A Comparison of Two Mathematics Problem-Solving Strategies

The Journal of Educational Research, 104:381–395, 2011Copyright C© Taylor & Francis Group, LLCISSN: 0022-0671 print / 1940-0675 onlineDOI:10.1080/00220671.2010.487080

A Comparison of Two MathematicsProblem-Solving Strategies: Facilitate

Algebra-ReadinessYAN PING XINPurdue University

DAKE ZHANGClemson University

JOO YOUNG PARKKINSEY TOM

Purdue University

AMANDA WHIPPLENorthrop Grumman

LUO SIPurdue University

ABSTRACT. The authors compared a conceptual model-based problem-solving (COMPS) approach with ageneral heuristic instructional approach for teachingmultiplication–division word-problem solving to elementarystudents with learning problems (LP). The results indicatethat only the COMPS group significantly improved, frompretests to posttests, their performance on the criteriontest (that involves equal groups and multiplicative compareproblems) and the prealgebra model expression test. Thestudy results suggest that elementary students with LP canbe expected to move beyond concrete operations and toalgebraically represent mathematical relations in conceptualmodels that drive the solution plan for accurate problemsolving.

Keywords: algebra readiness, conceptual model, elementaryschool, instructional strategies, learning disabilities, mathe-matics, problem solving

R ecently released, the final report of the NationalMathematics Advisory Panel (2008) indicates that“American students have not been succeeding in

the mathematical part of their education at anything likea level expected of an international leader” (p. xii). Al-though American students are struggling with many aspectsof mathematics, the panel sees “algebra as a central concern”(National Mathematics Advisory Panel, 2008, p. xiii).

Interestingly, American students may enjoy school math-ematics at the early elementary grades. However, they be-gin to experience difficulty and to dislike mathematics afterGrade 4, when learning becomes more abstract or symbolicand involves more algebraic thinking (Cai et al., 2004).According to the panel, mathematics achievement in theUnited States decreases significantly in the late middlegrades when students are expected to learn algebra, whichraises the essential question: “How students can be best pre-

pared for entry into algebra?” (National Mathematics Ad-visory Panel, 2008, p. xiii). No doubt, the panel’s reportunderscores the importance of algebra-readiness instruction.

Indeed, algebra readiness has been characterized as serv-ing a gate-keeping function for secondary and postsecondaryeducation (Cai & Knuth, 2005; Maccini, McNaughton,& Ruhl, 1999). In fact, the National Council of Teachersof Mathematics (NCTM) has endorsed “algebra as a K–12enterprise” (Moses, 1997, p. 264) and set up the goal that allstudents, including those with special needs, learn algebraor succeed in high-level mathematics (NCTM, 2000).In line with the NCTM, the National Research Council(Kilpatrick, Swafford, & Findell, 2001) called teachers to in-troduce basic algebraic concepts in early elementary gradesto lay a foundation for algebra instruction in later years.Students should be taught to think algebraically “beforethey are expected to be proficient in manipulating algebraicsymbols” (Kilpatrick et al., 2001, p. 13). The good news isthat elementary students, those with learning disabilitiesor problems (LP) in particular, are able to benefit fromexperiences that prepare them for algebra readiness in theirelementary mathematics learning (Carraher, Schliemann,Brizuela, & Earnest, 2006; Xin, 2008; Xin, Wiles, & Lin,2008). In the next section, we characterize problem solvingand word-problem solving as well as algebraic thinkingin problem solving. Next, we briefly review interventionresearch in word-problem solving with students with LPand then introduce a word-problem solving model thatemphasizes algebraic thinking and readiness.

Address correspondence to Yan Ping Xin, Purdue University, BeeringHall of Liberal Arts and Education, Department of Educational Studies,100 North University Street, West Lafayette, IN 47907-2098, USA.(E-mail: [email protected])

382 The Journal of Educational Research

Algebra Thinking in Problem Solving

Problem solving has been defined as higher order cognitiveprocess that requires detecting steps or processes “betweenthe posing of the task and the answer” (Goldin, 1982, p. 97).When information about the problem is presented as textrather than in mathematical notation, the problem becomesa word problem (Verschaffel, Greer, & De Corte, 2000).As word problems often involve a narrative or story, theyare also called story problems (Moyer, Moyer, Sowder, &Threadgill-Sowder, 1984).

Problem solving is a relevant and significant perspectiveand context through which to introduce students to algebra(Bednarz & Janvier, 1996). Algebra is essentially “a system-atic way of expressing generality and abstraction” (Kilpatricket al., 2001, p. 256). In algebra, the focus is on expressionor representation of relations (Carpenter, Levi, Franke, &Zeringue, 2005). Representation is one type of activity thatinvolves algebraic thinking, through “translating verbal in-formation into symbolic expressions and equations,” such asgenerating “equations that represent quantitative problemsituations in which one or more of the quantities are un-known” (Kilpatrick et al., 2001, pp. 256–257). Within thecontext of arithmetic problem solving, algebraic thinking“involves the use of symbols to generalize certain kinds ofarithmetic operations” (Curcio & Schwartz, 1997, p. 296)and represent relations (Charbonneau, 1996).

Word-Problem Solving Instruction for Students with LP

Xin and Jitendra (1999) conducted a meta-analysis of 25published and unpublished research studies to specificallyevaluate the effectiveness of word-problem solving strategyinstruction for students with LP. Strategies investigated inthe obtained published and unpublished studies includedrepresentation techniques (i.e., any procedure that permitsthe interpretation or representation of ideas or informationgiven in a problem by using concrete manipulatives, verbal orlinguistic training, pictorial diagramming, or schema-basedmapping instruction), strategy training (i.e., any explicitproblem-solving heuristic procedure involving directinstruction, and cognitive and metacognitive instruction),computer-aided instruction (CAI), and other strategies(e.g., keyword instruction, problem sequence instruction,attention only). Results of Xin and Jitendra’s meta-analysisindicated that representation techniques, especially thoseprocedures emphasizing semantic structure understandingor schema knowledge-mediated diagramming, are moreeffective than other strategies (e.g., keyword instruction,sequential instruction only, metacognitive instructiononly) to promote students’ mathematics problem-solvingperformance.

In the past decade or so, schema-based instructionhas shown potential benefits in teaching mathematicsproblem solving to students with and without disabilities.Specifically, researchers have investigated the effects

of schema-based instruction on teaching elementaryarithmetic word-problem solving (e.g., Fuchs et al., 2008;Jitendra, Griffin, Deatline-Buchman, & Sczesniak, 2007;Jitendra et al., 1998) and transferring learned problemsolution rules to novel problems (e.g., Fuchs et al., 2003;Fuchs, Fuchs, Finelli, Courey, & Hamlett, 2004). However,most of these studies focused on addition and subtractionword problem solving and emphasized decision makingon the choice of operation (e.g., add or subtract) that wasconfined to specific word problem situations.

Conceptual Model-Based Problem Solving

Modeling involves translation or representation of au-thentic problems into mathematical expressions or modelsthat include real objects, formulas, algebraic expressions, oralgebraic representations. Mathematical models are an es-sential part of all areas of mathematics including arithmetic,and should be introduced to all age groups including ele-mentary students (Mevarech & Kramarski, 2004). Contem-porary approaches to story-problem solving have emphasizedthe conceptual understanding of the story problems beforeany solution attempts that involve selecting and applying anarithmetic operation for solution (Jonassen, 2003). Becauseproblems with the same problem schema share a commonunderlying structure requiring similar solutions (Chen, 1999;Gick & Holyoak, 1983), it is suggested that students needto learn to understand the structure of the mathematicalrelationships in word problems and that students should ex-hibit this understanding through creating and working withmeaningful representation of the structure (Brenner et al.,1997) or modeling (Hamson, 2003).

The representations that model underlying problem struc-ture facilitate solution planning and accurate problem solv-ing. For instance, factor–factor–product (or factor × factor =product) is a generalizable conceptual model in multipli-cation and division arithmetic word problems where fac-tor, factor, and product are the three basic elements. Itshould be noted that the three basic elements in thefactor–factor–product model have unique denotations whena specific problem subtype applies. For example, in an equalgroups problem type (e.g., “A school arranged a visit to themuseum in Lafayette Town. It spent a total of $667 buy-ing 23 tickets. How much did each ticket cost?”), the costof each ticket (the unknown) and the number of ticketsbought are the two factors, whereas the total money spentis the product. In contrast, in a multiplicative compare prob-lem (e.g., “Cameron has 242 marbles. Isaac has 22 times asmany marbles as Cameron. How many marbles does Isaachave?”), the marbles that Cameron has and the multiple re-lation (i.e., 22 times when Isaac is compared with Cameronon the number of marbles they have) are the two factors,whereas the number of marbles Isaac has (the unknown) isthe product.

In short, a conceptual model that recognizes and reor-ganizes the deep structure of the problem (i.e., problem

The Journal of Educational Research 383

schemata) needs to be constructed before solution plan-ning. More importantly, the conceptual model should drivethe development of a solution plan that involves selectingand applying appropriate arithmetic operations. Building oncontemporary approaches to story problem solving and torespond to the call for algebra readiness, Yan Ping Xin re-cently developed a conceptual model-based problem-solving(COMPS) approach that emphasizes prealgebraic conceptu-alization of mathematical relation to teach basic arithmeticword-problem solving to elementary students with LP (Xin,2008; Xin, Wiles, & Lin, 2008). Unlike a rule-driven orarithmetic-oriented approach, COMPS requires expressionof mathematics relation in a generalizable conceptual model(e.g., factor–factor–product); it does not rely on solutionrules to make decisions on choice of operation. Instead, themodel expression directly drives the selection of the opera-tion for solution.

The results of preliminary studies that evaluated COMPSusing single-subject design (Kazdin, 1982) indicate thatthere is a functional relationship between the interventionand students’ improved performance on researcher-designedcriterion tests that involve simple addition, subtraction, mul-tiplication, and division problems (e.g., Xin, Wiles, & Lin,2008) and on problems involving irrelevant information ormultiple steps (e.g., Xin & Zhang, 2009). The results alsoshow that students improved their prealgebra concept andskills after the COMPS instruction.

Purpose of the Study

The purpose of the present study was to extend Xin, Wiles,and Lin’s (2008) and Xin’s (2008) studies, which employedsingle-subject designs. The single-subject design does nothelp clarify whether the study findings are attributable tothe specific nature of the COMPS instruction or the gen-erally carefully designed small-group intensive instruction.Specifically, the purpose of the present study was to eval-uate and compare the effectiveness of two problem-solvinginstructional procedures, COMPS and a general heuristicinstructional approach to problem solving (GHI) in teach-ing multiplication and division word problems to elementarystudents with LP.

The study was designed to answer the following researchquestions:

Research Question 1: What were the differential effects ofCOMPS and GHI on the criterion tests designed to as-sess student performance in solving multiplication anddivision word problems targeted in this study?

Research Question 2: Did students maintain the acquiredproblem-solving skills following the termination of theintervention?

Research Question 3: What were the differential effects ofCOMPS and GHI on the prealgebra test designed to assessstudents’ grasp of prealgebra concepts and skills?

Research Question 4: What were the differential effects ofCOMPS and GHI on a norm-referenced standardizedmeasure?

Method

Design

A pretest–posttest, comparison group design with randomassignment of participants to groups was used to examinethe effects of the two word problem-solving instructionalapproaches: COMPS and GHI. Both groups also took amaintenance test 1 to 2 weeks following the terminationof the intervention.

Participants

Participants included a group of 29 students with LP fromtwo elementary schools in the midwestern United States. Asnot enough special education students were available in oneschool to satisfy the sample size requirement for this groupcomparison study, we had to recruit students from two ele-mentary schools in the same school district. The average ageof the students in the COMPS group was 10.32 years old (SD= 0.91) and 10.30 years old (SD = 0.86) for the GHI group.This included 16 students with learning disability (LD) orwith other disabilities, and 13 were school-identified at riskfor mathematics failure (e.g., failed the high-stake tests inmathematics). The special education eligibility criteria usedby participating schools define LD as a severe discrepancybetween the student’s academic achievement and normal ornear-normal potential. Of the six students who were iden-tified as other disabilities, two were with communicationdisorders, two were with mild mental retardation, one waswith attention deficit hyperactivity disorder (ADHD), andone was orthopedically impaired.

Specifically, participant selection was based on (a) teacheridentification of students experiencing substantial problemsin mathematics word-problem solving and (b) scores of 70%or lower on the criterion test involving multiplication anddivision word-problem solving. To determine sample size,a power analysis using a Cronbach’s alpha level of .05 andan effect size of .76 based on existing related research stud-ies (e.g., Jitendra et al., 1998) was conducted, which in-dicated that a minimum of 14 participants in each groupwas sufficient to obtain a power of .90 for a 2 × 3 repeatedmeasures analysis of variance (ANOVA; Friendly, 2000).Table 1 presents demographic information with respect toparticipants’ gender, grade, age, ethnicity, special educationclassification, IQ level, and standardized achievement scoresin mathematics and reading.

Dependent Measures

Criterion word-problem solving tests. The equivalent formsof criterion tests used during pre- and postintervention


TABLE 1. Participating Student Demographics, by Condition

COMPS group GHI group

Variable n M SD n M SD

GenderMale 6 6Female 9 8

Grade3 5 44 7 65 3 4

Age (months) 123.86 10.92 123.64 10.35Race

EA 6 6AA 4 1Hispanic 5 6Multiracial 0 1

ClassificationLD 5 5ADHD 0 1NL 6 7Other 4 (2 CDs, I MiMH, 1 OI) 1 MiMH

IQFull scale 84.16 13.55 87.36 11.99Verbal 83.08 10.77 88.25 15.87Performance 91.33 16.17 88.91 15.70

AchievementMathematics PR 24.44 33Reading PR 23.89 35.57

Note. LD = learning disability; ADHD = attention deficit hyperactivity disorder; CD = communication disability; MiMH= mild mental disability; NL = not labeled; OI = orthopedically impaired; EA = European American; AA = AfricanAmerican; PR = percentile rank. IQ scores were obtained from the Wechsler Intelligence Scales for Children–FourthEdition (Wechsler, 2003). Achievement scores in mathematics and reading were obtained from the Northwest EvaluationAssociation (NWEA) Test, or Woodcock-Johnson Psychoeducational Battery Test of Achievement–Third Edition (WJTA-III).

assessment were computer generated (Xin, Wiles, & Lin,2008). Each test involved 12 multiplication and divisionword problems that represented a range of equal groups(EG) and multiplicative compare (MC) problems (Van deWalle, 2004). These criterion tests were designed in align-ment with the NCTM (2000) standards, which emphasizevarying construction of word problems for assessing concep-tual understanding of mathematics problem solving (Cawley& Parmar, 2003). Table 2 presents sample problems to illus-trate each variation. As shown in Table 2, the constructionof each word-problem item was systematically varied in ref-erence to the unknown position so that a range of wordproblems was represented. Parallel form reliability betweentwo randomly selected forms was .84 for the sample in thisstudy. Cronbach’s Alpha of the criterion test was .86, andthe test–retest reliability was .93.

Prealgebra tests. Because the COMPS approach (Xin,Wiles, & Lin, 2008) applied in this study emphasizes al-gebraic expression of mathematical relations in problem

representation, prealgebra tests were administered pre- andpostintervention to both groups to examine whether thetwo groups were different on prealgebra concept and skilldevelopment. The prealgebra tests had two subtests: solveequations and model expression (Xin, Wiles, & Lin, 2008).The solve equations test included six items that requiredstudents to find the value of an unknown quantity (i.e., let-ter a) that makes the equation true (e.g., 196 = a × 28).Positions of the unknown were systematically varied acrossthree terms in the equation (i.e., the multiplicand, multi-plier, and product). The model expression test included fiveitems (e.g., “Antoni has collected 84 autographs. He filled 14pages in his new autograph album. Each page holds an equalnumber of autographs. Write an equation with a variableto model this problem.”). These items were directly takenfrom the commercially published mathematics textbook be-ing adopted by the participating schools (Maletsky, 2004).Cronbach’s alpha was .70 for the solve equations test and.62 for the model expression test, and test–retest reliabilitywas .90 for the prealgebra test.


TABLE 2. Sample Problems in Probes: Multiplication–Division

Problem type Sample problem situations

Equal groupsUnit rate unknown A school arranged a visit to the museum in Lafayette Town. It spent a total of $667 buying

23 tickets. How much does each ticket cost?Number of units (sets) unknown There are a total of 575 students in Centennial Elementary School. If one classroom can

hold 25 students, how many classrooms does the school need?Product unknown Emily has a stamp collection book with a total of 27 pages, and each page can hold 13

stamps. If Emily filled up this collection book, how many stamps would she have?Multiplicative compare

Compared set unknown Isaac has 11 marbles. Cameron has 22 times as many marbles as Isaac. How many marblesdoes Cameron have?

Referent set unknown Gina has sent out 462 packages in the last week for the post office. Gina has sent out 21times as many packages as her friend Dane. How many packages has Dane sent out?

Multiplier unknown It rained 147 inches in New York one year. In Washington, DC, it only rained 21 inchesduring the same year. The amount of rain in New York is how many times the amount ofrain in Washington, DC?

Note. It should be noted that MC problems in Table 2 only include those with multiple but NOT part relations such as “2/3” because the participantsdid not know operations with fractions during the intervention of this study. Problems presented in this table are derived from test materials in Xinet al. (2008) study.

KeyMath Revised Normative Update. The problem-solving subtest of the KeyMath Revised Normative Update(KeyMath-R/NU; Connolly, 1998) was administeredbefore and after the intervention to evaluate the fartransfer effect of the intervention on a standardized test.The KeyMath-R/NU is a norm-referenced, individuallyadministered diagnostic test. The content meets theNCTM standards especially for primary school-age students.The KeyMath-R/NU provides two alternate forms, andrequires little reading and writing. It has excellent internalconsistency for subtests (e.g., .89 for the problem-solvingsubtest across the Grades 4 and 5) and total test (.98). Itsalternate-form reliability for the problem-solving subtestranged from .67 (grade scores) to .71 (age scores).

Scoring

The percentage of problems solved correctly was used asthe dependent measure and calculated as the total pointsearned divided by the total possible points. Specifically,items on the criterion word-problem solving tests werescored as correct, and 1 point was awarded if the correctanswer was given. A half-point was given if only the math-ematics sentence or equation was correctly set up and theanswer was not correct due to calculation errors.

Items on the prealgebra solve equations test were scoredas correct and awarded 1 point if the numerical answer wascorrect. Items on the prealgebra model expression test werescored as correct and awarded 1 point if a correct model(or equation) was presented to express the mathematicalrelations of the given information in the problem. Three re-search assistants who were enrolled in the doctoral program

of special education scored all tests using an answer key.A graduate student who was unfamiliar with the purposeof the study rescored 33% of the tests. Interrater reliabilitywas computed by dividing the number of agreements by thenumber of agreements and disagreements and multiplyingby 100, which resulted in a median interrater reliability of100% (range = 75–100%).

Procedures

A stratified random-sampling procedure based on stu-dents’ grade, gender, and pretest scores was used to assignstudents into the two conditions. Two research assistants(one with 1 year of regular education teaching experienceand the other with 3 years of special education teaching ex-perience) and two participating school teachers (one with26 years of teaching experience and one with 16 years ofteaching experience) volunteered to be the instructors forthe two conditions. All instructors received two 1-hr train-ing sessions on the two instructional strategies. Yan PingXin developed the teaching scripts that were studied by allof the instructors to prepare for teaching these lessons. Thetwo research assistants taught the COMPS condition first,and the two school teachers taught the GHI condition first.To control for teacher effects, each pair of instructors (i.e.,one research assistant and one school teacher at each schoolsite) switched treatment groups midway through the inter-vention. It should be noted that despite the changes of theinstructors, the students in the COMPS condition continuedto receive the COMPS instruction and the students in theGHI condition continued to receive the GHI instruction. As


such, there was no change in the type of instruction studentswere receiving in either the COMPS or GHI condition.

Students in both conditions engaged in the assigned strat-egy learning three times a week, with each session lastingfor approximately 30–45 min (one school with 30 min andthe other school with 45 min). The COMPS group receivedthree sessions on introduction, six sessions each on EG andMC problem structure and problem solving, and three ses-sions on mixed review. Although students in the GHI groupalso received 18 sessions of instruction, they engaged insolving both types of problems in each session. Unlike theCOMPS group, students in the GHI group did not receiveinstruction in recognizing the two different word-problemtypes (i.e., EG or MC). Students in both conditions solvedthe same number and type of problems. Calculators were al-lowed in both groups throughout the study to accommodateparticipants’ skills deficits in calculation.

COMPS Condition

The research team designed a PowerPoint presentation(based on the teaching script developed by Yan Ping Xin)with animation for use during the intervention. Instructionsfor the COMPS condition was carried out in three phases: (a)the introduction session for the understanding of the conceptof equal groups, (b) the instruction on EG and MC problemsolving using the model-based problem-solving approach,and (c) mixed review for solving the EG and MC problems.During Phase II, instruction on EG and MC problem solv-ing was delivered in two parts: model representation andproblem-solving instruction. During model representationinstruction, students learned to detect the problem type andto map mathematical relations onto respective conceptualmodel diagrams (see Figure 1 for conceptual model diagramsfor EG and MC problem types) using story situations with nounknowns. The purpose of presenting story situations withno unknowns was to provide students with a complete repre-sentation of the mathematical relation in a specific problemtype so that generalized mathematical relations in the modelcould be visualized.

Model representation instruction was followed byproblem-solving instruction. During problem-solving in-struction, word problems with an unknown quantity werepresented. Yan Ping Xin designed a four-step DOTS(Detect–Organize–Transform–Solve) checklist (see Fig-ure 2) to guide students’ problem-solving process. In step 1,students detect the problem type based on the problem struc-ture they learned during model representation instruction.In step 2, students organize the information through rep-resentation of mathematical relations in conceptual modeldiagrams. Yan Ping Xin developed respective word-problemstory grammar prompting cards (Xin, Wiles, & Lin, 2008)for the EG and MC problem types (critical features for EGand MC problems are described in next section) to guidestudents’ problem representation (see Figure 1). Studentswere allowed to use any letter they preferred to represent

the unknown quantity. In step 3, students transform the di-agram to a meaningful mathematics equation by peeling offboxes and labels in the conceptual model diagram to makea true mathematics equation. In step 4, students solve forthe unknown quantity through equation manipulations. Inaddition, students were asked to provide a complete answerto the problem and to check the answer to make sure it madesense.

Word-problem story grammar questions (see Figure 1)were designed to facilitate understanding of word-problemstory structure through identification of three key elementsin each problem type. An EG problem describes a numberof equal sets or units. The placement of the unknown canbe on the unit rate (number of items in each unit or unitprice), number of units or sets, or the product (see the threevariations of EG problems in Table 2). An MC problemcompares two quantities and involves a compare sentencethat describes one quantity as a multiple or part of the otherquantity. The placement of the unknown can be on thecompared set, the referent unit, or the multiplier (see thethree variations of MC problems in Table 2).

Overall, the instruction was delivered through explicitstrategy explanation and modeling, dynamic teacher–student interaction, guided practice, performance moni-toring with corrective feedback, and independent practice.During independent practice, students were provided witha six-item independent worksheet to solve one of the twoproblem types (i.e., EG or MC) they had just learned.It should be noted that the conceptual model diagramswere provided on all modeling, guided, or independentpractice worksheets during the intervention. However, theywere gradually faded out on the worksheets when studentsworked on solving mixed word-problem types and were notprovided during postintervention assessment.

GHI Condition

The GHI was guided by a general heuristic five-step problem-solving checklist, SOLVE (Search–Organize–Look–Visualize–Evaluate). SOLVE was taken from the par-ticipating schools’ enacted curriculum and teaching practice.SOLVE required students to (a) search for the question, (b)organize the information, (c) look for a strategy, (d) visual-ize and then work the problem (e.g., draw a picture, make atable, write an equation), and (e) evaluate the answer.

For the first step, search for the question, the instructorasked students to find the question and then read the prob-lem twice. Students took turns to read the problem aloudor the teacher read with the students together, and thenthe group discussed about what they were asked to solve tohighlight the question. For the second step, organize the in-formation, the instructor guided students to highlight thekey words (e.g., times as in MC problems; each or per as inthe EG problems). For the third step, look for a strategy,the instructor asked students to think about the best way tosolve the problem and specifically which operation to use.


FIGURE 1. Conceptual models for (a) EG and (b) MC problem types (adapted from Xin, Wiles, & Lin, 2008).


FIGURE 2. DOTS checklist (adapted from Xin, Wiles,& Lin, 2008; color figure available online).

For the fourth step, visualize and then work the problem, stu-dents were engaged in visualizing the problem situation. Stu-dents were also instructed that they could draw a picture (todescribe the given information in the problem), make a table(to organize the given information), or write an equation ormathematics sentence. The instructor had the flexibility touse multiple strategies (e.g., draw a picture or write a math-ematics sentence or equation) in the teaching. It should benoted that PowerPoint presentations were also used in thiscondition. However, unlike the COMPS condition, GHIused the PowerPoint presentations to represent problem sit-uations with concrete or semiconcrete pictures only, but notsystematically introduce the EG and MC conceptual modelsas in the COMPS. For the last step, evaluate the answer, theinstructor guided students to ask self-check questions, suchas “Did you answer the question?”; “Does your answer makesense?”; and “Did you label your answer?” During the in-struction, students were provided with the SOLVE checklistto facilitate their problem solving (see Figure 3).

Distinction Between the Two Conditions

The distinction between the two conditions was thatCOMPS provided students with a well-defined model (i.e.,unit rate × # of units = total or product for the EG type; orunit × multiplier = product for the MC type) in which analgebraic expression of mathematical relation is emphasized.Further, the model expression served to drive the selectionof the operation to solve for the unknown. In contrast, theGHI condition had the flexibility of using multiple strategies,such as drawing concrete pictures to represent the problemor using “ × = ” to set up a mathematics sen-tence. Although equations could also be used in the GHIgroup, students were asked to set up the mathematics sen-tence by filling in the numbers. That is, three elements orterms in the equation (i.e., factor, factor, product) were not

FIGURE 3. SOLVE checklist used in the GHIcondition.

defined within specific problem contexts (i.e., unit rate ×# of units = total or product in EG problem situations; orunit × multiplier = product in MC problem situations).

Treatment Fidelity

For each instructional condition, a checklist that con-tained critical instructional components was used to assessthe instructors’ adherence to the assigned strategy instruc-tion. 67% of the COMPS sessions and 67% of the GHIsessions across two schools were videotaped. A doctoral stu-dent in special education viewed the videos and evaluatedthe adherence of the instructors’ teaching to the assignedinstructional strategy based on the components listed in thechecklist by judging the presence or absence of each crit-ical component. Treatment fidelity was calculated as totalcomponents present divided by the total possible compo-nents (which varied across conditions and problem types)in the checklist. Overall, fidelity was 100% for the GHI and86% for the COMPS condition (range = 75%–100%). Ona few occasions, the instructors skipped the step, “name thereferent unit (benchmark) in the diagram,” in solving theMC problems. Students were instructed to map the numbersinto the diagram based on the relational statement in MCproblems without naming the referent unit in the diagram.

Results

Pretreatment Group Equivalency

Table 3 presents results on pre- and postinterventionmeasures. Separate one-way ANOVAs were used to exam-ine pretreatment group equivalency on criterion and transfer


TABLE 3. Means and Standard Deviations, by Treatment Condition with Effect Sizes

M SD n

Test COMPS GHI COMPS GHI COMPS GHI ES

Pretest (%) 23.86 42.31 24.04 34.89 14 13 −0.6158Posttest (%) 83.57 67.93 19.21 31.42 14 13 0.6006Maintenance (%) 80.77 65.69 15.64 28.47 15 13 0.6563Pre-SE (%) 19.64 27.75 22.41 29.54 14 12 −0.3092Post-SE (%) 74.86 74.83 22.43 27.15 14 12 0.0012Pre-ME (%) 2.14 11.15 8.18 22.74 14 13 −0.5272Post-ME (%) 51.79 28.69 27.20 26.55 14 13 0.8594Pre-KeyM (SS) 8.64 9.31 2.62 2.90 14 13 −0.2429Post-KeyM (SS) 10.50 10.69 2.71 2.98 14 13 −0.0667

Note. Pretest, Posttest, Maintenance = pretest, posttest, maintenance test scores on the criterion tests; Pre-SE = solve equations pretest; Post-SE =solve equations posttest; Pre-ME = model expression pretest; Post-ME = model expression posttest; ES = effect size by Cohen’s d (calculated as thetwo conditions’ mean differences divided by the pooled standard deviation; a positive sign in ES indicates the effect favors the COMPS group and anegative sign indicates the effect favors the GHI group; Hedges & Olkin, 1985); Pre-KeyM = KeyMath-R/NU pretest; Post-KeyM = KeyMath-R/NUposttest; SS = scaled score.

assessments. Results indicated no statistically significantdifference between the two groups on either the criteriontest, F(1, 27) = 2.003, p = .172; the prealgebra solveequations test, F(1, 26) = 0.558, p = .462; the prealgebramodel expression test, F(1, 27) = 1.906, p = .179; and theKeyMath-R/NU, F(1, 27) = 0.295, p = .591, although theeffect sizes seemed to favor the GHI group on all measuresduring the pretests (.616 on the criterion test, .309 on thesolve equation test, .527 on the model expression test, and.243 on the KeyMath-R/NU problem-solving subtest).

Acquisition and Maintenance Effects of Word-Problem SolvingInstruction

A 2 (group) × 3 (time of testing: pretest, posttest, andmaintenance test with the criterion test) ANOVA with a re-peated measure on time was performed to assess the effects ofinstruction on students’ word-problem solving performance.It must be noted that two students (who moved out of theschool district in the middle of the project) did not completethe post and maintenance criterion tests. As such, this anal-ysis was based on the data for the 13 students in the GHIgroup and the 14 students in the COMPS group. Overall,the 2 × 3 ANOVA indicated a significant main effect fortime of testing, F(2, 50) = 27.145, p = .000, and there wasno significant main effect for group, F(1, 25) = 0.136, p =.715. More importantly, the results indicated a significantinteraction effect between group and time of testing, F(2,50) = 4.499, p = .016 (Figure 4). Further post hoc analysesby paired-samples tests indicated that the COMPS groupsignificantly improved its performance (M percentage pointdifference = 59.71, SD = 32.57) from pretest to posttest,t(13) = 6.86, p = .000, and maintained the improvement(M percentage point difference = −2.96, SD = 18.81) from

posttest to maintenance test, t(13) = −0.59, p = .565. Incontrast, the GHI group did not significantly improve itsperformance from pretest to posttest (M percentage pointdifference = 25.62, SD = 46.35), t(12) = 1.993, p = .069,and its performance on the maintenance test was similar toits posttest performance (M percentage point difference =−2.23, SD = 23.94), t(12) = −0.337, p = .742. Between-group effect sizes were .601 on the posttest and .656 on themaintenance test, all favoring the COMPS condition (seeTable 3).

To further examine whether the COMPS group improvedmore than the GHI group following the intervention, wealso used gain score as the measure to examine the groupdifference. With pretest–posttest gain score as the measure,

FIGURE 4. COMPS (red) and GHI (blue) groups’performance on the criterion tests across three times oftesting (color figure available online).


FIGURE 5. COMPS (red) and GHI (blue) groups’performance on two prealgebra measures, (a) solveequations and (b) model expression, during pretest andposttest (color figure available online).

results indicated a significant group difference favoring theCOMPS group, F(1, 25) = 4.950, p = .035; t(25) = 2.225,p = .035. The COMPS (gain score M = 0.5971, SD =0.0870) improved more than the GHI group (gain score M =0.2562, SD = 0.1285) with an effect size (ES) of 3.1067.

Effects on Prealgebra Concept and Skills and a StandardizedMeasure

Two (group) × 2 (time of testing: pretest and posttest)ANOVAs with repeated measures on time were performed toexamine the two groups’ performance on the prealgebra testsand the KeyMath-R/NU. For the prealgebra solve equationstest, the analysis was based on 26 students who completed thepre- and posttests. Results indicated a main effect for time,F(1, 24) = 105.714, p = .000. The effects for the group, F(1,24) = 0.219, p = .644, or interaction between group andtime, F(1, 24) = 0.668, p = .422, were not significant (seeFigure 5a).

For the prealgebra model expression test, the analysiswas based on 27 students who completed the pre- andposttests. Results indicated a main effect of time, F(1, 25)= 37.039, p = .000, but no effect of group, F(1, 25) =.125, p = .299. There was a significant interaction effectbetween group and time of testing, F(1, 25) = 8.459, p

FIGURE 6. COMPS (red) and GHI (blue) groups’performance on the far transfer measure(KeyMath-R/NU) during pretest and posttest (colorfigure available online).

= .008. A post hoc paired-samples test indicated that theCOMPS group significantly improved their performancefrom pretest to posttest (M percentage point difference =49.64, SD = 27.09), t(13) = 6.856, p = .000, whereas theGHI group did not significantly improve their performancefrom pretest to posttest (M percentage point difference =17.54, SD = 30.26), t(12) = 2.089, p = .059 (see Figure5b). Between-group effect size was .859 during the posttest,favoring the COMPS condition (see Table 3).

To further examine whether the COMPS group improvedmore than the GHI group on the prealgebra model ex-pression test following the intervention, we also used gainscore as the measure to examine the group difference. Withpretest–posttest gain score as the measure, results indicateda significant group difference favoring the COMPS group,t(25) = 2.908, p = .008. The students in the COMPS group(gain score M = 0.4964, SD = 0.2709) improved more thanthe students in the GHI group (gain score M = 0.1754, SD= 0.3026).

Finally, results on the standardized test (KeyMath-R/NU)indicated a main effect of time F(1, 25) = 23.062, p = .000,but no main effects of group, F(1, 25) = 0.175, p = .679, orinteraction between group and time, F(1, 25) = 0.49, p =.49 (see Figure 6) with an ES of 1.1178.

Discussion

The purpose of the present study was to evaluate and com-pare the effectiveness of COMPS and the GHI in teachingmultiplication and division word problems to elementarystudents with LP. Overall, the results indicated that theCOMPS group improved significantly more than the GHIgroup from pre- to posttest on the criterion tests following therespective intervention. Both groups of students maintainedtheir posttest performance 1 to 2 weeks following the termi-nation of the instruction. In addition, the results indicatedthat the COMPS group improved significantly more thanthe GHI group from pre- to posttest on the prealgebra model


expression test. Results also indicated that there was no sig-nificant difference between groups on their improvementfrom pre- to posttest on the far transfer measure, a norm-referenced diagnostic assessment in mathematics problemsolving.

Effect on the Criterion Tests

Results showed that the COMPS group improved sig-nificantly more than the GHI group from pre- to postteston the criterion word problem-solving tests. These findingssupport and extend previous research regarding the effec-tiveness of the COMPS instruction in solving arithmeticword problems (e.g., Xin, 2008; Xin, Wiles, & Lin, 2008). Itseems that elementary school students with LP benefit fromconceptual model expression of mathematical relationships,which served to drive the solution plan including selectionof operation.

Examination of students’ work during the pretests indi-cated that some students applied addition or subtractionacross the board. Some students seemed to apply an opera-tion based on the size of the number. That is, if a big numberwas given, then division might be used; if two small numberswere given, multiplication might be applied. A few studentswere able to solve rate times quantity problems (i.e., an EGproblem with product as the unknown); some students wereable to solve fair share problems (i.e., an EG problem withunit rate as the unknown). None of the students were ableto solve the MC problems with the reference unit as theunknown. Overall, it seemed that students relied on guessand check (according to the size of the numbers given in theproblems) in making decisions on the choice of the oper-ation. Clearly, there were no precise mathematical modelsguiding their problem solving. This observation is supportedby existing literature in mathematics education. That is,students might select an operation based on “syntactical”or “surface clues” (Greer, 1992, p. 285) to produce an an-swer for the solution. Building on findings from researcherssuch as Sowder (1988), Greer (1992) insightfully summa-rized possible problem-solving strategies used by students:

Look at the numbers; they will tell you which operations touse. Try all the operations and choose the most reasonableanswer. Look for key words or phrases to tell which operationto use. (p. 285)

In contrast, examination of students’ work during theposttest revealed that all students (except for two studentsin the posttest and three in the maintenance test) used alge-braic equations to express the relationship before solving forthe unknown (Figure 7 presents student sample work beforeand after the intervention). Two of the three students whodid not use the model expression before attempting to find asolution were reported constantly absent from the program.

The COMPS strategy emphasizes conceptual under-standing and expression of the mathematical relation inan equation. Focusing on relationships between elements

FIGURE 7. Student sample work before (a) and afterthe COMPS intervention (b and c).


of the problem (rather than key word or surface cluesfor operation selection and procedures for calculating theanswers) may not only improve students’ skills in arithmetic,but also provide them with the foundation for access toalgebra (Carpenter et al., 2005). The more individualspay attention to the meaning or mathematical relation,the closer the connection between arithmetic and algebrabecomes (Carpenter et al., 2005).

As the model expression in COMPS drives the selection ofan operation, the COMPS strategy does not merely instructthe learner in what to do, but helps the learner to under-stand why individuals do what they do. It facilitates efficientand generalizable problem-solving skills. That is, choice ofthe operation (multiplication or division) for solving var-ious multiplication and division word problems is derivedfrom a single generalized algebraic expression, factor × fac-tor = product. Specifically, for EG problem solving, when theproduct or total is the unknown, the EG model expression(i.e., unit rate × number of units = product or total) tells thatmultiplying the two factors (i.e., number of units and unitrate) gives the solution for the unknown product. When theunit rate or number of units is the unknown, the expression(i.e., number of units × unit rate = product or total) tellsthat dividing the product by one known factor may solvefor the unknown factor. For MC problem solving, when thecompared set or the product is the unknown, the MC modelexpression (i.e., referent unit × multiplier = compared orproduct) tells that multiplying two factors (referent unitand multiplier) solves for the compared quantity or prod-uct. When the referent unit or multiplier is the unknown,the model expression tells that dividing the product by theone known factor solves for the unknown. As such, thereis no ambiguity about the algorithm to use to solve forthe unknown. Students do not need to rely on key or cuewords to gamble on the operation or to remember solutionrules to figure out the choice of operation.

Effect on the Prealgebraic Tests

The results of this study showed that the COMPS groupimproved significantly more than the GHI group on theprealgebra model expression test. Because the COMPSstrategy emphasizes algebraic expression of multiplicativerelations in the multiplication and division word problems,it makes sense that students in the COMPS group improvedsignificantly more (i.e., a mean percentage point increase of49.64 from pretest to posttest) than the GHI group (i.e., amean percentage point increase of 17.54) on the prealgebramodel expression test following the intervention. Theresults of this study supported and extended existing studiesin teaching model expression (Xin, 2008; Xin, Jitendra,& Deatline-Buchman, 2005; Xin, Wiles, & Lin, 2008).Elementary students with LP can be expected to movebeyond concrete operations and to think symbolically oralgebraically. Algebraic conceptualization of mathematical

relations and problem solving can be taught through explicitand systematic strategy instruction.

Traditionally, teaching for understanding has seemed toinvolve concrete object manipulations or representationsthat are “away from symbolic formalisms” (Sherin, 2001,p. 524). However, the use of symbolic expressions can in-volve significant understanding because students’ mappingof information in the mathematical model (e.g., factor ×factor = product) is based on conceptual understandingof the three elements involved and the relations amongthem (i.e., unit rate × # of units = total or product as inEG problems; or unit × multiplier = product as in MCproblems). The COMPS strategy, which promotes algebraicexpression of mathematical relations in arithmetic word-problem solving, may facilitate algebra readiness as promotedby NCTM (2000) and the National Mathematics AdvisoryPanel (2008).

As for the solve equations test, as both groups used thefactor × factor = product model to set up the equationand solve for the unknown variable in the equation, it wasexpected that students in both groups would improve theirperformance from pretest to posttest on the solve equationstest. This might be an explanation for the nonsignificantgroup difference when an ANOVA was conducted on thesolve equations test.

Effect on the Far Transfer Measure

On the standardized diagnostic test (KeyMath-R/NU), al-though it appeared that the COMPS group improved morethan the GHI group from pretest to posttest, the improve-ment was relatively small and the difference between groupswas not significant. Careful examination of the problem-solving subtest of KeyMath-R/NU indicated that only four(22%) of 18 items were related to the problem types includedin this study (i.e., EG problem type). Most of the problemsin the KeyMath-R/NU problem-solving subtest are additionand subtraction. This might have contributed to the non-significant results obtained on this measure.

Limitations and Directions for Future Research

First, the present study may be limited because the twocomparison groups showed a slight difference in demo-graphic characteristics, although there were no statisticallysignificant differences on the pretest performance. For ex-ample, the students in the comparison condition (GHI) hadhigher percentile ranks in reading and mathematics achieve-ment based on their school record on norm-referenced as-sessments. No doubt, reading comprehension contributes tomathematical word problem-solving skills. This seemed tobe evidenced by participating students’ pretest scores acrossall measures (see Table 3). Nevertheless, following the in-tervention, the COMPS group outperformed the GHI groupon all measures. As indicated in the Method section, the


COMPS and the GHI students were accommodated whenthey experienced difficulties in reading the problems (i.e.,the problems were read to them). Future researchers mayneed to make an effort to control for participants’ read-ing ability, as this may serve as a confounding variable. Bycontrolling participants’ reading ability variable, the resultsof the COMPS condition may be even better than whatwas found in this study, as reading ability involves not onlydecoding skills (which was accommodated by reading theproblems to the students) but also comprehension skills.

Second, the treatment fidelity of the COMPS conditionwas only 86%, compared with 100% for the GHI condition.It is important that the treatment was implemented as in-tended. As the COMPS approach was new to all the instruc-tors, in few occasions, the instructors skipped an importantcomponent (“name the referent unit [benchmark] in the di-agram”) in their MC problem-solving instruction. It is rea-sonable to hypothesize that if the COMPS instruction werefully adhered to the intended curriculum, the performanceof the COMPS group would be even better than what wasfound in this study. Future researchers need to make sure thatthe instructors are trained to mastery to ensure the treatmentfidelity and therefore reduce possible confounding variables.

Third, the study was limited in that the GHI conditionwas allowed to use multiple strategies. One of the strate-gies occasionally used by the GHI was filling the number inthe equation. When this was the choice of the strategy, theinstructor also had to teach students how to solve for theunknown in the equation. This may explain finding no dif-ference on the measure of solve equations. Future researchersmay compare COMPS with only one strategy so that the dis-tinction between the conditions can be pinpointed.

Fourth, this study was also limited by having students solveproblems with clear EG and MC problem structure only. Fu-ture researchers should extend the problem pools to includemore complex real-world problems (problems with irrele-vant information or multiple steps) to facilitate skill transfer.Future researchers may also enhance the COMPS interven-tion program by including student problem posing based onthe model in addition to problem solving to promote thestudents’ construction of the concept of unit rate and num-ber of units. It was found that at the end of the present studythat a few students were still confused about the quantity ofunit rate and number of unites when representing the prob-lem in the model equation; however, it did not affect theirsolutions when they switched the two factors in the modelequation due to the commutative property of multiplication.

Collaborating with colleagues in mathematics education(Ron Tzur) and computer science (Luo Si), we are presentlydeveloping a computerized modeling system for nurturingstudents’ multiplicative reasoning (Xin, Tzur, & Si, 2008;supported by the National Science Foundation). Advancedcomputer science technology will be used to model students’thinking on the basis of what students know and provide thetasks that is within students’ zone of proximal development

(Vygotsky, 1978). Advanced computer science technologywill be used to facilitate not only representations of mathe-matical relations in a problem but also differentiated instruc-tion that is tailored to individual students’ learning profiles.

Implications for Practice

Multiplication is one of the most important concepts thatchildren develop progressively throughout their mathemat-ics education years (Harel & Confrey, 1994). A key differ-ence between additive and multiplicative reasoning is thatin the former, the same units are combined (12 apples + 3apples = 15 apples), whereas in the latter one unit is dis-tributed over a second unit to create a third unit (e.g., 12apples/basket × 3 baskets = 36 apples). One difficult part inteaching the multiplicative conceptual model is the conceptof unit rate (e.g., 12 apples/basket). To solve EG problems inthis study, students were guided to identify the three essen-tial quantities in an EG problem and to represent them in themodel equation (unit rate × # of units = total or product).However, the conception of unit rate was not as concrete orobvious as we had expected with the population included inthis study. Visual representation of the unit rate and num-ber of units pertinent to concrete problem situation wouldbe helpful for studies to establish the connection betweenthe concrete and the abstract concept. Instead of tellingstudents, “12 is the unit rate because each basket holds 12apples,” we used a PowerPoint animation in this study torepresent the concrete problem situation along with the ab-stract model equation so that students could make sense fromthe abstract model.

To conclude, students in the COMPS condition improvedsignificantly more than the GHI group following the in-tervention. The GHI group did not significantly improvetheir performance following the general heuristic approachin which three elements (i.e., factor, factor, and product)were not delineated in the conceptual model and the rela-tion among them was not made explicit. The present studysupported research in special education in that students withLP learn better when the strategy is explicitly taught andwhen conceptual understanding is the focus. With system-atic instruction, students with LP are able to think alge-braically through representation in the conceptual model,which serves to drive the solution plan for accurate problemsolving.

ACKNOWLEDGMENTS

This research was partially supported by Synergy Grant from College ofEducation at Purdue University and by Grant 0749462 from the NationalScience Foundation.

The authors would like to thank the administrators, teachers, and stu-dents at Edgelea and Miami Elementary Schools who facilitated this study.The authors also thank Casey Hord for proofreading the earlier draft of thisarticle.


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AUTHORS NOTE

Yan Ping Xin is an Associate Professor of Special Ed-ucation at Purdue University. Her current research inter-ests include effective instructional strategies in mathematicsproblem solving with students with learning disabilities anddifficulties, conceptual model-based problem solving, andalgebra readiness.


Dake Zhang is an Assistant Professor of Special Educa-tion at Clemson University. Her current research interestsinclude mathematics learning disabilities and scientific rea-soning of students with disabilities.

Joo Young Park is a PhD student in special education atPurdue University.

Kinsey Tom is a PhD student in special education at Pur-due University.

Amanda Whipple received a master’s degree in humanresource management from the Krannert School of Man-agement at Purdue University. She is currently working asa metrics and reporting analyst for Northrop Grumman anduses her talents in education to tutor at-risk students in LosAngeles.

Luo Si is an Assistant Professor of Computer Science atPurdue University.

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A Comparison of Two Mathematics Problem-Solving Strategies