthe WPS-CRT, F(1, 54) = 3.35, ns, WPS-F, F(1, 54) = 0.00,ns, and computation pretest scores, F(1, 54) = 0.92 ns, werenot significant prior to the study.

Posttreatment Differences Between Classrooms onMathematics Performance

Results of ANOVA applied to the pretest and posttestdata yielded no significant classroom by time of testinginteraction for WPS-CRT, F(1, 54) = 1.39, ns, indicatingthat classroom had no effect on changes in correct respons-es from pretest to posttest.

However, the analyses yielded a significant main effectfor time of testing for WPS-F, F(1, 54) = 10.93, p < .00, andfor computation measures, F(1, 54) = 22.81, p < .000, indi-cating improvement from pretest to posttest for the entiresample of students regardless of classroom condition (aver-age for WPS-F = 2.19; SD = 4.90; computation = 2.18, SD= 3.45). Effect sizes for time at posttest when comparedwith pretest were small to moderate for the WPS-F (ES =0.31) and computation measures (ES = 0.34). The maineffect for time of testing on the WPS-CRT was not signifi-cant, F(1, 54) = 1.68, p = .20. We found no significantmain effect for classroom for all three tests: WPS-CRT, F(1,


TABLE 4. Student Demographic Information and Mathematics Performance by Classroom: Site 2

Classroom 1 Classroom 2

Variable n % M SD n % M SD ES

Age (in months) 106.69 4.35 107.26 3.21Gender

Male 13 44.82 14 51.85Female 16 55.17 13 48.15

RaceEA 23 79.31 21 77.77AA 3 10.34 2 7.40Hispanic 2 6.90 4 14.81Asian 1 0.03 0 0.00

ITBS Mathematics subtests

PSDI 554.01 114.55 565.99 100.64Computation 550.07 99.79 543.67 84.14

Reading subtestsVocabulary 568.22 95.34 582.00 85.95Comprehension 558.04 95.86 563.14 91.86

WPS-CRT (25/50)Pretest 30.05 10.61 25.50 7.65 +0.24Posttest 29.98 12.01 28.15 7.18 +0.09Growth −0.07 7.96 2.65 6.82 −0.18

WPS-F (16/32)Pretest 19.76 8.45 19.67 6.40 +0.01Posttest 22.59 8.50 21.17 3.97 +0.10Growth 2.83 5.03 1.50 4.74 +0.13

Computationa (25/43)Pretest 31.79 5.55 33.48 7.50 −0.12Posttest 33.45 5.88 36.22 6.57 −0.21Growth 1.66 4.04 2.74 2.65 −0.15

Note. EA = European American; AA = African American; ITBS = Iowa Test of Basic Skills (Houghton Mifflin, 1999); PSDI = problem solvingand data interpretation; WPS-CRT = word problem solving criterion test; WPS-F = word problem solving fluency measure; ES = effect size. Par-enthetical numbers separated by slashes are total number of test items and total possible score.aDigits correct.


54) = 1.80, ns, for WPS-F, F(1, 54) = .18, ns, and for com-putation, F(1, 54) = 1.84, ns.

Pretreatment Differences Between Groups (LO, AV, and HI)on Mathematics Performance

Results indicated that, as expected, differences betweengroups on the WPS-CRT, F(2, 53) = 5.68, p < .000 and com-putation, F(2, 53) = 5.68, p < .01, were significant prior tothe study.

LSD follow-up tests indicated that mean scores for HIwere significantly different than were those for AV (p <.00) and LO (p < .00) on the WPS-CRT (HI > AV > LO).However, the difference between LO and AV was not sig-nificant. We found large effect sizes of 1.14 and 1.15 for HIwhen compared with AV and LO, respectively. For theWPS-F measure, follow-up tests indicated that the meanscores for HI were significantly different than those for AV(p < .000) and LO (p < .000; HI > AV > LO). In addition,the difference between LO and AV was significant (p <.05), favoring the AV group. On the WPS-F, we found largeeffect sizes of 1.26 for HI compared with AV, 2.51 for HIcompared with LO, and 0.94 for AV compared with LO.

Finally, follow-up tests on the computation measurerevealed that mean scores for HI were significantly differ-ent than were those for LO (p < .00; (HI > LO), but notbetween HI and AV (p = .15). The difference between LOand AV on the computation measure was also significant (p= .05); the AV group scored higher. We found large effectsizes of 1.67 and 0.91 for HI compared with the LO, and forAV compared with LO, respectively.

Results of ANOVA applied to the WPS-CRT pretestand posttest data yielded no significant group by time oftesting interaction, F(2, 53) = 1.19, ns. That finding indi-cated that group status (HI, AV, LO) had no effect onchanges in correct scores from pretest to posttest. In addi-tion, there was no significant main effect for time, F(1, 53)= 0.26, ns, indicating lack of improvement from pretest toposttest for the entire sample of students, regardless ofgroup status. However, the analysis yielded a significantmain effect for group, F(2, 53) = 12.45, p < .000. The LSDfollow-up test indicated that WPS-CRT mean scores for HIstudents were significantly different from AV (p < .000)and LO (p < .000). We found effect sizes of 1.05 and 1.48for HI students when compared with AV and LO students,respectively, on the WPS-CRT. The difference between LOand AV, however, was not significant (p = .12).

Results for WPS-F indicated the following significanteffects: group by time of testing interaction, F(2, 53) =3.19, p < .05, time of testing, F(1, 53) = 17.37, p < .000,and group status, F(2, 53) = 16.72 p < .000. Because thegroup by time of testing interaction supercedes maineffects, we conducted follow-up tests only for the signifi-cant interaction. Follow-up tests indicated that frompretest to posttest, averaged across all groups, the LO stu-dents improved reliably more than did the HI students (p =

.02); the trend of improvement for LO students comparedwith AV students approached significance (p = .07); thegrowth of the AV and HI students was comparable (averagegrowth for LO students = 5.61; SD = 5.81; AV students =2.13, SD = 4.04; HI students = 0.96, SD = 4.88). The effectsize for LA growth compared with AV growth was 0.76; forLA compared with HI, 0.91; for AV compared with HI,0.26.

The analysis of computation data yielded no significantgroup by time of testing interaction, F(1, 54) = 0.36, ns,indicating that group status (HI, AV, LO) had no effect onchanges in correct scores from pretest to posttest. Howev-er, results indicated a significant main effect for time, F(1,53) = 18.84, p < .000, indicating improvement from pretestto posttest for the entire sample of students, regardless ofgroup status (M = 2.19; SD = 4.90). The effect size for timefrom posttest to pretest was 0.34. In addition, there was asignificant main effect for group, F(2, 53) = 5.74, p < .01.The follow-up test for group indicated that mean scores forHI students were significantly different than that for LOstudents (p < .00) but comparable to those for AV students(p = .20). In addition, the difference between LO and AVstudents was significant (p < .05). We found effect sizes of1.39 and 0.29 for HI students when compared with LO stu-dents, and for AV students compared with LO students,respectively.

Discussion

Students in Study 2 made small-to-moderate improve-ments in their word problem-solving and computation per-formance despite unforeseen problems that occurred duringthis field-based research. The benefits of a SBI were partic-ularly apparent for the low-performing students in the twoheterogeneously grouped third-grade classrooms. As inStudy 1, third-grade teachers delivered instruction inwhole-class arrangement without special instructionaladaptations for low-performing students. Despite the lackof individualized instruction, those students showed thegreatest amount of growth from pretest to posttest on twoof the three measures. Conversely, the results indicatedthat the high-performing students made the least amountof improvement. The findings support previous studiesregarding the importance of explicit instruction for strug-gling learners (e.g., Swanson, 1999) and its triviality forhigh performers. Even young children with high intelli-gence are capable of using strategies spontaneously to per-form given tasks without explicit instruction (Cho & Ahn,2003).

Results for the WPS-CRT, in particular, were disap-pointing. Our hypothesis that there would be a significanteffect for time from pretest to the posttest was not realized.The lack of improvement over time may be attributed toseveral reasons. First, the length of the WPS-CRT task(i.e., 25 one-step and two-step word problems) in conjunc-tion with administration of the posttest only 2 weeks before

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May/June 2007 [Vol. 100 (No. 5)] 299

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school ended for the summer may not have been motivat-ing for students, in spite of explicit instructions to do theirbest. Teachers reported students’ less-than-determinedattempts to complete the test. As such, that problem couldhave negatively affected student performance on the WPS-CRT posttest, which is not uncommon to interventionresearch (Dunst & Trivette, 1994).

Another plausible explanation for a lack of improve-ment was that the high-performing students may haveknown how to solve third-grade word problems before thestudy began and did not require extensive strategy instruc-tion. A closer look at the data suggests that that may be thecase. Fifteen of the high-performing students scored closeto an accepted criterion level of 80% correct before thestudy began, indicating possible ceiling effects (M = 39.91,SD = 3.99; range = 34 to 47 on the 50-point WPS-CRT).As such, pretest to posttest improvement was not evidentfor the students, and ceiling effects limited the ability toferret out differences among higher scoring students.

Results on the WPS-F measures were more encouraging.All students improved their word problem-solving perfor-mance on the shorter, 8-item, probes from pretest toposttest. The LO group made the most progress on thismeasure, as evidenced by the large effect size of 0.79 whencompared with effect sizes of 0.34 and 0.20 for the AV andHI groups, respectively. Given that the measure includedeasier items than did the WPS-CRT, was shorter in length,and required only 10 min to complete each probe, therebyreducing fatigue, it may have positively influenced theresults of the WPS-F.

The pretest-to-posttest results for the computation mea-sure revealed a pattern similar to those for the WPS-F. Allgroups improved significantly over time; however, the LOgroup demonstrated the largest gains from pretest toposttest, with a medium effect size of .57 when comparedwith small-to-moderate effect sizes of .36 and .32 for theAV and HI students, respectively. Again, word problemsolving seems to play a role in the development of numberoperations, such as computational skills (Van de Walle,2004), particularly for low-performing students.

As in Study 1, we learned ways to enhance curriculum,instruction, and measures. For example, whereas teachersfrom both sites were concerned with the length of theintervention, Study 2 teachers were less concerned aboutcoverage of other topics. Instead, teachers from Study 2reported the need to reduce what they considered redun-dancy in the curriculum as well as highly directive instruc-tion because the majority of their students (i.e., averageand high performers) occasionally appeared unmotivatedduring the instructional sessions. Considerable evidenceshows that pacing curriculum and instruction to match theneeds of the student is one way to ensure that the needs ofhighly able students are addressed (Rogers, 2002).Attempts to replicate SBI in heterogeneous classrooms maynecessitate changes to differentiate the instruction formixed-ability groupings (e.g., Tomlinson, 1995).

Our observations in Study 2, like those in Study 1, indi-cated that LO students benefited from the explicit model-ing, guided practice, use of schematic diagrams, and check-lists that are characterized by SBI. We observed that LOstudents tended to participate more than was typical duringSBI instruction, and their teachers stated that instructionhad clear benefits for them. However, as a group, the per-formance of LO students on the three measures was incon-sistent and highly variable. They performed well on theWPS-F and computation measure but not on the WPS-CRT. On the WPS-CRT (pretest: M = 22.00, SD = 11.18;posttest: M = 19.89, SD = 12.97), performance of LO stu-dents was highly variable, as indicated by large standarddeviations. That result did not occur for the WPS-F(pretest: M = 12.06, SD = 4.66; posttest: M = 17.67, SD =8.91) or computation test (pretest: M = 27.00, SD = 4.06;posttest: M = 29.44, SD = 4.50). The length and difficultyof the WPS-CRT appeared to affect the performance of theLO group most directly; variability decreased when themeasures were shorter or less challenging.

Finally, although the two teachers who participated inStudy 2 reported that they had few problems implementingSBI instruction and had high levels of treatment fidelity(M = 98%), in hindsight, the teachers may have had moreto reveal had they received our consistent support. Provid-ing teachers with more frequent opportunities to conversewith us may have allowed for instructional adaptations thataddressed varying student needs. Instruction that meets theneeds of heterogeneous groups of students is not easilydesigned, and, consequently, may require more structured,collaborative efforts between teachers and researchers.

General Discussion

Design studies have the potential to allow researchers todevelop in-depth understandings of teaching and learningin classroom environments, that in turn, can assist them inthe design of experimental or quasi-experimental studies(Gersten et al., 2000). Specific examples of those benefitsinclude enhanced knowledge about (a) instructional designfeatures of an intervention, (b) student and teacherresponses to the intervention, (c) measures used, and (d)the role of the researcher in facilitating the study. Like pilotstudies, design experiments allow researchers to determinethe feasibility of an experiment before implementing it ona larger scale and under more carefully controlled condi-tions (Gersten, in press).

We undoubtedly learned much about SBI for promotingmathematics word problem solving by conducting the twodesign studies reported here. Although we may have a fewunanswered questions, neither the knowledge gained northe uncertainties revealed would have occurred withoutcarrying out the investigative work of these studies. Themost notable insights revealed to us from conversationswith the teachers and from our own observations includedthe benefits of strategy fading to promote independent stu-



dent use of SBI. Encouraging independent, self-regulateduse of strategies has been supported in the reading literaturefor years and is perceived as an important long-term goal ofstrategy instruction (Pressley et al., 1992). We suggest thatsupporting the independent use of strategies is an impor-tant aspect of mathematics word problem-solving instruc-tion as well.

We also realized the need for changes in the compareproblem instruction by collaborating first with the teachersat Site 1, and later with teachers at Site 2. We observedthat the vocabulary words and compare schema diagramwere difficult and confusing for students at all achievementlevels. Having the opportunity to alter academic interven-tions on the basis of student and teacher input is a power-ful outcome of design studies and will strengthen theinstruction when examined in subsequent studies. We alsolearned about the importance of providing extensive sup-port to teachers during the study implementation. We areconvinced that the frequent and consistent support provid-ed to teachers in Study 1 allowed their students to performbetter. Alternatively, if we had offered more support inStudy 2, the teachers may have shared additional com-ments with us about the implementation of the study.Those comments could have prompted changes in theinstruction, which may have better addressed the academ-ic needs of the students and improved their performance.

Furthermore, Study 2 helped reveal the complexities ofworking in heterogeneously grouped classrooms. With thestrengthened Least Restrictive Environment (LRE) provi-sion in 1997 mandating that students with disabilities haveaccess to, and make progress in, the general education cur-riculum (U.S. Department of Education, 1997), more stu-dents with disabilities may be part of general educationclassrooms. Consequently, teachers will be increasinglychallenged to create instruction that addresses the needs ofstudents in academically heterogeneous classrooms. Themethod of doing that well is still unclear to us. However,we conjecture that some aspects of SBI will likely benefithigh-, average-, and low-performing students differentially,and others may improve their performance as a group.

Low-performing students in both studies most likely ben-efited from direct, explicit, and systematic instruction.High-performing students may have enjoyed the paired-learning opportunities provided during SBI. Research thatspans more than a decade suggests that the learning ofhigh-achieving students, such as those in Study 2, is facili-tated through peer interaction in learning (e.g., Swing &Peterson, 1982; Webb, 1991; Webb, Troper, & Fall, 1995).The use of schema-based diagrams may have contributed tothe word problem-solving and computation improvementamong many of the participating students, regardless ofability level. Using what we learned from the design stud-ies will allow us to strengthen the instruction before it isexamined in future research studies. In the interim, prelim-inary findings reveal the potential of SBI in mathematicsword problem solving with third-grade students.

NOTES

This research was supported by Grant H324D010024 from the U.S.Department of Education, Office of Special Education Programs.

The authors thank Dr. Russell Gersten for his insightful feedback dur-ing the study. A version of this article was presented at the 2004 annualmeeting of the Council for Exceptional Children in New Orleans,Louisiana.

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Among Those Present

STEVEN B. SHELDON is an associate research sci-entist, Center on School, Family, and CommunityPartnerships, Johns Hopkins University. He studies thedevelopment of family and community involvementprograms in schools, along with the impact of theseprograms on student outcomes. He also conductsresearch into the influences of parent involvement,including parent beliefs and social relations, and schooloutreach.

KERRY HOLMES is an assistant professor of ele-mentary education, School of Education, the Univer-sity of Mississippi. Her research interests focus onclosing the reading achievement gap through soundearly childhood reading practices and the motivationof reluctant readers. SARAH POWELL is an honorstudent in the School of Education at the same uni-versity. She graduated with honors from the SallyMcDonnell Barksdale Honors College. STACYHOLMES is an associate professor of engineering atthe University of Mississippi. He is interested inbringing quantitative techniques to the study of read-ing acquisition. EMILY WITT is a fourth-gradeteacher at Oxford Elementary School. She graduatedwith a master’s degree in elementary education fromthe University of Mississippi in 2005.

ASHA K. JITENDRA is professor, College of Edu-cation, Lehigh University. Her areas of research inter-ests include mathematics instruction and assessment.Other interests include reading interventions, text-book analysis, and curriculum-based assessment.CYNTHIA C. GRIFFIN is an associate professor,Department of Special Education, University of Flori-da. She is interested in beginning special educatorsand inclusive practice in mathematics and reading.

ANDRIA DEATLINE-BUCHMAN is a specialeducation consultant, Easton Area School District,Pennsylvania. She provides academic support to ele-mentary, middle, and high school special educationstudents. Her areas of research interests include writ-ing instruction and curriculum-based assessment.EDWARD SCZESNIAK is also a special educationconsultant with the Easton Area School district. Heprovides behavioral support to elementary, middle,and high school special education students by con-ducting functional behavioral assessments and devel-oping behavioral intervention plans. He is interestedin mathematics instruction and assessment.

HERBERT WARE is associated with George MasonUniversity, Fairfax, Virginia. As a former public schoolteacher, research coordinator, and administrator, hisinterests lie with the application of quantitative meth-ods to the study of school climate. He is particularlyinterested in the use of national data sets as a way ofstudying the relationship of school climate to studentachievement. ANASTASIA KITSANTAS is an asso-ciate professor, College of Education and HumanDevelopment, also at George Mason University. Herresearch interests focus on social cognitive processes,self-regulated learning, and motivational beliefs.

KELLIE J. ANDERSON is a school psychologist, Stu-dent Services Center, Anne Arundel County PublicSchools, Maryland. Her areas of interest include familyand school collaboration, crisis intervention, and ado-lescent development. KATHLEEN M. MINKE is aprofessor, School Psychology Program, University ofDelaware. Her research interests include family–schoolcollaboration, self-concept, and professional issues inschool psychology.

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