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Pagliaro, C. M., & Ansell, E. (2012). Deaf and hard of hearing students’ problem-solving strategies with signedarithmetic story problems. American Annals of the Deaf 156(5), 438–458.

DEAF AND HARD OF HEARING STUDENTS’PROBLEM-SOLVING STRATEGIES WITH SIGNED

ARITHMETIC STORY PROBLEMS

Problem solving, defined here as thesolving of story problems or what havetraditionally been called word prob-lems, is a valuable mathematical activ-ity through which mathematics islearned and created, and by whichmathematics understanding can be ex-amined and measured. More than thesimple use of a procedure, problemsolving can be further defined as asense-making process whereby a syn-thesis of knowledge and procedures isused as a means to devise a meaning-ful interpretation of a problem situa-tion—the “story” in the story problem(Carpenter, Ansell, Franke, Fennema,& Weisbeck, 1993). Successful prob-lem solvers are inventive, practical,flexible, and reflective in their use of avariety of solution strategies (Baroody& Dowker, 2003; Franke & Carey,1997; Hegarty, Mayer, & Monk, 1995;

Schoenfeld, 1985). Problem solvingthus becomes a “window” throughwhich one can see not only what asolver knows about mathematics, butalso, through its application, the depthand quality of that knowledge.In the problem-solving process, the

solver must first understand the prob-lem situation. At the most basic level,this requires that the solver have fullaccess to the problem—that is, that theproblem be in a language and mode,and at a level of complexity, that matchthe skill and knowledge of the solver.The solver then devises a plan or solu-tion strategy based on what she or heknows about the situation and aboutrelated mathematical concepts andprocedures, and, finally, acts on thatplan, thinking logically about its out-come. Skilled problem solvers, how-ever, do more than simply translate the

HE USE OF problem-solving strategies by 59 deaf and hard of hearingchildren, grades K–3, was investigated. The children were asked to solve9 arithmetic story problems presented to them in American Sign Lan-guage. The researchers found that while the children used the samegeneral types of strategies that are used by hearing children (i.e., model-ing, counting, and fact-based strategies), they showed an overwhelminguse of counting strategies for all types of problems and at all ages. Thisdifference may have its roots in language or instruction (or in both),and calls attention to the need for conceptual rather than proceduralmathematics instruction for deaf and hard of hearing students.

CLAUDIA M. PAGLIARO AND

ELLEN ANSELL

PAGLIARO IS AN ASSOCIATE PROFESSOR,DEPARTMENT OF COUNSELING, EDUCATIONAL

PSYCHOLOGY, AND SPECIAL EDUCATION,MICHIGAN STATE UNIVERSITY, EAST LANSING.ANSELL IS AN ASSOCIATE PROFESSOR,DEPARTMENT OF INSTRUCTION AND LEARNING,UNIVERSITY OF PITTSBURGH, PITTSBURGH, PA.

T

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syntax of a problem (e.g., key words)directly into symbols and then operateon them—they analyze the semanticsof a problem situation and, as part ofthe solution process, may decide totransform the problem situation into aform in which it will be easier to solvebefore they represent it mathemati-cally. A conceptual rather than proce-dural understanding of mathematicsallows solvers to draw on all of theirknowledge and skills as they addressthe present situation. By contrast,those who have learned a set proce-dure that is attached (mentally or in-structionally) to a specific problemtype are stymied by any situation thatdoes not follow the learned structure.These solvers tend to be rigid and un-able to succeed at “true problem solv-ing” (Hegarty et al., 1995; Kelly, Lang,& Pagliaro, 2003; Reed, 1999).Researchers continue to find that

many children have not been giventhe opportunity to develop the con-ceptual understanding of the mathe-matics one needs to be a successfulproblem solver. Instead, many havebeen taught and rely on rote proce-dures for solving story problems thatare not based on sense making buton “surface-level analysis” (Kelly et al.,2003; Pagliaro, 1998b); such studentsscan the story problem for numbersand verbal cues (such as key words)that will “tell them what to do” (Garo-falo, 1992; Hegarty et al., 1995; Wiest,2003). Although these proceduresmay often result in a correct answer(and thus persist), they prevent thesolver from making sense of the prob-lem and from truly understanding therelationship between the situationand the mathematics that representsit (Parmar, Cawley, & Frazita, 1996).While these rote methods may pro-vide limited success in school mathe-matics, a grade of “A” on a test, forexample, they build a weak founda-tion for addressing more complex

problems (Baroody & Hume, 1991)and are devastatingly inadequate formeaningful problem solving outsidethis setting, that is, in the real world.Research on children’s problem

solving has shown that most (hearing)children in the early elementary grades(i.e., K–3) attend to the semantics of astory problem (the actions and rela-tions depicted in the story situation)and use “intuitive analytic modelingskills” (Carpenter et al., 1993; De Corte& Verschaffel, 1987) to solve them.That is, young children tend to engagein mathematics in the same way theyapproach the world in general at thisage. As stated by the National Councilof Teachers of Mathematics (2000),“Problem solving is natural to youngchildren because the world is new tothem, and they exhibit curiosity, intelli-gence, and flexibility as they face newsituations” (p. 116). However, researchalso has shown that as they get older,children seem to rely more on roteprocedures for solving story problems(Garofalo, 1992).

Problem Types and Solution StrategiesEvidence for the sense-making aspectof children’s problem solving, and thetheoretical basis for the present inves-tigation, is rooted in studies that havelooked at the relationship betweenstory problem types and the strategiesyoung children use to solve them(Carpenter & Moser, 1984; De Corte& Verschaffel, 1987). In particular,when arithmetic story problems (i.e.,those involving addition, subtraction,multiplication, and division) are cate-gorized on the basis of their semanticstructure, there are matches betweenthese structures and how young chil-dren model the problem situations asthey solve the problems. Describedbelow is the problem type–solutionstrategy framework that has beenused to describe and explain young

(hearing) children’s strategy develop-ment (Carpenter, 1985; Carpenter &Moser, 1984), and that serves as a toolfor teachers to understand and suc-cessfully build on their students’ think-ing (Carpenter, Fennema, Franke, Levi,& Empson, 1999; Carpenter, Fennema,Peterson, Chiang, & Loef, 1989; Fen-nema et al., 1996).

Problem TypesResearchers in mathematics educa-tion agree in general on a frameworkof arithmetic story problem types thattarget the semantic structure of theproblems, not the operations typicallyused to solve them. Table 1 shows aclassification of typical story problemsused in the early elementary grades.The classification is based on theproblems’ semantic structure of thesituation (addition and subtraction asjoining, separating, comparing, andpart-whole; multiplication and divi-sion as grouping, partitioning, andmeasuring) and the position of theunknown value. There are a numberof similar iterations of this scheme(e.g., Carpenter et al., 1999; Kintsch &Greeno, 1985). Though there are mi-nor differences between the iterations(some have more categories), theyshare the critical dimensions of the se-mantic distinctions and the unknownquantity. For traditionally labeled ad-dition and subtraction problems, theframework distinguishes the prob-lems first by whether or not an actionis involved in the situation. In actionproblems, there is some change inquantity (through either joining orseparating) over time. In static or no-action problems, however, all the in-formation is available at the start andno change occurs; the problem is pre-sented more as a moment in timethan as an event taking place overtime. The second level of distinction isthe placement of the unknown. In anaction problem, the unknown can be

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the result, the change, or the startquantity. In Part-Whole problems,which are static problems, the un-known can be placed as the whole orone of the parts, while in Compareproblems, the unknown is either thedifference between the sets, the com-pared set, or the referent.In traditional multiplication and di-

vision problems, the framework ex-tends into “grouping,” in which thetotal number altogether is unknown(traditional multiplication), “measure-ment division,” in which the numberof groups is unknown, and “partitivedivision,” in which the number ofitems in each group is unknown (Car-penter et al., 1999).

Studies of young (hearing) chil-dren’s solutions to such problems,when these children are given accessverbally, show that successful problemsolvers use strategies that reflect thestructure of the problem situations(the stories), not necessarily the oper-ations or procedures that adults andolder children may use to find an an-swer. For example, while an adult maysolve the Join Change Unknown prob-lem in Table 1 with a subtraction oper-ation (11 – 3 = ?), a young (hearing)child, paying attention to the joiningor additive nature of the story (Angiegetting more crayons than she alreadyhas), might solve the problem as 3 + ?= 11 (adding on to 3 and stopping at

11). Although this may seem compli-cated and less efficient to an adultwho might have a more mature un-derstanding of mathematical opera-tions, the young problem solver isusing a solution that logically followsthe storyline, and would be confusedby the subtractive operation, as it“does not make sense.”The classification framework also

reflects the relative difficulty of theproblem types. In general, problemdifficulty increases across the table(moving from Result Unknown, toChange/Part Unknown, to Start Un-known), and down, with action prob-lems tending to be easier thanno-action problems. The level of infer-

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Operation Problem type Unknown placement

Table 1

Problem Type Framework

Addition andsubtraction

ACTION

NO ACTION

Join

Separate

Part-whole

Compare

Grouping

ResultAngie has 3 crayons. Jadagives her 8 more crayons.How many crayons doesAngie have now?

Result

Angie has 11 crayons. Shegives 3 to Jada. How manycrayons does Angie havenow?

WholeAngie has 3 red crayonsand 8 blue crayons. Howmany crayons does shehave?

DifferenceAngie has 8 crayons. Jadahas 3 crayons. How manymore crayons does Angiehave than Jada?

Multiplication (total)

Angie has 3 packages ofcrayons. Each packagehas 6 crayons. How manycrayons does Angie have?

ChangeAngie has 3 crayons. Jadagives her some morecrayons. Now Angie has 11crayons. How many crayonsdid Jada give her?

ChangeAngie has 11 crayons. Shegives some to Jada. NowAngie has 8 crayons. Howmany crayons did Angiegive to Jada?

Part

Compare quantityJada has 3 crayons. Angiehas 5 more crayons thanJada. How many crayonsdoes Angie have?

Partitive division (numberin each group)Angie has 18 crayons. She wants to put them into3 packages with the samenumber in each package.How many crayons will bein each package?

StartAngie has some crayons.Jada gives her 8 morecrayons. Now Angie has 11crayons. How many crayonsdid Angie have to start?

StartAngie has some crayons.She gives 3 to Jada. NowAngie has 8 crayons. Howmany crayons did Angiehave to start?

ReferentAngie has 8 crayons. Shehas 5 more than Jada. Howmany crayons does Jadahave?

Measurement division(number of groups)Angie has 18 crayons. She can put 6 crayons ineach package. How manypackages can she make?

Angie has 11 crayons. Three are red and the rest are blue.How many blue crayons does Angie have?

Multiplication and division

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ence involved in solving the problemaffects its relative difficulty at both lev-els of distinction (Carpenter et al.,1993; De Corte & Verschaffel, 1987;LeBlanc & Weber-Russell, 1996). Forexample, a problem can be written sothat the action or relation is more orless explicit, or comparisons can bemore or less linked to setting up aone-to-one correspondence to solvethe problems. Compare the followingproblem to the Compare DifferenceUnknown problem in Table 1: Thereare 8 birds and 3 worms. How manybirds will not get a worm? This prob-lem encourages the solver to use amatching solution, while the idea ofmatching to solve the problem inTable 1 is less explicit and thus re-quires the solver to infer or make useof information that is implied.

Solution StrategiesStudies in mathematics education ofelementary school students show thatwhen given the choice, children make

use of three broad strategy types tosolve arithmetic story problems suchas those in Table 1 (Carpenter et al.,1999): (a) modeling, (b) counting,and (c) fact based. Each of thesestrategies is briefly described andcontrasted below in regard to the im-plications for the types of problemschildren who use these strategiestend to be able to solve, and are exem-plified in Table 2. A child who uses amodeling strategy would representeach quantity in the problem sequen-tially and act on the quantities accord-ing to the action or relation depictedin the problem, keeping track of onlyone quantity at a time. Because of thissequential approach, children who di-rectly model problems are generallyunable to solve problems with the“start” unknown, and may find prob-lems with no action (e.g., Compareproblems) more challenging, as theproblem situation itself does not ex-plicitly indicate a solution action.In contrast to modeling, in which

all the quantities and actions are repre-sented, a child using a counting strat-egy would use one quantity within theproblem as a starting point withoutphysically representing it, and use theother quantity only to mark or keeptrack of the action taken on the firstquantity. Counting strategies requiresome form of double counting to si-multaneously keep track of how manycounts have been made and when tostop the counting sequence. For exam-ple, in the counting strategy describedfor the Separate Result Unknown prob-lem presented in Table 2, the solvermaintains a backward counting se-quence (10, 9, 8) while also conductinga forward counting sequence to keeptrack of the number of counts made(1, 2, 3) either mentally or by meansof some physical marker (fingers ormanipulatives). The use of countingstrategies also involves a conception ofnumber as parts and wholes. This con-ception supports a view of operationsas inverses, so that, for example, a sep-

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Problem

Table 2

Exemplification of Strategy Types

Modeling

Counting

Fact: Recall

Fact: Derived

Join Change UnknownAngie has 3 crayons. Jada gives her some more crayons.Now Angie has 11 crayons. How many crayons did Jadagive her?

The child counts out 3 cubes. S/he continues to add 1more cube, continuing the count (“4, 5, 6 . . .”) until s/hereaches 11. S/he then counts the number of cubes s/hehas added and answers “8.”

The child starts with the number 3 as the beginning ofher/his counting sequence and continues to count untilthe number 11 is reached, keeping track of how manycounts s/he has made (e.g., pulling out one cube foreach count or making a tally mark for each count). Thechild counts the number of cubes/marks and answers “8.”

The child answers “8” because s/he knows the fact that 3 + 8 = 11.

The child answers “8” because s/he knows that 3 + 7 = 10and 1 more makes 11, and that 7 + 1 = 8.

Separate Result UnknownAngie has 11 crayons. She gives 3 to Jada. How manycrayons does Angie have now?

The child counts out 11 cubes, then from that removes 3cubes. S/he counts the remaining cubes and answers “8.”

The child, starting with 11, marks 3 counts backward inthe counting sequence (“10, 9, 8”). S/he answers “8,” thelast number in the sequence.

The child answers “8” because s/he knows the fact that11 – 3 = 8.

The child answers “8’ because s/he knows the fact that10 – 3 = 7, and deduces then that 11 – 3 = 1 more than7, therefore, 8.

Strategy type

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arating problem may be solved bycounting up as well as by countingdown. An understanding of the inverserelationship of the addition and sub-traction operations would then sup-port reversing the action depicted in problems and thus allow one tomore easily solve start unknown prob-lems. Thus, children who use countingstrategies show more sophisticatedcounting skills and a conceptual knowl-edge of number.There are two types of fact-based

strategies that children use, recall,where the fact is known, and derived,where the needed fact is indirectly de-duced from the recall of other knownfacts. Derived facts are often based ona knowledge of doubles (e.g., “I knowthat 5 plus 5 equals 10, and 1 moreequals 11, so 5 plus 6 equals 11”) orfacts related to 10 (e.g., “I know that 8plus 3 equals 11 because I know that 8plus 2 equals 10 and 1 more equals11”). Children learn (not memorize)facts through problem solving suchthat a child who uses modeling orcounting may, depending on the num-ber combination in a given problem,use a fact-based strategy. Eventually,children will predominantly use fact-based strategies. Their ability to do so,however, depends on their expandedunderstanding of number relation-ships (Carpenter et al., 1999). Thesestrategies, then, can be thought of asdevelopmental, with some “variabil-ity in the ages at which children” usethem (Carpenter et al., 1999, p. 109).Studies of young hearing childrenshow that they generally move frommodeling as they enter and movethrough kindergarten and first gradeto counting during first grade, and fi-nally to fact-based strategies, althoughnot in such a clean and linear fashion.Children may use both modeling andcounting strategies for some time (ifallowed), often depending on the

problem and its action. A few knownbasic facts such as 1 + 1 = 2 or 5 + 5= 10 may be used early, but the con-sistent use of recall and derived factstrategies develops over time (Car-penter et al., 1993, 1999).As described above, these three

strategy types differ in the countingskill and number knowledge that sup-port their use. The use of countingstrategies requires more sophisticatedcounting skills than the counting re-quired when children model prob-lems. The strategy types can also bedescribed as moving from less to moreabstract, with modeling being the leastabstract and fact based being the most.In general, studies show that younghearing children tend to use less ab-stract strategies before more abstractstrategies, depending on their under-standing of the given problem, theirunderlying conceptual knowledge ofthe mathematics entailed in the strat-egy, their related comfort or confi-dence in using the strategy itself, andthe efficiency of a particular strategy(Carpenter et al., 1999).

Research on ProblemSolving and Strategy UseAmong Deaf and Hard ofHearing StudentsStudies of similarly aged deaf andhard of hearing students’ problemsolving have shown similar strategytype use, though occurring at laterages. This suggests the possibility ofa delay in these students’ problem-solving performance (Chien, 1993;Frostad, 1999; Secada, 1984). Resultsof these studies, however, were basedon presented stimuli that were eitherstripped of context (i.e., computationproblems) or presented in contextsthat were potentially limiting. Prob-lems given in picture form, for exam-ple, cannot completely illustrate thenature of the problem situation (e.g.,

showing action), and therefore mustassume a shared experience or accu-rate inference.The majority of studies involving

deaf and hard of hearing studentssolving problems “in context” havepresented the problems in writtenlanguage (Kidd & Lamb, 1993; Kidd,Madsen, & Lamb, 1993; Serrano Pau,1995). Not surprisingly, given thegreat difficulties deaf and hard ofhearing students have with under-standing the written mode of lan-guages such as English (Allen, 1995;Traxler, 2000), these studies foundthat the problem-solving perform-ance of deaf and hard of hearing stu-dents was significantly behind that ofhearing students, with a high positivecorrelation between success in prob-lem solving and reading proficiency(Kelly & Gaustad, 2007; Kelly, Lang,Mousley, & Davis, 2003; Kelly & Mous-ley, 2001; Serrano Pau, 1995). Subse-quent research, then, has focusedprimarily on English-language issuesand reading strategies, not on themathematical processes deaf and hardof hearing students use when solvingstory problems.An investigation by Frostad and

Ahlberg (1999) of contextual problemsolving (presented via an animatedcomputer game) by Norwegian deafand hard of hearing children, ages6–10 years, revealed three ways inwhich the children addressed thistask: (a) by focusing solely on thenumbers in the problem, (b) by ap-proaching the problem as a “takeaway” circumstance, and (c) by ap-proaching the problem as a part-whole association. While the lattertwo ways are conceptually orientedand logically appropriate, with thechildren focusing on the situation andthe relationship of the quantitieswithin that situation, the first is notconceptually based, and is therefore

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less effective and less reliable. Frostadand Ahlberg described this approachas one in which the children “simplypicked the available numbers andcombined them rather coincidentallyaccording to well-known procedures”(p. 288). Here, the children were notmaking use of their understanding ofthe problem situation or of the con-cepts of mathematics. The proceduresand strategies they employed werenot used purposefully as conceptuallylinked tools, but, rather, often indis-criminately, as hopeful solutions toproblems.Investigations of deaf and hard of

hearing children’s solution strategieswhen they are presented with compu-tation problems (e.g., 6 + 3 = ?) insign language (Chien, 1993; Frostad,1999; Secada, 1984) suggest that theproduction of the counting string insign may influence the children’s abil-ities with counting and number. Inmany signed languages, the countingsequence is a series of systematicrules that follow a simple pattern to acertain point of change. In AmericanSign Language (ASL), for example, thecounting sequence for numbers 1–5 issuch that a finger is “added” to a par-ticular handshape: the number 1 being signed with the index finger ex-tended only; 2 having the index fingerand the middle finger extended; 3having the index and middle fingers,as well as the thumb, extended; 4 hav-ing all fingers except the thumb ex-tended; and 5 adding the thumb sothat all fingers are extended, all withpalm facing in. At the sign for 6, how-ever, the base handshape completelychanges, with the palm facing out andthe thumb touching the pinky finger.The series 7–9 basically starts again,with 7 signed palm out but with thethumb touching the ring finger, 8 withthe thumb touching the middle fin-ger, etc. The base handshape then

changes again at 10, with the thumbonly out and shaken slightly. Othersigned languages present numberssimilarly, though the major changesmay take place at different terms(Frostad, 1999; Leybaert & Van Cut-sem, 2002; Nunes & Moreno, 1998). Inaddition, most signed languages havesome level of embedded cardinality intheir number signs. In ASL, for exam-ple, the signs for numbers 1–5 inher-ently show the number of elementswithin the set being represented. Thesign for 2 has two fingers raised; thesign for 5 has five fingers raised. It isthis ease of moving one finger over, orraising an additional finger for eachnumber, and this inherent cardinality,that researchers have found to possi-bly influence counting, and thereby,problem solving.Secada (1984) found that while

deaf and hard of hearing children be-tween the ages of 3 and 8 years whowere native signers of ASL were morelikely than their hearing peers to cor-rectly state the next number in acounting sequence, they were lesslikely to correctly perform a rotecounting exercise, because they mis-constructed the signs (i.e., had diffi-culty physically forming them) or didnot (yet) understand the productionrules. Likewise, Leybaert and Van Cut-sem (2002) found that deaf and hardof hearing 3-to-7-year-old signers ofBelgium French Sign Language weremore accurate than matched hearingchildren in counting objects and cre-ating sets, but showed significant de-lays in sign-counting production, witherrors mostly occurring at the pointof major change. Nunes and Moreno(1998) found similar confusion withthe British Sign Language countingsystem among young deaf and hard ofhearing children as these children at-tempted to use a signed algorithm(double counting) to solve mathemat-

ical problems. Frostad (1999), how-ever, suggests that the counting systemin sign language (in this case, Norwe-gian Sign Language) could have a pos-itive impact on deaf and hard ofhearing children’s success in mathe-matics problem solving, acting as an“extra ‘aid’ to the production system”and affording these children a “richerrepertoire of strategies” (p. 146).Specifically, Frostad points to the cardi-nality found in the signing of lowernumbers such as 1–5. He does caution,however, that the ease and efficiencyof counting in sign language may facili-tate a procedural understanding, as op-posed to the more desirable and usefulconceptual knowledge of mathematicsconcepts and skills, if not properly sup-ported with a true understanding ofnumber. For a more complete discus-sion of language and mathematics, seePagliaro (2010).No known study to date has fo-

cused on the problem-solving strate-gies or strategy development of deafand hard of hearing students whenthey are given full, linguistic access tocontextually based arithmetic storyproblems. In light of the importanceof problem solving in the mathemat-ics curriculum and the historicallypoor achievement of deaf and hard ofhearing students in mathematics,specifically in problem solving (Allen,1985; Traxler, 2000), the present studyinvestigated the mathematics prob-lem-solving success and strategy useof primary-level (grades K–3) deaf andhard of hearing students when pre-sented with arithmetic (addition, sub-traction, multiplication, and division)story problems in ASL. The study wasspecifically designed to parallel thestudies by Carpenter and colleaguesoutlined above (Carpenter, 1985; Car-penter et al., 1993, 1999, 1989; Car-penter & Mosher, 1984) in sampleage, procedure, and content so as to

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structure a comparison of sorts be-tween results. Four research questionsguided the study:

1. What strategies do primary-leveldeaf and hard of hearing stu-dents who are ASL signers use tosolve arithmetic story problemswhen these problems are pre-sented in ASL?

2. What is the progression ofstrategy use exhibited amongprimary-level deaf and hard ofhearing students who are ASLsigners across grade levels whenthese students are given arith-metic story problems in ASL?

3. How does strategy use differ be-tween more and less successfulprimary-level deaf and hard ofhearing students who are ASLsigners when these students aregiven arithmetic story problemsin ASL?

4. How does the strategy use ofprimary-level deaf and hard ofhearing students who are ASLsigners differ from the strategyuse found among hearing stu-dents in parallel studies?

MethodologyTo answer the research questions, wedesigned a descriptive study usingstructured interviews for data collec-tion, with frequency counts and someparametric statistics for data analysis.

SampleA total of 232 K–3 children from 9schools for the deaf and hard of hear-ing across the United States partici-pated in the present study. Theschools were selected to maximizethe number of children who wouldlikely have good comprehension ofASL. An initial list of schools were cho-sen from the reference issue of theAmerican Annals of the Deaf on thebasis of their use of ASL as a primary

language of communication and theirranking by number of students ingrades K–3. A letter of interest wassent to the top 22 schools on this list.Of these, 16 (73%) expressed interestin the study. Visits were arranged to 14of the 16 interested schools. The pur-pose of the visits was twofold: (a) forthe researchers to observe and infor-mally evaluate student sign commu-nication skills so as to maximize thenumber of student participants, and(b) for the schools’ teachers and administrators to understand the re-sponsibilities of participation, includ-ing scheduling. Based on informationgathered during these visits, 9 schoolsbest met our criteria for inclusion inthe sample with respect to communi-cation, diversity (e.g., geographically,racially), and the presence of a signifi-cant number of students in gradesK–3 (a total of 260 students), andwere thus invited to participate in thestudy. These 9 schools were situatedin both urban and rural settings in thenortheastern, middle Atlantic, mid-western, and southwestern regions ofthe United States.After parent consent was obtained,

all participating students were admin-istered three receptivity tasks fromthe American Sign Language Assess-ment Instrument (ASLAI; Hoffmeister,1999). At the time of the presentstudy, there were no standardized in-struments to formally measure devel-opment of ASL parallel to thoseinstruments for spoken and writtenEnglish. The ASLAI, however, is such ameasure under development at theCenter for the Study of Communica-tion and the Deaf at Boston University.Consisting of subtests that assess age-related receptive and expressive skillsin ASL, the ASLAI has been used inseveral large-scale research studies, forexample, Schick, J. deVilliers, P. deVil-liers, and Hoffmeister (2002), and P.deVilliers, Blackwell, and Hoffmeister

(1988) (as cited in Hoffmeister, 2009).The three receptive subtests of theASLAI used in the present study wereSynonyms, Antonyms, and Plurals andArrangement. These subtests eachdemonstrate at least “acceptable” in-ternal consistency ratings. Their co -efficient alphas are Synonyms .87,Antonyms .81, Plurals and Arrange-ment .70. Their Guttman split-half coefficients are Synonyms .87, An -tonyms .81, Plurals and Arrangement.69 (Hoffmeister, 2009).Based on the results of these sub-

tests, data on 59 of the 232 childrenwho participated in the study were se-lected for analysis. These children hadscored at or above the mean for deafchildren of deaf parents (i.e., nativesigners) in each of their respective agegroups (ages 5, 6, 7, 8, and 9 years) inthe larger sample. Thus, the 59 chil-dren were at or above the level of theaverage native signer of their age inASL comprehension in the presentsample. The children ranged in agefrom 5 to 9 years (n = 11, 14, 15, 15,and 4, respectively), with a mean ageof 7.4 years, and spanned grade levelskindergarten through three (n = 9,16, 18, and 16 respectively). (Chil-dren’s mathematics levels as indicatedby their teachers were highly corre-lated with grade level, p< .000; there-fore, grade level was used for allanalyses.) The subsample was splitnearly equally between male and fe-male (29 and 30, respectively) andspanned the four hearing-loss levels(mild: 3, moderate: 5, severe: 18, pro-found: 33). Thirty-four children had atleast one deaf parent, and 21 had nodeaf parents; parental hearing statusfor 4 children was not known (2) ornot reported (2). Seven schools wererepresented in the subsample; no stu-dents from two of the nine schools inthe full sample achieved the ASL recep-tivity score necessary to be included inthe present analysis.

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ProcedureChildren’s problem-solving strategieswere elicited in a one-on-one inter-view conducted by a Deaf adult profi-cient in ASL. During the interview, thechildren were shown as many as nineaddition, subtraction, multiplication,and division story problems. Theproblems were representative of typi-cal story problems that children intheir age group are exposed to inschool. The order of the problemsprovided for what we thought (basedon studies of hearing children) wouldbe an increase in difficulty while stillgiving the children some successalong the way. Given the age levelsand relative attention span of oursample, as well as what we knew fromthe studies of hearing children, we feltthat these nine problem types pro-vided the best access to mathematicsthinking while also giving us a broadspectrum from which to see problemdifficulty.The problems were presented by a

Deaf signer in ASL on videotape. Thewritten-English translations of thenine problems along with an Englishgloss of their ASL versions (see Baker-Shenk & Cokely, 1991, for gloss key)are shown in Figure 1 (numbers inbrackets refer to Set B, as explainedbelow). The problems are comparableto those used in the studies by Car-penter and colleagues referencedabove involving hearing children (Car-penter et al., 1993; Carpenter & Moser,1984). A panel of four Deaf, nativesigners of ASL who had experiencewith children in schools translated theproblems from written English to ASLin such a way that three factors weretrue of the signed versions: (a) Theywould be appropriate for primary-levelchildren; (b) they would maintain themathematical structure of the originalproblems; (c) they would follow thelinguistic rules of ASL. Attention to themathematical structure of the original

problem in translation is critical giventhe well-defined set of problem typesdescribed in the mathematics educa-tion literature and results from a studyby Ansell and Pagliaro (2001), whichshowed possible changes in problemtype depending on how a story prob-lem was signed. An interpreter certi-fied by the Registry of Interpreters forthe Deaf translated the signed prob-lems back to English, confirming thetranslations. Two versions of eachproblem that differed only in the num-ber combination used were recorded,Set A having numbers 1–20 and Set Bhaving numbers 1–10. So that none ofthe children would be encumbered bynumbers that were out of their count-ing range, each child was assigned ei-ther Set A or Set B depending on thatchild’s knowledge of number as as-sessed through a series of relatedtasks. Rote counting skill was assessedby first having the children count “ashigh as they could” beginning with 1,then by having them continue tocount upward from 14. Cardinal un-derstanding was assessed by havingthe children count a set of eight cubeswhich were subsequently coveredwith a sheet of paper. The childrenwere then asked how many cubeswere hidden. If a child could notcount up to 20 correctly and effi-ciently, or could not correctly statethat there were eight cubes hidden,that child was given Set B. All childrenwere at least able to rote count up to10, and thus all were assigned to ei-ther Set A (n = 32) or Set B (n = 27).Upon completion of the number

knowledge tasks, the problem-solvinginterview was administered. The in-terviewer explained this task as a series of stories on videotape thatcontained a question for the child toanswer. Throughout, the interviewerperiodically reminded the child thatshe or he could use the available ma-terials (counting cubes of two differ-

ent colors, paper and markers), theirfingers, or “just thinking” in order toanswer the questions. The child couldwatch the videotape of the signedstory as many times as she or hewanted. The interviewer did not par-ticipate in any way in the child’s prob-lem solving. If the child could notremember a detail of the story, the in-terviewer asked the child if she or he wanted to see the video again. Asthe child solved each story problem,the interviewer marked the child’sstrategy choice on a coding sheet. If achild’s solution strategy was not obvi-ous to the interviewer, the child wasasked to explain his or her actions.The same response by the interviewerwas given regardless of whether thechild’s response was correct or incor-rect. The interviews ranged in lengthfrom 20 to 45 minutes and werevideotaped.

Coding and AnalysisSolutions to each of the nine prob-lems were coded for correctness andstrategy use. Coding of the strategiesincluded general type (modeling,counting, fact based), specific strategywithin the types, correctness, and via-bility. A viable strategy is one that isappropriate to solve a specific prob-lem and is free of systematic error,such as those described in Table 2.(An example of a nonviable strategyfor the Separate Result Unknownproblem in Table 2 would be if a childadded the two shown quantities.) If achild changed strategies while solv-ing a particular problem, the last strat-egy used was coded so as to matchthe solution strategy with the finalanswer. Codings were originally doneby the interviewer; we subsequentlywatched the interview videotapes tocheck and, as necessary, change thecodings. Because there was a verystrong correlation in the data betweenstrategy viability and correct answer

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Figure 1

Problems in English and English Gloss, in Order Given

Problem 1: Join Result Unknown (JRU)

English:

Aaron had 3 [2] cars. Jumal gave him 8 [6] more cars. How many cars does Aaron have

altogether?

Gloss: ___t____ ____t____________________ INDEX-lf BOY NAME A-A-R-O-N INDEX-lf HAVE THREE CAR THREE-lf- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - __t________________________________ INDEX-rt BOY J-U-M-A-L INDEX-rt HAVE EIGHT CAR EIGHT - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - _____puff ck____________ INDEX-rt rt-GIVE-TO-lf rt-EIGHT-lf ____________________whq________________________________ ALTOGETHER-lf HOW-MANY CAR INDEX-lf HOW-MANY

Problem 2: Separate Result Unknown (SRU)

English:

There were 11 [6] children on the playground. 7 [4] children went home. How many children

were still on the playground?

Gloss: ___t_______ ____________rh-q_____________ ______nod______________ YOU KNOW SCHOOL OUTSIDE PLAY 5↓-CL-ctr ‘area’ THAT-ctr 5 -CL-ctr ‘area’ _______mm/puff ck________ ELEVEN CHILDREN PLAY (2h)5 -CLwg-(ctr) ‘mixed play’ SEVEN GO-AWAY HOME ______________________________________________________________ whq__________ (2h)HOW-MANY CHILDREN STILL (2h)5 -CLwg-(ctr) ‘mixed play’ (2h)HOW-MANY

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Figure 1

(Continued)

Problem 3: Part Unknown (PU)

English:

Megan has 13 [9] balloons. 8 [6] are red and the rest are blue. How many blue balloons does

Megan have?

Gloss: GIRL INDEX-lf NAME M-E-G-A-N INDEX-lf HAVE THIRTEEN BALLOON+ THIRTEEN S-CL-lf “grab and hold strings of balloons; look up”----------------------------------------------------------------------------- _nod_ INDEX-sweep-lf ‘up at balloons’ SOME RED SOME BLUE -------------------------------------------------------------------------- _t__ INDEX++ “counting balloons in air; look up” RED EIGHT ------------------------------------------------------------------------------------------- ____________whq____________________________________________ HOW-MANY BLUE INDEX-sweep-lf ‘up at balloons’ HOW-MANY

Problem 4: Whole Unknown (WU)

English:

There are 4 [2] girls and 9 [7] boys playing soccer. How many children are playing soccer?

Gloss: _____________t_________ _________mm________ CHILDREN PLAY SOCCER (2h)5-CL wg-ctr ‘mingle’ __t__ GIRL FOUR-lf/ctr - - - - - - - - -- - - - - - - - - - - - - __t__ ___nod___ BOY NINE-rt/ctr ________________________whq_______________________ HOW-MANY CHILDREN PLAY SOCCER HOW-MANY

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Figure 1

(Continued)

Problem 5: Compare Difference Unknown (CDU)

English:

Rachel built a tower 8 [4] blocks high. Pat built a tower 14 [7] blocks high. How much higher is

Pat’s tower than Rachel’s?

Gloss: _t__ ___t___ ___________mm____________ GIRL NAME R-A-C-H-E-L INDEX-rt PLAY (2h)RECT-CL++ “blocks” [(2h)alt.o→-CL-rt “build tower” ______mm_________ (2h)B→-CL-rt “tower”

EIGHT-rt _t__ ______prsd lips____ INDEX-lf GIRL NAME P-A-T (2h)alt.o→-CL-lf “build tower” (2h)B→-CL-lf “tower” FOURTEEN-lf --- ----------------- EIGHT-rt lower B:-CL-rt lower ------------------------------------------------------------------------------ __________________________whq______________

B:-CL-lf higher HOW-MANY higher-1→ -CL-lower HOW-MANY Problem 6: Join Change Unknown (JCU)

English:

Bob wants 15 [8] worms. He has found 9 [5] already. How many more worms does he need to

find?

Gloss:

___t__ ___nod__ BOY NAME B-O-B INDEX-lf WANT* WORM FIFTEEN-lf higher--------------------------------------------------- ------------------------------------------------- ______t_____ FIND FINISH NINE-rt lower----------- -------------------------------------------------------------- FIFTEEN-lf higher---------------------- _______________________________________________whq________________________________ HOW-MANY NEED B:CL wg “move up to 15” EQUAL HOW-MANY

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Figure 1

(Continued)

Problem 7: Multiplication (MLT)

English:

Kelly has 3 [3] bags of candy. There are 4 [2] candies in each bag. How many candies does

Kelly have?

Gloss: __t_ GIRL NAME K-E-L-L-Y HAVE THREE (2h)C-CL ‘bag’ (2h)C-CL ‘bag’++-rt-ctr-lf __t___ CANDY FOUR++-rt-ctr-lf ______________________________whq____________________ ALTOGETHER (2h)HOW-MANY CANDY (2h)HOW-MANY Problem 8: Partitive Division (Share)

English:

Jake had 12 [6] cookies to sell. He put the cookies into 4 [2] bags with the same number of

cookies in each bag. How many cookies were there in each bag?

Gloss: __t_ __t____ _rhq_ BOY NAME J-A-K-E INDEX-lf MAKE TWELVE COOKIE FOR SELL TWELVE-lf

___t_ _____t____ _____mm_____________________ BAG COOKIE+-lf PUT+++-rt “into each finger of 4-HS” FOUR-rt-------------------------------------------------------------------------------------------------------- UNDERSTAND MUST SAME-rt “across fingers of 4-HS” --------------------------------------------------------------------------- _______________________________________whq_____________ HOW-MANY EACH+++-rt “onto fingers of 4-HS” HOW-MANY -------------------------------------------------------------------------------------

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(r = .935, p < .01), and because ourfocus is on strategy use, success is de-scribed here by the viability of thestrategies employed by the children.The use of viability provides a moreaccurate indication of success withthe problems, as an error in counting,for example, would not count againsta successful strategy. Level of successis thus described by the percentage ofparticipants across and within gradelevel and problem type who used vi-able strategies and, within viable

strategies, with respect to the threegeneral strategy types (modeling,counting, and fact based). The relativefrequency of the different strategytypes across problem type was alsocompared across grade level so thatwe might investigate the progressionor development of strategy type use.Finally, the relative frequency of strat-egy types was determined for moreand less successful problem solvers.Based on the mean number of viablestrategies used (4.4), the sample was

further split into two groups—thosewho were relatively more successful atsolving the problems (i.e., those whohad a viable strategy on more thanfour of the problems, n = 29) andthose who were relatively less suc-cessful (i.e., those who had a viablestrategy on four or fewer of the prob-lems, n = 30). Both groups includedchildren from all grade levels K–3, andwith both deaf and hearing parents.(See Table 3 for demographic infor-mation by grade level.)

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At least one parent Deaf Gender Student age (years) Hearing loss

Grade Deaf or hard

level Deaf only of hearing F M 5 6 7 8 9 Mild Moderate Severe Profound

K 4 4 4 5 9 — — — — — 1 3 5

1 9 10 4 12 2 12 2 — — — 2 7 7

2 12 12 14 4 — 2 10 6 — 2 — 3 13

3 8 11 8 8 — — 3 9 4 1 2 5 8

Total 33 37 30 29 11 14 15 15 4 3 5 18 33

Table 3

Demographics of the Sample, by Grade Level (N = 59)

Figure 1

(Continued)

Note. The signer was left-handed.

Problem 9: Measurement Division (MS)

English:

Paul had 15 [8] caterpillars. He put 3 [2] caterpillars in each jar. How many jars did he put

caterpillars in?

Gloss: __t_ BOY NAME P-A-U-L INDEX-lf HAVE FIFTEEN CATAPILLAR FIFTEEN __t_ ______mm__________ JAR+ C-CL+++ “jars in a row” B↑-CL-------------------- _________t____ ______________mm____ ____________________________whq______ CATAPILLAR-lf THREE++ “in row of jars” (2h)HOW-MANY JAR+ (2h)HOW-MANY B↑-CL-------------------------

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ResultsStrategiesThe strategy types used by the pri-mary-level deaf and hard of hearingstudents to solve arithmetic storyproblems presented to them in ASLare presented here in the order ofproblem difficulty as determined bythe children’s success (the percent-age of viable strategies per problem).Problems with a higher percentageof viable strategies were consideredto be easier than problems with alower percentage of viable strategies.(For details on the determination ofrelative problem difficulty, see Ansell& Pagliaro, 2006.)As was the case in prior studies, all

three strategy types (modeling, count-ing, and fact based) were used by thechildren in the present study. Figure 2shows the distribution of strategy typesby problem difficulty. For each prob-lem, all codable strategies, both viableand nonviable, are represented. Twoimportant pieces of information arerevealed in these results. First, on all problems, counting strategies werepredominately used while fact-basedstrategies were used minimally, regard-less of problem type or difficulty. Sec-ond, modeling strategies were morefrequently used on more difficult prob-lems—an inverse relationship to theuse of counting strategies. For exam-ple, for the easiest problem, Join ResultUnknown (JRU), about 82% of thestrategies used were counting strate-gies. Modeling strategies accountedfor only about 4% of the strategiesused. In contrast, the strategy distri-bution for the more difficult SeparateResult Unknown (SRU) problem, whilestill showing a majority of countingstrategies (approximately 58%), in-cluded about 33% modeling strategies.The percentage of fact-based strategieswas similar for the two problems: 14%and 10%, respectively.In order to understand the relation-

ship between modeling and problemdifficulty, and in particular to deter-mine if the problems were more diffi-cult because of the use of modeling,we calculated the percentage of viablestrategies within each strategy type.Figure 3 shows the distribution of viable strategies for modeling andcounting only. (The low percentage offact-based strategies and the codingcriteria for viability make a reliabilitycomparison of this strategy type mean-ingless.) The results show that whilecounting strategies are still prominent,they are less reliable, that is, less likelyto be appropriate or viable for solvingthe problem. This is especially the casewith the increase in problem difficultyas shown by the lower percentage ofviable counting strategies in the moredifficult problems. For example, in theJRU problem, half the modeling strate-gies used were viable, while 95% ofthe counting strategies were viable.This means that for this problem-andlikewise for the Whole-Unknown (WU)problem—counting strategies weremore reliable, that is, more likely tolead to success, than modeling strate-gies. In contrast, in the SRU problem,

just over 69% of the modeling strate-gies were viable, as compared to 52%of the counting strategies. Thus, forthis more difficult problem, modelingstrategies were more reliable. In fact,across all problems (and in each ex-cept the two easiest problems, JRUand WU), modeling strategies weremore reliable than counting strategies(about 69% of modeling strategieswere viable, compared to about 62% ofcounting strategies), although model-ing strategies were used far less often(20%) than counting strategies (about69%). (It should be noted that ASLAIreceptive scores showed a moderatepositive correlation with total viablestrategies for the nine problems, r =.641, n = 59, p = .000.)A closer investigation of the count-

ing strategies showed that the majority(65.8%) were “count-on” strategies.When using a count-on strategy, thechild would take one of the quantitiesin the problem and continue thecounting sequence by 1’s from thatpoint based on the other quantity orquantities in the problem. This strat-egy is appropriate for problems inwhich the solution is the sum of

Figure 2

Distribution (by Percentage) of Strategy Type Use (Viable and Nonviable), byProblem Difficulty (Left to Right)

Notes. JRU, Join Result Unknown. WU, Whole Unknown. MLT, Multiplication. PU, Part Unknown. JCU,Join Change Unknown. SRU, Separate Result Unknown. Share, Partitive Division. MS, MeasurementDivision. CDU, Compare Difference Unknown.

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numbers—for example, JRU, WU, andmultiplication (MLT). (It should benoted that for the multiplication prob-lem, a distinction was made betweenthe count-on strategy that studentsused inappropriately—i.e., simplyadding the two numbers given in theproblem—and the count-on strategythat is appropriate where one num-ber is repeatedly added on to (by 1’s)a set number of times as indicated bythe other number). In the presentstudy, however, many children used acount-on strategy regardless of prob-lem type, even when it was inappro-

priate to do so. For example, childrenused this strategy appropriately in theJRU problem (see Figure 1 for storyproblem). They would take the 8 andcontinue counting up three times (inreference to the other quantity in theproblem), stopping at and answering“11.” For the SRU problem, however, acount-on strategy is not appropriate;even so, more than a third (34.8%) ofthe counting strategies used in thisproblem were count-on. In this case,these children found the sum of thetwo numbers given in the problem,rather than the difference between

them. In fact, count-on strategiesmade up 45% of the counting strate-gies used across the problems forwhich it is not an appropriate strategy.

Progression of Strategy UseThe progression of strategy use wasdetermined by analyzing the distribu-tion of strategy types by grade level.As shown in Figure 4, as grade level in-creased, the use of modeling strate-gies increased from kindergarten tosecond grade, while the use of count-ing strategies decreased by gradelevel, up to and including third grade.(The decrease in the use of modelingstrategies by third graders can be ex-plained by the increase in the use offact-based strategies among thesechildren.) Chi-square analyses showthe distribution of strategy type acrossgrade levels to be statistically different(p = .000) for all grade-level compar-isons except the comparison betweengrades 1 and 2, which was significantat p= .011, even though a similar pat-tern is evident—that is, countingstrategies were used predominately,fact-based strategies were used spar-ingly, and modeling strategies wereused moderately (and especially inmore difficult problems).

Strategy Use Difference: More Successful and LessSuccessful Problem SolversIn order to obtain a more detailedunderstanding of strategy use amongdeaf and hard of hearing children, weanalyzed the subsample on the basisof their success in problem solving(again, here, “success” is defined bysolution viability), trying to discernwhether those who were relativelysuccessful were unique in terms ofbackground variable or strategy use.The subsample analyses of back-ground variables showed that the twogroups were not significantly differentin any way (e.g., parental hearing

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Figure 3

Distribution (by Percentage) of Viable Strategy Type Use, by Problem Difficulty

Notes. JRU, Join Result Unknown. WU, Whole Unknown. MLT, Multiplication. PU, Part Unknown. JCU,Join Change Unknown. SRU, Separate Result Unknown. Share, Partitive Division. MS, MeasurementDivision. CDU, Compare Difference Unknown.

Figure 4

Distribution (by Percentage) of Strategy Type Use (Viable and Nonviable), by Grade Level

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status, level of hearing loss), with twoexceptions: (a) The more successfulproblem solvers were older than theless successful problem solvers by amean of 1.1 years (p = .000), thoughthe complete range of ages (5–9 years)was represented in both groups; and(b) the more successful problemsolvers scored significantly higherthan the less successful problemsolvers on the ASL receptivity task (M = 32.38 vs. M = 26.83, respec-tively; p = .000 ).With regard to strategy use, Figures

5 and 6 show that both groups usedall three strategy types, with countingstrategies again predominating andfact-based strategies used sparingly.(Since success was based on viability,both viable and nonviable strategieswere included in this analysis.) A strik-ing difference between the strategyuse of these two groups, however, isthat the more successful problemsolvers made significantly more use ofmodeling strategies than the less suc-cessful problem solvers. Overall, themore successful problem solvers wereabout twice as likely to use modelingstrategies as the less successful prob-lems solvers (see Table 5). This wasparticularly true on the more difficultproblems, as can be seen by compar-ing Figures 5 and 6, which show thedistribution of strategy type use byproblem among the more successfuland less successful problem solvers,respectively.

DiscussionMuch as was found in previous stud-ies (Frostad, 1999; Kelly & Mousley,2001), our results indicate that deafand hard of hearing students makeuse of the same general types of solu-tion strategies as hearing students do.However, our findings raise questionsas to whether the data indicate a delayin strategy type use among these stu-dents, as concluded in previous stud-

ies, or show evidence of the existenceof an entirely different pattern of de-velopment. A delay implies that thesame pattern of development existsfor deaf and hard of hearing studentsas for hearing students, but at a latertime. Thus, if the developmental se-quence for strategy types used by deafand hard of hearing children weresimilar to that used by hearing chil-dren, then there would be an increasein the more abstract strategy types,

with an increase in grade level; that is,the younger students would use moremodeling strategies while the olderstudents would use more countingand fact-based strategies. Further, ifthere existed a delay in strategy typeuse among deaf and hard of hearingstudents, the data should show a lateonset of strategy type use, with themore abstract strategy types (count-ing and fact-based) appearing later, atthe second- or even third-grade level,

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Figure 5

Distribution (by Percentage) of Strategy Type Use (Viable and Nonviable) by MoreSuccessful Problem Solvers

Notes. JRU, Join Result Unknown. WU, Whole Unknown. MLT, Multiplication. PU, Part Unknown. JCU,Join Change Unknown. SRU, Separate Result Unknown. Share, Partitive Division. MS, MeasurementDivision. CDU, Compare Difference Unknown.

Figure 6

Distribution (by Percentage) of Strategy Type Use (Viable and Nonviable) by LessSuccessful Problem Solvers

Notes. JRU, Join Result Unknown. WU, Whole Unknown. MLT, Multiplication. PU, Part Unknown. JCU,Join Change Unknown. SRU, Separate Result Unknown. Share, Partitive Division. MS, MeasurementDivision. CDU, Compare Difference Unknown.

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and perhaps not at all in the earliergrades (kindergarten and first grade).As shown in Table 4, the data on strat-egy type use by grade level do notsupport either case. Results from thepresent study show that deaf and hardof hearing children who are users ofASL not only use an overwhelmingnumber of counting strategies for alltypes of problems, but also at all ages,including very early on.We originally expected that, like

hearing children, the deaf and hard ofhearing children would use more con-crete strategies (e.g., modeling) at theyounger grades and shift to more ab-stract strategies (e.g., counting) asthey progressed through the gradelevels. But this was not the case:Counting strategies, while represent-ing the majority of strategies used ateach grade level, decreased in use asgrade level increased. Conversely, theuse of modeling strategies increasedwith grade level. Fact-based strategieswere used minimally (less than 10%)across grades K–2, reaching an appre-ciable proportion (20%) in grade 3.Thus, while studies of young hearingchildren show them to have a generalprogression of strategy type usefrom modeling to counting to factbased (Carpenter, 1985), data fromthe present study suggest that deafand hard of hearing children who areASL users may possess a different pro-gression in which counting strategiesare among the first to occur and con-tinue throughout the lower elemen-tary grades. The progression thenwould perhaps be something like this:counting to modeling/counting to factbased/counting.We suggest two hypotheses to ex-

plain such a difference. First, it couldbe the result of a linguistic influence.Aspects of ASL, and signing in general,such as the ease of use of the count-ing system, the benefits of its manual

nature, and its inherent cardinality,may naturally lead children to count-ing strategies. As Frostad (1999) sug-gests, the ease of use of the countingsystem may be related to countingstrategies being used as the defaultstrategy by these deaf and hard ofhearing children.In addition, because ASL is manual,

the language allows children to hold aseparate number on each hand; thus,children can simultaneously manipu-late two separate counting sequencesto aid them in their solution strate-gies. For example, in Problem 2 (seeFigure 1), where the situation in-volved taking 7 from 11, some of thechildren who used a double countingstrategy started with ELEVEN on onehand and SEVEN on the other, thenproceeded to count back by one stepon each hand until the hand that orig-inally indicated SEVEN had counteddown to zero (ELEVEN/SEVEN, TEN/SIX,NINE/FIVE, etc.). They then looked to theother hand (initially ELEVEN, now FOUR)for the remaining quantity and the an-swer. By contrast, children using anoral language, such as English, wouldneed to remember their place in each

sequence as they alternated betweenthe two, counting up or down. Forthose using ASL, their place in the se-quence would always be present, andthe counting fluency (that Frostad,1999, observed) may have helpedthem to maintain the sequence, mak-ing their procedure more efficient andperhaps more effective.Finally, the children in the present

study made use of the cardinality as-pect in some of the number signs inASL. In these cases, children “toggled”between using the number sign as alabel for the set and using the raisedfingers within the sign as manipula-tives or elements of the set. For ex-ample, in Problem 1 (see Figure 1),where the sum of the two quantities(8 and 3) is unknown, most childrenput out the sign EIGHT on one handand the sign THREE on the other. Theythen made use of the three raised fin-gers on the THREE sign as markers tokeep track of the number of countsneeded to increment the sequence onthe other hand (9, 10, 11). The use ofthis aspect was seen across problems,particularly when one of the givenquantities included the numbers 1–5

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Distribution by grade level (% in parentheses)

Strategy type K 1 2 3

Modeling 1 (3.7) 16 (17.8) 40 (27.0) 23 (17.2)

Counting 24 (88.9) 67 (74.4) 100 (67.6) 84 (62.7)

Fact based 2 (7.4) 7 (7.8) 8 (5.4) 27 (20.1)

Total 27 (100.0) 90 (100.0) 148 (100.0) 134 (100.0)

Table 4

Frequency of Strategy Type Use, by Grade Level

Strategy type

Problem solvers Modeling Counting Fact based

More successful 24.8 62.4 12.8

Less successful 12.1 79.6 8.3

Table 5

Distribution (by Percentage) of Strategy Type Use by More Successful and LessSuccessful Problem Solvers

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for which the cardinality of the num-ber is explicit in its sign. For thoseproblems in which both quantitieswere greater than 5, some childrenseparated one of the quantities(whole) into two smaller quantities(parts) to facilitate the use of toggling.For example, in Problem 2 (see Figure1), some children split the quantity 7into parts, first putting out the signFOUR and counting back from 11 oneach finger, or toggling (10, 9, 8, 7),then putting out the sign THREE andcontinuing to count back, markingthe count on each finger (6, 5, 4). Useof the cardinality of the number signsensured the correct number of mark-ers and gave the children a more effi-cient procedure with less chance oferror or miscount. Indeed, it may bethat, at least at the earlier grade levels,using the cardinality found within thenumber system of ASL may actuallyform a hybrid “modeling/counting”strategy of sorts whereby the fingerscan represent the number as well asmarkers or counters, thus reducing therisk of error.We caution those who would inter-

pret this hypothesis to mean that theyshould teach these aspects of lan-guage as procedures for solving aproblem (whether a computation or astory problem), however. If the childdoes not have a conceptual under-standing of number (that is, does nottruly understand the concept, for ex-ample, that the number of fingersraised actually represents the numberof elements in the set for signs 1–5,but not for the signs 6–9), the childwill erroneously use the procedure re-gardless of cardinality within the num-ber sign. We did see some evidence ofthis in our data. One boy, for example,wished to add 6 and 9 for Problem 3,Set B (see Figure 1). Upon holdingup the signs for each number on hishands, he proceeded to count on

from 6 the number of fingers raised inthe sign for 9 (i.e., 3), getting 9 as theanswer. The child was satisfied withhis answer, giving no indication thathe had done anything incorrect, northat his answer made no sense mathe-matically (the sum of two nonzeronumbers would be the value of one ofthe numbers) or in the context of theproblem situation. He seemed simplyto follow a procedure—a rule—thathe had learned. His actions indicatedno distinction between the numbersign (label) and what it signified (thequantity). This behavior indicates aprocedural orientation to problemsolving similar to those described inthe studies cited above (Frostad &Ahlberg, 1999; Garofalo, 1992; Hegartyet al., 1995; Parmar et al., 1996; Wiest,2003).Our second hypothesis to account

for the overwhelming and early use ofcounting strategies relates to instruc-tion. Studies reveal a very traditionalapproach to mathematics instructionwithin the education of deaf and hardof hearing students that includes em-phasis on direct instruction, memo-rization, and practice exercises overconceptually based learning, in whichtrue problem solving fosters both thedevelopment and use of higher-ordercognitive functions and critical think-ing skills (Kelly et al., 2003; Pagliaro,1998b, 2010; Pagliaro & Ansell, 2002;Pagliaro & Kritzer, 2005). It may be,then, that educators are encouraging,if not outwardly teaching, the use ofprocedures for all mathematics (com-putation as well as problem solving)without first establishing a conceptualunderstanding of mathematics con-cepts such as number, as Frostad(1999) cautioned against. Teachers,for example, may have students makeuse of the cardinality in ASL for num-bers 1–5 as a quick and easy means ofadding or subtracting. Because these

adults understand the separation be-tween the sign or label for the numberand the elements in the set repre-sented by the raised fingers, and canmove easily between the two, theymay assume that their students un-derstand in the same way. However,the students may not, seeing only thelabels and a procedure that says “takeone sign and continue to count (up ordown) on the extended fingers of theother hand”; hence, they likely willcontinue to use the procedure onquantities, and with signs that do notshow cardinality.In addition, research has shown

that most teachers within the edu-cation of deaf and hard of hearingstudents have not been adequatelyprepared in mathematics and/or math-ematics education (Dietz, 1995; Kellyet al., 2003; Kluwin & Moores, 1985;Pagliaro, 1998a, 2010; Pagliaro &Kritzer, 2005). Therefore, even if theyare otherwise high-quality teachers,they may not be aware of the need fortrue problem solving nor have theconfidence or knowledge base neces-sary to guide students in their devel-opment of mathematics concepts andproblem solving. With the best inten-tions, they may believe that they areadvancing students’ achievement byencouraging the use of countingstrategies such as, for example, thecount-on strategy, when in reality thismay do more harm than good, as itseffect may be to skip the benefits ofthe more concrete modeling and limitthe choice of strategies the child haswhen faced with a problem to solve.Finally, the fact that deaf and hard

of hearing students already show agap in their mathematics perform-ance in the early elementary grades(Kritzer, 2009; Traxler, 2000) mayworsen the bias in favor of a proce-dural approach to teaching mathe-matics. Teachers facing the pressure

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to move students through the cur-riculum and “close the gap” may notfeel that they have the time to allowstudents to explore various solutionstrategies on their own as recom-mended by the National Council ofTeachers of Mathematics (2000) andthe National Action Plan for Mathe-matics Education Reform for the Deaf(Dietz, 1995), and as suggested in thestudies by Carpenter and colleagues(Carpenter & Moser, 1984; Carpenteret al., 1993, 1999, 1989).We propose that all children, espe-

cially those who are deaf or hard ofhearing, be taught in a way that devel-ops a conceptual understanding ofmathematics and of problem solving inparticular. Students should discoverand work with various strategies, par-ticularly more concrete strategies(modeling), in order to develop thisunderstanding—even if that modelingcomes in the form of the sign countingsystem—making appropriate use ofcardinality, for example. This sugges-tion is supported in the comparison ofmore and less successful problemsolvers. While both groups still madeextensive use of counting strategies,the more successful problem solversvaried their strategy use, choosing, forexample, to use a modeling strategy onmore difficult problems. In fact, onechild explained that she was “not goodat counting back” as she reached forthe available cubes and proceeded tosolve the problem by modeling. Thischild was very proficient at the count-on strategy but knew her limitationswith counting backward; thus, not feel-ing confident with this counting strat-egy, she chose the more reliable (forher) modeling strategy. In contrast,many of the less successful problemsolvers, when faced with a more diffi-cult problem, tended to forge aheadwith the count-on strategy regardlessof its appropriateness. These students

did not seem to recognize the valueof modeling in problem solving, anapproach that research suggestsleads to viable solutions across abroad range of problems for hearingchildren as young as kindergarten age(Carpenter et al., 1993).We suggest that teachers use care

not to push students to bypass moreconcrete strategies for more abstractones and/or limit students by impos-ing one strategy or strategy type, but,rather, allow students the opportunityand the time to experiment with vari-ous strategies and strategy types sothat when faced with a difficult prob-lem, they have a “menu” of approachesfrom which to choose the more ap-propriate and reliable. While this maynot be a short-term solution to theproblem of time pressure, in the longterm, conceptual understanding andthe ability to solve problems success-fully may prove to be the factor thatlessens or even eliminates the gap inachievement.

Recommendations forFurther StudyGiven the small number of studies inthis area and their various limitations,there is critical need for further inves-tigations of problem solving amongdeaf and hard of hearing students.This includes studies of the linguisticinfluence of a visual language onmathematics instruction and learning,and of the instructional practices thatlead to problem-solving success. It isimportant that future research go be-yond the narrow population consid-ered in the present study as well.Participants in our study were fromschools for deaf and hard of hearingstudents, which enroll approximately20% of these students in the UnitedStates (Gallaudet Research Institute,2008). In addition, we focused ongathering students from only those

schools that used ASL as the primarylanguage of instruction. While this maylimit the study’s implications, it wasnecessary in order to allow these stu-dents full access to the problems withno language barrier. Additional stud-ies, however, are needed to investi-gate the problem-solving strategies ofprimary-level deaf and hard of hearingstudents who are in other school set-tings and do not primarily use ASL, asresults may vary.Finally, it is imperative that profes-

sionals focus on all deaf and hard ofhearing children as unique learners,documenting what they do and howthey do it. Possible differences in adeaf or hard of hearing child’s experi-ences, language, and/or cognitive or-ganization may dictate a differentiatedapproach to instruction that does notfollow the progression of a traditional,general education curriculum. Know-ing how deaf and hard of hearingstudents solve problems, regardlessof their communication preferencesor educational placement, providesinsight into their understanding ofmathematics concepts as well as theirthought processes. This, in turn, mayhave implications for other academicareas.

NoteThe research for the present studywas supported through a grant fromthe U.S. Department of Education,Office of Special Education Programs.—The Authors.

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