© Division of Chemical Education  • www.JCE.DivCHED.org  •  Vol. 85 No. 3 March 2008  •  Journal of Chemical Education 381

In the Classroom

Problem solving is an important part of most science courses. Methods of improving the effectiveness of various methods of teaching problem solving continue to be an active area of research (1–5; see also references in these papers). The main objectives of problem solving are to • Clarify and reinforce theconcepts,principles, applica-

tions,andlawsofthedomainofstudy • Improvestudents’competenceinintellectualskills,strate-

gies,andprocedures,andthuspromotetheirintellectualdevelopment

Theseobjectivesareoftennotfullyachievedbecausemanystu-dents use memorized routine procedures (6) to solve problems withoutmuch thoughtor analysis.Theymuddle through totheanswers,placingemphasisonobtainingthecorrectanswersratherthanlearningfromthesolutions:theyareanswer-orientedandnotprocess-oriented.Thus,problem-solvingcannotthenbeexpectedtooptimallyachieveitsgoals,anditseffectivenessasalearning tool is therefore diminished.

Thedesiredobjectivesofproblem-solvingmaybeachievedbyrequiringstudentstodrawamapthatshowsboththe: • Sequenceofallthestepsinthesolution • Principlesand lawsused,andanyassumptionsandap-

proximations made during the solution

Wewillrefertothesemapsasproblem-solutionmaps.Thestu-dentsdrawthesemapsafter they know the solution to a problem (whichtheymaysolvebythemselvesorafterassistance).

Drawingaproblem-solutionmaprequiresmentaleffort.Studentsneedtocriticallyanalyzeandreflectontheproblemsolutions.Suchanalysisandreflectionhelpprovide(7) greater insight and understanding of both the subject content and the intellectualprocesses associatedwith the solution, andmayalsohelpinrecognizingthereasonsforanyinitialdifficultiesencounteredwhen trying to solve theproblemthefirst time.Thesemapsdisplaytheanalysisofthesolutioninoneplaceandshouldthereforehelpthestudenttoobtainanoverviewofthesolution and assist in searching for alternative paths.

Creating Problem-Solution Maps

Three examples of solution maps from different areas ofchemistryareconsidered in this section.Theexamplesaregivenintheorderofincreasingcomplexity.Sincethispaperisnotconcernedwithhowtosolveproblemsbutwithanalysisofsolutions (after theproblemshavebeen solved), adiscussionofthedifferentapproachesandmethodsthathavebeenusedfor solving problems is not relevant to this paper. Whatever method isused, theprinciples involved in the solutionof a

particularproblemwillbethesame;onlythesequenceofstepswillbedifferent.Theproblem-solutionmapsinthispaperareconstructedfromthesolutionsobtainedbytheuseofastep-by-stepprocedureoutlinedbyoneoftheauthors(8)wherethefirststepstartswithanappropriateequation(oftenthedefiningequa-tion) for thequantity tobecalculatedandproceeds stepwisetoreplaceeachunknownquantityinanequationwithknownquantities.Inthiswaywearriveatafinalequationthatwillrelatetherequiredquantitytothequantitiesgiveninthedata.Astep-by-stepprocedurehastheadvantageofreducingcognitiveload(9)andwouldthereforehelpovercomeourlimitedcapabilityforsimultaneouslyhandlingmanyitemsofinformation.Intheprocedureused,thelaststepis“reviewandlearnfromthesolu-tion”,andthedrawingoftheproblem-solutionmapsisincludedin this step.

Example 1Aclosedvessel at27 °Ccontains3.00gof an ideal gas

whosemolarmassis32.0gmol‒1.Ifthepressureofthegasis 2.52×105Pa,calculateitsdensity(R =8.314JK‒1 mol‒1).

A Solution to Example 1Westart the solutionwiththedefiningequation for the

requiredquantity(density,d)whichis

d

mV

(1)

Tousethisequationtocalculated,weneedtoknowthemassm and its associated volume V. The data give us m but not V. V can be related to the pressure pgiveninthedatabytheidealgasequation:

V

nRTp

(2)

To calculate Vusingtheaboveequationweneedvaluesofn, R, T and p.Ofthesetheonlyunknownquantityistheamountn,whichisrelatedtothemolarmassMgiveninthedataby

n

mM

(3)

Equations1,2,and3maybecombinedtogive

d

M pRT

(4)

Theaboveequationcanbeusedtocalculatetherequiredquan-titysinceallthequantitiesrequiredforitscalculation(M, p, R, T)aregiveninthedata.InSIunits,M=32.0×10‒3kg∙mol, p=2.52×105Pa,R =8.314JK‒1 mol‒1,T =300K.Calculationafter substitution of these values gives us d =3.23kgm‒3.

Using Problem-Solution Maps To Improve Students’ Problem-Solving SkillsMailoo SelvaratnamScience Foundation, North-West University, Mmabatho, South Africa 2735

Sebastian G. Canagaratna*Department of Chemistry, Ohio Northern University, Ada, OH 45810; *s-canagaratna@onu.edu

382 Journal of Chemical Education  •  Vol. 85 No. 3 March 2008  • www.JCE.DivCHED.org  • © Division of Chemical Education

In the Classroom

Figure1istheproblem-solutionmapforthissolution.Itorganizesconciselyallrelevantaspectsofthesolution,showingclearlyhowtheproblemissolvedstep-by-step,fromthebegin-ning,byreplacingunknownquantitieswithknownquantities.Themap includes the equationsneeded to solve eachof thethree stepsneededto solve thisproblem.Thethreeequationsrequired,oneforeachstep,arethedefiningequationfordensity(d = m∙V),theidealgasequation(pV = nRT),andthedefiningequationformolarmass(M = m∙n). Because the map organizes theentiresolutioninoneplace,itshouldhelpstudentsinterre-latethevariousstepsandobtainaclearoverviewofthesolution.Themapshowsthatthemassmappearsintwoplaces,onceasanumeratorandthenasadenominator.Itcancelsoutandthemassdoesnotappearinthefinalequation,eq4,usedtocalculatethedensity.Thisisexpected:densityisanintensivequantity(10) andcannotdependonthemassorsizeofasample.Anextensivequantitythatappearsintwoplacesinasolutionmapwillgener-ally canceloutandthereforeitsvalueneednotbeknown.Thusifwestartwithm =?inoneplaceandlaterendwithm =?,wecouldconcludethatwehavereachedtheendofouranalysis.Wewillseethisfeatureinthenextproblemaswell.

Example 2Asolutionofconcentratedhydrochloricacidcontains36%

bymassofhydrogenchloride(HCl).Thesolutionhasadensityof 1.18 g cm‒3.CalculatetheconcentrationofHClinthesolu-tion.ThemolarmassofHClis35.5gmol‒1.

A Solution to Example 2AsindicatedinFigure2,westartwiththedefiningequa-

tionfortherequiredquantity(concentrationofHCl,cHCl):

c

nVHCl

HCl

solution (5)

We see thatweneed toknow the amountofHCl in aknownvolumeofsolutiontobeabletouseeq5tocalculatetheconcentrationofHCl.SincecHClisanintensivequantity,itdoesnotdependonthevolumeofsolution:wemaythereforeselectanyvalueforthevolumeofsolution.VsolutionmaythereforeberegardedasaknownquantityandourproblemistocalculatenHClinaknownvolumeofsolution.

The term nHCl is related to themolarmass,MHCl, givenintheproblem’sdatabythedefiningequationformolarmass(MHCl = mHCl∙nHCl),fromwhich weget

n m

MHClHCl

HCl (6)

Replacement of the nHCltermineq5byeq6gives

c m

M VHClHCl

HCl solution (7)

To calculate cHClusingthisequationweneedtocalculatethemassofHCl,mHCl,presentinaknownvolumeofsolution.mHCl isrelatedtothemassfraction(expressedasapercent)ofHCl,fHCl,giveninthedatabythedefiningequationformasspercent( fHCl = mHCl∙msolution×100%)fromwhichweobtain

m

f mHCl

HCl ssolution

100% (8)

Figure 1. Solution map for calculation of density from Example 1.

Figure 2. Solution map for calculation of concentration of HCl from Example 2.

cHCl = ?

dsolution Vsolution

Calculation of msolution requires dsolution and Vsolution;

equation is:

msolution = dsolution Vsolution

msolution = ?fHCl

Calculation of mHCl requires fHCl and msolution; equation is:

mHCl = fHCl msolution

Calculation of nHCl requires mHCl and MHCl; equation is:

nHCl =mHCl

MHCl

mHCl = ?MHCl

nHCl = ?

Calculation of cHCl requires n and V; equation is:

cHCl =nHCl

Vsolution

Vsolution

d = ?

V = ?m

Calculation of d requires m and V; equation is:

d =m

V

n = ? pTR

Calculation of V requires n, R, T and p; equation is:

V =n RT

p

Mm

Calculation of n requires m and M; equation is:

n =m

M

© Division of Chemical Education  • www.JCE.DivCHED.org  •  Vol. 85 No. 3 March 2008  •  Journal of Chemical Education 383

In the Classroom

nNO2,eq = ?V , R , T

Equation to calculate Kp is:

pN2O4,eq = ?

Kp =pNO2,eq

2

pN2O4,eq2

Equation to calculate pNO2,eq is:

pNO2,eq =nNO2,eq RT

V

pNO2,eq = ? ptotal,eq

ptotal,eqV , R , ,T

ntotal,eq =ptotal,eq V

RT

Equation to calculate ntotal,eq is:

nNO2(formed) = ?nNO2

(initial) = 0

Equation to calculate nNO2,eq is:

nNO2,eq =

+

nNO2(initial)

nNO2(formed)

Equation to calculate pN2O4,eq is:

ptotal,eq = +pN2O4,eq pNO2,eq

change in ntotal = ?

Equation to calculate nNO2(formed) is:

nNO2(formed) = change in ntotal

Equation to calculate nNO2(formed) is:

ntotal,eq = ?ntotal(initial)

change in ntotal =ntotal,eq – ntotal(initial)

pNO2,eq = ?

Kp = ?

IfwenowreplacethemHClineq7byeq8weget

HCl

HCl solution

HCl solution100%c

f mM V

(9)

Inthisequationmsolutionisnotknown.Itcan,however,berelatedtothedensityofthesolution,dsolution,giveninthedata:

sm oolution solution solutiond V (10)

Replacement of the msolutiontermineq9byeq10andsimplify-ingyields

HCl

HCl solution

HClc

f dM100%

(11)

Thisisthefinalequation,whichrelatescHCltothequantitiesgivenintheproblem’sdata.Calculationusingthisequationshowsthat

cHCl=0.0120molcm-3,whichis12.0molL‒1.Asexpected,theterm Vsolutionisabsentineq11becauseitcancelsoutduringthederivation.Figure2 showsthedetailsof the solutionmapforcalculatingtheconcentrationofHClinExample2’sproblem.

Anotherapproachoftenusedistosolvetheproblemnu-merically.Afixedvolumeofsolution(e.g.,1.00dm3 ) is consid-eredandtheamountofHClpresentinitiscalculated.

Example 3Aquantityof0.310molofdinitrogentetraoxide(N2O4)

ismaintainedat50°Cina5.00dm3 vessel. Part of it then dis-sociatesaccordingtotheequationN2O4(g) ↔2NO2(g).Iftheequilibriumpressureis2.20×105Pa,calculatetheequilibriumconstant Kp for thedissociation in termsofpartialpressures,keepinginmindthatR =8.314JK‒1 mol‒1.

ThesolutiontothisexampleisgivenonlyasasolutionmapinFigure3,whichshowssequentiallythestepsinthesolution,

Figure 3. Solution map for calculation of Kp from Example 3.

384 Journal of Chemical Education  •  Vol. 85 No. 3 March 2008  • www.JCE.DivCHED.org  • © Division of Chemical Education

In the Classroom

startingwiththedefiningequationforKp.Manystudentshavedifficultywiththisproblembecauseofafailuretorealizethatstoichiometrygivesinformationnotonlyaboutthechangesinthepartialpressuresofreactantsandproducts,but also about the change in the total pressure.Theequilibriumstateisthefinalstate,andtheinitialstatecontainsonlydinitrogentetraoxide.The total pressure of the initial state is therefore the pressure of the dinitrogen tetraoxide in the initial state.

Discussion

Problem-solutionmapshavebeenusedbyoneofus formanyyearswithfirst-yearstudentsattheuniversity level in an attempt to enhance the usefulness of problem solving. The text used in the course (11)describesafive-stepprocedureforprob-lemsolvingandusesitintheworkedexamplesinthebook.Thefifthstepis“reviewandlearnfromthesolution”andthedrawingofproblem-solutionmapsisincludedinthisstep.Manysolutionmapsaredrawnandexplainedintheworkedexamplesinthebook:thiswouldprovideguidancetostudentsonhowtodrawnewmaps.Studentsaretaughttodrawsolutionmapsforsomeoftheproblemstheysolveintutorialsessionsforsmallgroupsofstudents.Initially,moststudentshavedifficultyindrawingthesemapsalthoughtheirperformancebecomesprogressivelybetterwithpractice.Theybenefitfromthementaleffortandstruggleneeded to identify sharply the steps andprinciplesinvolvedinthesolution.Suchabenefitwouldbeexpectedfromtheconstructivist theoryof learning(12).Without students’makingthementalefforttoconstructtheirownknowledgebase,meaningfullearningmaynotbepossible.

Drawingproblem-solutionmapscanbeeasilyincorporatedinto theprocess-oriented, guided inquiry earning (POGIL)approach,cooperativelearning,orpeer-ledlearning.Ofcourseinstructorsshouldprovideexamplestostudentsofhowtodrawproblem-solutionmaps.Sinceaproblem-solutionmapmerelymaps out the solution of a problem afterthesolutionisknown,thedrawingofsuchmapsshouldnotbedifficultifoneclearlyunderstandsthesolution.Theeasewithwhichonecandrawasolution-mapmaybeconsideredtoprovideameasureofone’sunderstanding of the solution.

Students’difficultiesindrawingsolutionmapsmaybeusedasamonitortochecktheirunderstandingofallaspects(bothprocess aspects and contentaspects)ofthesolution.Ifstudentsex-periencedifficultiesindrawingsuchmaps,orfindthatthetaskislaborious,itimpliesthattheyhaveconceptualdifficulties.Many

studentshaveconceptualdifficultieswithproblemsthatmayap-peartobesimple.Consider,forexample,theproblemgiveninTextbox1.Thisproblemhasbeentestedonabout200first-yearuniversitystudents(atentryintothecourse),overaperiodofthreeyears(13).About85%ofthestudentsdidnotsolvetheproblemcorrectly,whichsuggeststhattheyhad difficultieswiththeconceptsofpureliquid,solution,andconcentration.

Research (9)hasshownthatteachersareoftenunableto predict successfullyhowdifficultatopicwouldbeforstudents.Thestep-by-stepapproachaswellasthefactthatstudents are workingfromthesolutiontoaproblemshouldfacilitatelearn-ing.Adrop inperformancewasnoted(14) if studentswereaskedtoprovideanequation,balanceit,andthenuseittosolveaproblem,withasuccessrateofabout30%comparedtoasuc-cessrateof70%whenstudentsweregivenabalancedequation.Concentratingonlyondrawingthesolutionmapwithouthav-ingtoworryaboutsolvingtheproblemwouldbeexpectedtodecreasethedemandonstudents’workingmemory.

Drawingproblem-solutionmapsaftertheproblemhasbeensolvedgivesstudentsanopportunityto:

• Reflectonhowtheyanalyzedandsolvedtheproblem.Thisthinkingaboutthinking,ormetacognition,isknowntoimprovelearning,andisbelievedtobeanimportantdifferencebetweennovicesandexperts.

• Identifytheequations,principles,laws,approximations,andassumptionstheyusedinthesolution,andevaluatetheir importance to the solution process.

• Evaluateandcomparealternativepathsthatcouldbeusedfor the solution.

• Analyzeanydifficultiestheyhad,andunderstandclearlyhowthedifficultieswereovercome.

Todraw aproblem-solutionmap, studentsmust iden-tifytheprinciplesandlawsinvolvedinthesolution.Failingtoidentifytheprinciplesinvolvedmayleadtoerrorsinproblemsolving.Thefirst twoexamples in theonline supplement il-lustratethis.Theseexampleshavebeentestedonmanygroupsoffirst-yearuniversitystudents.Identificationoftheprinciplesinvolvedwouldalsoleadtoeasiersolutionsandmayalsohelptoaddressthedifficultyoftransfer(15)ofproblem-solvingskillslearned from the solution of a problem to other problems. The last example (16) in the online supplement illustrates the idea that identificationof theprinciples involvedwould lead toamore logical and easier solution.

The approach outlined here could also be applied in other areasofchemistry;forexample,studentscouldbeaskedtoana-lyzetheirsolution(orasolutiongiventothem)ofaprobleminorganicsynthesis.Thiswouldhelpfocustheirattentiononthetransformationseffectedineachstep.

Wehaveusedequationsforsolvingtheproblems.IntheU.S., problem solving in some areas of chemistry is largelyundertaken throughdimensional analysis, inwhichunits aremanipulatedtoobtaintheanswers.Ifstudentsdothiswithoutmuchunderstanding,drawing solutionmapswillbedifficultforthem,becauseitisnecessarytoidentify,fromtheunits, the correspondingphysicalquantitiesandalsotheequationrelatingthesequantities.Studentswhostringtogetherunitstoarriveatananswer(17)would thereforeparticularlybenefit fromtheexerciseofdrawingsolutionmaps.

Example Problem

1.0 mol of ethanol (a liquid) is dissolved in 1.0 dm3 of water. The concentration of ethanol in the solution obtained will be:

(a) 1.0 mol dm−3

(b) 0.5 mol dm−3

(c) <1.0 mol dm−3

(d) >1.0 mol dm−3

Textbox 1. First-year college chemistry sample test question.

© Division of Chemical Education  • www.JCE.DivCHED.org  •  Vol. 85 No. 3 March 2008  •  Journal of Chemical Education 385

In the Classroom

Canstudents subvert theobjectiveofdrawingproblem-solutionsmapsandmakeitaroutine,mechanicalprocedure,astheyoftentrytodowhentheysolveproblems?Thismaybedif-ficultbecausethesolutionsaredifferentfordifferentproblemsand also because a given problem can be solved in more than oneway.Forexample,theprobleminvolvingKpmaybesolveddirectlyintermsofpartialpressures,intermsofamounts,orintermsofmolefractions.Differentstudentsmaythereforehavedifferentproblem-solutionmapsforthesameproblem.Studentswillthereforeneedtofocussharplyontheirsolutionstodrawsolution maps.

Conclusions

Drawingsolutionmapstakestime.Inourview,thetimespentisworthwhilebecauseitensuresmentaleffortbystudents,andalsohelpsintheclearidentificationofallaspectsthatareimportant for the solution of a problem. We believe that it is better for students toanalyze indetail, andunderstandthor-oughly,thesolutionofafewcarefullyselectedproblemsratherthanforthemtomuddlethrough,withoutmuchthoughtorunderstanding,thesolutionoflargenumbersofproblems.Itisourviewthatdrawingproblem-solutionmapsisausefultooltohelp improve problem-solving abilities of students and to build theirknowledgebaseandself-confidence.

Literature Cited

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1163. 4. Jeon,K.;Huffman,D.;Noh,T.J. Chem. Educ. 2005, 82, 1558–

1564.

5. Hanson,D.;Wolfskill,T. J. Chem. Educ. 2000, 77, 120. 6. Selvaratnam,M.Educ. Chem. 1990, 27, 163. 7. Hanson,D.Instructor’s Guide to Process-Oriented Guided-Inquiry

Learning;PacificCrest:Lisle, IL,2006;http://www.pogil.org/downloads/POGIL_IG.pdf(accessedDec2007).

8. Selvaratnam,M.;Frazer,M. J. Problem Solving in Chemistry; HeinemannEducationalPublishers:London,1982.

9. Johnstone,A.H. J. Chem. Educ. 1983, 60, 968–971.10. Canagaratna,S.G. J. Chem. Educ. 1992, 69, 957–963.11. Selvaratnam,M.A Guided Approach to Learning Chemistry;Juta

Publishers:CapeTown,RepublicofSouthAfrica,1998.12. Bodner,G.M.J. Chem. Educ. 1986, 63, 873–878.13. Drummond,H.P.Students’CompetenceintheIntellectualSkills

andStrategiesNeededforLearningSouthAfricanMatriculationChemistryEffectively.Ph.D.Thesis,North-WestUniversity,Ma-fikeng,SouthAfrica,2003.

14. Johnstone,A.H.J. Chem. Educ. 1997, 74, 262–268.15. Sternberg,R. J.Essays on the Intellect,Link,FrancesR.,Ed.;

Association for Supervision andCurriculumDevelopment:Alexandria,VA,1985;p54.

16. Gagne,M.R.The Conditions of Learning;Holt,Rinehart andWinston:NewYork,1977.

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Supporting JCE Online Materialhttp://www.jce.divched.org/Journal/Issues/2008/Mar/abs381.htmlAbstractandkeywordsFull text (PDF) LinkstocitedURLsandJCE articlesSupplement Additionaldiscussionofwhetherstudentsidentifytheprinciples

thattheyuseforthesolutionofaproblem Anexampleproblemforstudentstosolve