Concepts for teaching and learning in class Deliverable 4.м

This project has been funded with support from the European Commission.This publication reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Key Competences for Lifelong Learning and Mathematics Education:

Concepts for Teaching and Learning Mathematics in Class Demetra Pitta-Pantazi, Constantinos Christou, Maria Kattou, Marios Pittalis, Paraskevi Sophocleous

1 Introduction

Thischapteraimstopresentaproposedframe‐workwhichshowshowbasicskillsandtransver‐salcompetencescanbedevelopedbymathemat‐icseducation.To thisend, section twopresentsthe Key Competences for Lifelong Learning ac‐cording to the European Parliament and theCouncil of the European Union (2006). Sectionthreeprovidesthemathematicalproceduresandpracticesthatstudentsshoulddevelopaccordingtovariousmathematicseducationorganizationsandresearchers.Sectionfour,havingasaspring‐boardthetwoprevioussections,discussesapro‐posed framework which supports the develop‐mentofstudents’keycompetencesinmathemat‐ics.Finally,insectionfivewepresentassessmentmethodsforstudents’keycompetencesinmath‐ematics.

2 Key Competences for Lifelong Learning

In the last tenyears a lotof attentionhasbeengivenattheEuropeanleveltokeycompetencesfor lifelong learning.Anumberofreportsdocu‐mentedtheimportanceofdevelopingkeycompe‐tences at school inEurope (seeEuropeanCom‐mission/EACEA/Eurydice,2012; Otten&Ohana,2009). In addition, a significant number of re‐searchprogramswereconductedintheframeofthekeycompetences(e.g.,Q4iproject:Qualityforinnovation in European schools, KeyCoNet pro‐ject:KeyCompetenceNetworkonSchoolEduca‐tion,EU‐27:KeyCompetencesandTeacherEdu‐cation). At the end of 2006, these competencesweredefinedbytheEuropeanParliamentandtheCounciloftheEuropeanUnion(Recommendation 2006/962/EC), after a five year collaborativework between specialists to find a common

frameworkofkeycompetencesforlifelonglearn‐ingattheEuropeanlevel.Pepper(2011)charac‐terizedthedevelopmentofthesecompetencesas‘an importantpolicy imperative forEUmemberstates’(Pepper,2011,p.335).EuropeanParliamentandtheCounciloftheEu‐ropeanUnion(2006)definedkeycompetencesasa multifunctional and transferable package ofknowledge, skills and attitudes, which ‘all indi‐vidualsneed forpersonal fulfillmentanddevel‐opment, active citizenship, social inclusion andemployment’(p.13).Inparticular,EuropeanPar‐liament and theCouncil of the EuropeanUnion(2006)identifiedeightkeycompetences:

(1) Communicationinthemothertongue(2) Communicationinforeignlanguages(3) Mathematicalcompetenceandbasiccompe‐

tencesinscienceandtechnology(4) Digitalcompetence(5) Learningtolearn(6) Socialandciviccompetences(7) Senseofinitiativeandentrepreneurship(8) Culturalawarenessandexpression

Thesecompetencesareequallyimportantandin‐terrelated. They are anticipated to be acquiredbothbystudentsattheendofcompulsoryeduca‐tion,aswellasbyadultsthroughaprocessofde‐velopingandupdatingtheirskills(EuropeanPar‐liament and Council of the European Union,2006).Thus,initialeducationshouldofferallstu‐dents the opportunities to develop key compe‐tencesinasufficientlevelthatwillequipthemforadultandworkinglife(EuropeanParliamentandCounciloftheEuropeanUnion,2006).Inthefol‐lowingsection,weprovideashortdescriptionoftheeightkeycompetences:

“Communication in the mother tongue” requiresanindividualtohaveknowledgeofwords,terms,

functionalgrammarandthefunctionsofmotherlanguage. At the same time, this competence isconnected to the acquisition of an individual'scognitiveabilitytointerprettheworldandrelatetoothers.Moreover,itinvolves‘theskillstocom‐municatebothorallyandinwritinginavarietyofcommunicative situations and to monitor andadapt their own communication to the require‐mentsofthesituation’(EuropeanParliamentandCounciloftheEuropeanUnion,2006,p.14).Thiscompetencealsoincludestheacquisitionofposi‐tiveattitudetocriticalandconstructivedialogue,anacceptanceofaestheticqualitiesaccompaniedwithanenthusiasmtotryforthem,andaninter‐est in interactionwithothers (EuropeanParlia‐mentandCouncilof theEuropeanUnion,2006;EuropeanCommunities,2007).

“Communication in foreign languages” requiresan individual to have knowledge of the words,terms, functionalgrammarandstructureof for‐eignlanguages.Itincludesskillsforcommunica‐tion in foreign languages. These skills are com‐prehensionofspokenmessages,makingconver‐sations, reading and producingwriting.Moreo‐ver,thiscompetenceinvolves‘theappreciationofculturaldiversity,andaninterestandcuriosityinlanguagesandinterculturalcommunication’(Eu‐ropeanParliamentandCouncilof theEuropeanUnion,2006,p.15).

“Mathematical competence and basic competences in science and technology”involvestwoessentialcompetences:mathematicscompetenceandsci‐enceandtechnologycompetence.‘Knowledge in mathematics involves a soundknowledgeofnumbers,measuresandstructures,basicoperationsandbasicmathematicalpresen‐tations,anunderstandingofmathematicaltermsandconcepts,andanawarenessofthequestionstowhichmathematicscanofferanswers.Anindi‐vidualshouldhavetheskillstoapplybasicmath‐ematical principles and processes in everydaycontextsathomeandwork,andtofollowandas‐sesschainsofarguments.Anindividualshouldbeabletoreasonmathematically,understandmath‐ematicalproofandcommunicateinmathematicallanguage,andtouseappropriateaids’(EuropeanParliament and Council of the EuropeanUnion,2006,p.15).Forscienceand technologycompetence, ‘essen‐tialknowledgecomprisesthebasicprinciplesofthe natural world, fundamental scientific con‐cepts, principles and methods, technology andtechnologicalproductsandprocesses,aswellasan understanding of the impact of science and

technology on the natural world. Skills includetheabilitytouseandhandletechnologicaltoolsandmachinesaswellasscientificdatatoachieveagoalortoreachanevidence‐baseddecisionorconclusion’(EuropeanParliamentandCounciloftheEuropeanUnion,2006,p.15).

“Digital competence” requires an individual tohaveknowledgeofthe‘nature,roleandopportu‐nitiesoftechnologyineverydaycontexts’(Euro‐peanParliamentandCounciloftheEuropeanUn‐ion,2006,p.16).Thiscompetence,also,includes‘the ability to search, collect and process infor‐mationanduseitinacriticalandsystematicway,assessing relevance and distinguishing the realfromthevirtualwhilerecognizingthelinks’(Eu‐ropeanParliamentandCouncilof theEuropeanUnion,2006,p.16).

“Learning to learn” requires ‘an individual toknowandunderstandhis/herpreferredlearningstrategies, the strengths and weaknesses ofhis/herskillsandqualifications,andtobeabletosearchfortheeducationandtrainingopportuni‐ties and guidance and/or support available.Learningtolearnskillsrequirefirstlytheacquisi‐tionofthefundamentalbasicskillssuchasliter‐acy,numeracyand ICTskills thatarenecessaryfor further learning.Buildingon these skills, anindividualshouldbeabletoaccess,gain,processandassimilatenewknowledgeandskills.Individ‐ualsshouldbeabletoorganizetheirownlearn‐ing,evaluatetheirownwork,andtoseekadvice,information and support when appropriate. Apositiveattitudeincludesthemotivationandcon‐fidence to pursue and succeed at learningthroughoutone'slife’(EuropeanParliamentandCounciloftheEuropeanUnion,2006,p.16).

“Social and civic competences” includes two es‐sentialcompetences:socialcompetenceandciviccompetence.‘Social competence involvesknowledgeofbasicconceptsrelatingtoindividuals,groups,workor‐ganizations,genderequalityandnon‐discrimina‐tion, society and culture. The core skills of thiscompetence include the ability to communicateconstructivelyindifferentenvironments,toshowtolerance, express and understand differentviewpoints,tonegotiatewiththeabilitytocreateconfidence,andtofeelempathy.Civic competence is basedonknowledgeof theconceptsofdemocracy,justice,equality,citizen‐ship,andcivilrights,includinghowtheyareex‐pressedintheCharterofFundamentalRightsof

the European Union and international declara‐tionsandhowtheyareappliedbyvariousinsti‐tutionsatthelocal,regional,national,Europeanand international levels. Skills for civic compe‐tence relate to the ability to engage effectivelywithothersinthepublicdomain,andtodisplaysolidarityandinterestinsolvingproblemsaffect‐ingthelocalandwidercommunity.Thisinvolvescritical and creative reflection and constructiveparticipationincommunityorneighborhoodac‐tivities as well as decision‐making at all levels,fromlocaltonationalandEuropeanlevel,inpar‐ticularthroughvoting’(EuropeanParliamentandCounciloftheEuropeanUnion,2006,p.17).

“Sense of initiative and entrepreneurship” re‐quiresindividualtobeabletoturnideasintoac‐tion and to assess and take risks tomaterializeobjectives. Individuals should also ‘be aware oftheethicalpositionofenterprises,andhowtheycan be a force for good’ (European ParliamentandCounciloftheEuropeanUnion,2006,p.17).Inadditiontothis,‘anentrepreneurialattitudeischaracterized by initiative, pro‐activity, inde‐pendenceandinnovation inpersonalandsociallife, asmuch as atwork’ (European ParliamentandCounciloftheEuropeanUnion,2006,p.18).

Finally, “Cultural awareness and expression” ‘in‐cludesanawarenessoflocal,nationalandEuro‐pean cultural heritage and their place in theworld.Skillsrelatetobothappreciationandex‐pression: the appreciation and enjoyment ofworksofartandperformancesaswellasself‐ex‐pressionthroughavarietyofmediausingone’sinnatecapacities.Skillsincludealsotheabilitytorelateone'sowncreativeandexpressivepointsofviewtotheopinionsofothersandtoidentifyandrealizesocialandeconomicopportunitiesincul‐turalactivity.Asolidunderstandingofone'sowncultureandasenseofidentitycanbethebasisforanopenattitudetowardsandrespect fordiver‐sityofculturalexpression’(EuropeanParliamentandCounciloftheEuropeanUnion,2006,p.18).

3 Mathematical Practices

Inthemathematicseducationdomainvariousor‐ganizations and educational authorities definedimportant mathematical procedures and prac‐tices that students should develop (NationalCouncilofTeachersofMathematics,2000;Stand‐ards for Mathematical Practice of the CommonCore State Standards Initiative, 2011). In addi‐

tion,thedebateregardingthepracticalperspec‐tive and the mathematical rigor in developingmathematical curricular has been extensivelydiscussed in various countries (Sullivan, 2011).In this framework,Ernest (2010)described thegoalsof thepracticalperspectiveandexplainedtheimportanceoflearningthemathematicsade‐quateforgeneralemploymentandfunctioninginsociety and mathematics necessary for variousprofessional and industry groups. Furthermore,the Shapeof theAustralianCurriculum:Mathe‐matics (Commonwealth of Australia, 2009) dis‐tinguished the practical and the specialized as‐pectsofthemathematicscurriculumandempha‐sizedtheneedtoeducatestudentstobeactive,tointerpret the world mathematically and usemathematics to make predictions as well as totake decisions regarding personal and financialpriorities. The Victorian Department of Educa‐tion and Early Childhood Development of theGovernment inAustralia (2009)underlined theimportanceofdevelopingstrategicskills,suchasknowingthatmathematicsmighthelp,adaptingmathematicstothecontext,knowinghowaccu‐ratetobe,andknowingiftheresultmakessenseincontext.Italsomadeexplicittwelvescaffoldingpractices that are appropriate to explore/makeexplicit what is known, challenge/extend stu‐dents’mathematicalthinkingordemonstratetheuseofamathematical instrument. Inparticular,thetwelvescaffoldingpracticesrefertoexcavat‐ing,modeling,collaborating,guiding,convincing,noticing, focusing,probing, orienting, reflecting,extendingandapprenticing.

Classroommathematical practices focus on thetaken‐as‐sharedwaysofreasoning,arguing,andsymbolizingestablishedwhilediscussingpartic‐ularmathematicalideas(Cobb,Stephan,McClain,&Gravemeijer,2011).Mathematicalpracticesde‐scribetheconditionsunderwhichstudentslearnmathematicswithdeep conceptual understand‐ing. The mathematical practices are not skill‐basedcontentthatstudentscanlearnthroughdi‐rect teaching methods, but emerge over timefromopportunitiesandexperiencesprovidedinmathematicsclassrooms(Hull,Balka,&Harbin‐Miles,2011).Thus,theseopportunitiesandexpe‐riences should be coordinated by mathematicsteachers and include challengingproblems, col‐laborativegroupsandinteractivediscourse.Thepracticesareinterdependent,inotherwordsarenotdevelopedinisolationfromoneanother,andhencemathematicseducatorsneedtocontinuallyassess student progress on these practices in aholisticfashion.Theintentisthattheseessential

mathematicalhabitsofmindandactionpervadethecurriculumandpedagogyofmathematics,inage‐appropriateways.

TheCommonCoreStateStandardsforMathemat‐icsdescribedbothContentStandardsandStand‐ardsforMathematicalPractice.TheStandardsforMathematicalPracticeoftheCommonCoreStateStandardsInitiative(2011)describedvarietiesofexpertisethatmathematicseducatorsatalllevelsshouldseektodevelopintheirstudents.Thethe‐oreticalfoundationofthemathematicalpracticeslies on important “processes and proficiencies”with longstanding importance in mathematicseducation. First, themathematical practices arerelatedtotheNCTM(2000)processstandardsofproblemsolving,reasoningandproof,communi‐cation,representation,andconnections.Inaddi‐tion,theytakeintoconsiderationthemathemati‐calproficiencyoftheNationalResearchCouncil’sreportAdding It Up:adaptivereasoning,strategiccompetence,conceptualunderstanding(compre‐hension of mathematical concepts, operationsandrelations),proceduralfluency(skillincarry‐ingoutproceduresflexibly,accurately,efficientlyand appropriately), and productive disposition.Thefollowingmathematicalpracticesdescribeina more mathematical‐oriented framework themajorityoftheAustralianscaffoldingpractices.

Make sense of problems and persevere in solving them. Thefirstpracticeinvolvesthediscussion,expla‐nation and solution of a problemwithmultiplerepresentationsandinmultipleways,aswellasstudents’persistenceduringthesolutionprocess.Students shouldbe able to analyze givens, con‐straints, relationships, and goals, make conjec‐turesaboutthesolutionandplanasolution,mon‐itorandselfevaluatetheirprogressandchangecourseifnecessary.Thus,thispracticeconsistsofthefollowing:workingtomakesenseofaprob‐lem,makingaplan, tryingdifferent approacheswhentheproblemisdifficult,solvingaprobleminmorethanoneway,checkingwhethertheso‐lutionmakessenseandconnectingmathematicalideasandrepresentationstooneanother.

Reason abstractly and quantitatively. Thesecondpracticerelates toconvertingsitua‐tions into symbols to appropriately solve prob‐lemsaswellasconvertsymbolsintomeaningfulsituations.Inotherwords,itinvolvestheappro‐priateuse,interpretationandexploitationofthemathematical language. Thus, two salient abili‐tiesareinvolved;decontextualizing–toabstract

a given situation and represent it symbolicallyand manipulate the representing symbols as iftheyhavealifeoftheirown,andcontextualizing,topauseasneededduringthemanipulationpro‐cess inordertoprobeintothereferentsforthesymbolsinvolved.Hence,thispracticeconsistsofthefollowing:(a)representingproblemsandsit‐uations mathematically with numbers, words,pictures, symbols, gestures, tables and graphs,and(b)explainingthemeaningofthenumbers,words, pictures, symbols, gestures, tables andgraphs.

Construct viable arguments and critique the rea-soning of others. Thethirdpracticeinvolvesjustifyingandexplain‐ing,withaccuratelanguageandvocabulary,whya solution is correct and comparing and con‐trastingvarioussolutionstrategiesandexplain‐ing the reasoning of others. Studentsmay con‐struct arguments using concrete referents suchasobjects,drawings,diagrams,andactions.Stu‐dentsshouldbeable to listenor read theargu‐mentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.Summingup,thispracticeconsistsofexplainingbothwhattodoandwhyitworksandworking tomake senseof others’mathematicalthinking.

Model with mathematics. Modelwithmathematicsdescribestheabilitytouseavarietyofmodels,symbolicrepresentationsanddigital tools todemonstrate a solution to aproblem,byidentifyingimportantquantitiesinapractical situation and mapping their relation‐shipsusingsuchtoolsasdiagrams, two‐wayta‐bles, graphs, flowcharts and formulas. In addi‐tion,studentsshouldbeabletoanalyzethosere‐lationshipsmathematicallytodrawconclusions.Thus, this practice consists of applying mathe‐matical ideas toreal‐worldsituationsandusingmathematicalmodels suchas graphs,drawings,tables,symbols,anddiagramstosolveproblems.

Use appropriate tools strategically. This practice involves students’ ability to com‐binevarious tools, includingdigital tools, toex‐plore and solve problems as well as justifyingtheirtoolselection.Toolsareusedtodeepenstu‐dents’ understanding of concepts. Studentsshouldbeabletomakedecisionregardingtheap‐propriatenessoftheusedtool.Thus,thispracticeconsistsofchoosingappropriatetoolsforaprob‐lemandusingmathematical toolscorrectlyandefficiently.

Attend to precision. Attendtoprecisionrelatestostudents’appropri‐ateuseofsymbols,vocabularyandlabelingtoef‐fectively communicate andexchange ideas. Stu‐dentsshouldbeabletousecleardefinitionswithothersandintheirownreasoning.Thispracticeconsists of communicatingmathematical think‐ing clearly and precisely, and being accuratewhensomeonecounts,measuresandcalculates.

Look for and make use of structure. This practice involves students’ ability to seecomplex and complicatedmathematical expres‐sionsascomponentparts.Theycanconceptual‐izecomplicatedthings,assingleobjectsorasbe‐ing composed of several objects. This practiceconsistsoffinding,extending,analyzingandcre‐atingpatternsandusingpatternsandstructurestosolveproblems.

Look for and express regularity in repeated rea-soning. Thispracticedefinesstudents’abilitytodiscoverdeep,underlyingrelationships, findandexplainsubtlepatternsandlookforregularityinproblemstructureswhensolvingmathematicaltasks.Thispracticeconsistsofusingpatternsandstructurestocreateandexplainrulesandshortcuts,usingproperties,rulesandshortcutstosolveproblemsandreflectingbefore,duringandaftersolvingaproblem.

Theabovemathematicalpracticeshadasignifi‐cant impact in themathematicseducationcom‐munity. For instance, the Mathematics Assess‐mentProject,collaborationbetweentheUniver‐sityofCalifornia,BerkeleyandShallCenterattheUniversity ofNottingham, developed an assess‐mentmaterial that brings into life and contrib‐utesinimplementinginrealclassroomstheCom‐monCoreStateStandardsforMathematicalPrac‐tice.

4 Proposed Framework: Development of Concepts for Teaching and Learning in Class

4.1 Mathematics Education and Key Com-petences

DespitetheimportanceofKeyCompetencesandthegreatemphasistotheirdevelopmentatEuro‐pean level, littleattentionhasbeengiventotheroleofmathematicseducation,asamainsubjectintheschoolsystem,indevelopingstudents’key

competences.Basedonthis,theKeyCoMathpro‐jectunderlinedthatthereisaneedtoinfuseac‐tivities that promote the development of keycompetencesinmathematicsteachingandlearn‐ing.Thus,theKeyCoMathprojecttacklesthechal‐lengeof implementingthe“EuropeanReferenceFrameworkofKeyCompetences”(EuropeanPar‐liament and Council of the European Union,2006)inmathematicseducationandfocusesonthedevelopment,implementationandevaluationofthemajorityoftheEuropeankeycompetencesinmathematics.Specifically, at thebeginningoftheproject,thefollowingkeycompetencesweredefined and elaborated in the framework ofmathematicsteachingandlearning.

Mathematical competence. KeyCoMath arguesthatstudents’mathematical thinkingcanbede‐velopedandenhancedthroughanactive,explor‐atorylearninginopenandrichsituations.

Communication in the mother tongue.KeyCoMathcloselyintertwinesdoingmathematicsandcom‐municatingwithothersorallyorinwrittenform.Forthis,KeyCoMathclaimsthatstudentsshouldbeencouragedtotalkaboutmathematics,todis‐cuss ideas, to write down thoughts and reflec‐tionsandtopresentresults.

Digital competence. KeyCoMath supports thatstudentsshouldbeengagedinlearningenviron‐ments that utilize digital media (e.g. spread‐sheets,dynamicgeometry,computeralgebra).Byworking mathematically they develop a confi‐dent,criticalandreflectiveattitudetowardsICT.

Learning to learn.KeyCoMathrecommends thatemphasisshouldbegiventostudents'self‐regu‐lated and autonomous learning. Thus, studentsshoulddevelopabilitiestomanagetheirlearning(both individually and in groups), to evaluatetheirworkandtoseekadviceandsupportwhenappropriate.

Social competences. KeyCoMath argues thatteachingofmathematicsshouldprovidestudentsopportunities to collaborate and communicate.They cooperatively do mathematics, discussideas, present findings and have to understanddifferentviewpointstoachievecommonmathe‐maticalresults.

Sense of initiative.KeyCoMathsupportsthatstu‐dentsshouldbeencouragedtobecreative,proac‐tiveandtoturnideasintoactionthoughexplora‐tory, inquiry‐based learning in mathematics.

Through this process students might developabilities to plan, organize, and manage theirwork.

4.2 The Proposed Framework

Inthefollowing,weoutlinethewayinwhichtheKeyCoMathprojectinvestigatedalltheaboveKeyCompetences and Mathematical practices andreachedanewframeworkforKeyCompetencesthat should be developed in the mathematicsclassroom.Thepurposeof theproposed frame‐workistosynthesizethetwostrandsofresearchinthedomainofkeycompetences,i.e.,theEuro‐peanframeworkofkeycompetencesandthere‐centefforts inmathematicseducationtoexplic‐itly define important mathematical practices.Thus,inanattempttoprovideapracticalframe‐workthatdefinesmathematicalpracticeswhichcontribute in enhancing students’ key compe‐tences, we suggest a five‐pillar comprehensivedescriptionofmathematicalpracticesandappro‐priatedidacticalapproaches.

Explore problems and persevere in solving them. Wesetofffromtheprinciplethatonelearnsalotofthingsinmathematicsifhe/sheisgiventheop‐portunitytosolvealotofproblems.Workingonproblems teaches students the importance ofpersistence when dealing with a mathematicalproblemandtheimportancetodevelopreasona‐ble arguments. Through problem solving stu‐dentslearntolearn.Through the exploration of problems studentsmayalsodevelopasenseofinitiativesincediffer‐ent approaches may be used. Students may beaskedtosolveaproblemorcarryoutanassign‐mentwherelimitedornospecificdirectionsaregiven.Suchscenarios,forcestudentstotaketheirowninitiative.

Approaches Useexplorationactivities:Mathematicsdis‐

ciplineisagoodenvironmentforexploring.This gives students the opportunity to ex‐ploreandlearnbythemselves.

Offerplethoraofproblems:Studentsshouldbe given the opportunity to solve differenttypes of problems, for instance, easy andhardproblems,problemsthatcanbesolvedifyouareworkingbackwards.Theyshouldtrytounderstandthemeaningofproblems,toanalyzeinformation,toidentifytheentrypointfortheirsolution.

Engagestudentswithproblemswithdiffer‐entsolutions.

Offerproblemswhichcanbesolvedwithdif‐ferentapproaches.

Useopenproblems. Askstudentstomakeconjectures:Students

shouldtrytomakeconjecturesandplanap‐proachesforsolutions.

Givestudentstime:Itisimportanttostressthat learning to learnneeds time, and timeneeds to be allocated to this purpose. Stu‐dentsshouldbegiventhetimetoreflectonactivitiesandlearnbythemselves.

Askstudentstoreflectonactivitiesthattheycarried out: Through problem solving stu‐dentsshouldbeaskedtoreflectontheirownlearning,monitorandevaluate theirproce‐duresandaskthemselveswhethertheirso‐lutionmakessense.

Communicate in mother tongue, construct viable arguments and critique the reasoning of others. Oneofthemainconcernsofmathematicsteach‐ingshouldbetheflexibletransitionfromevery‐daylanguagetothelanguageusedinmathemat‐ics: Eachchildhastohavethepossibilitytoex‐

press his/her own thoughts and feelingsthroughamathematicalapproach.

It is a long journey to reach formalmathe‐matics.Thismaystartfromearlyyearswhenstudentsareaskedtoaddforexample3+2.Insuchcasesstudentsmaybeaskedtotellastory which represents this mathematicalsentence. Teachers should try to break theseparation that exists betweennatural lan‐guageandmathematics.

When using mathematical language, stu‐dents need to realize that they need to bepreciseandaccurate.Theaimwillbetobringprecision in their language.This is of greatessenceandnecessarywhenstudentsmakeatransitiontomoretypicalandformalmath‐ematics.

Whenreferringtocommunicationitneedstobestressedthatthisshouldbeachievedbothinwrittenandspokenlanguage.

Communicationinmothertongueshouldbeused in the classroom for students to com‐municate and understand their teacher aswellastheirpeers.

Approaches Askstudentstoproducea learningjournal:

Students may be asked to write a journal

about the mathematics they learn in theclassroom.

Requirefromstudentstoproduceaposterorpresentation: Students should be given theopportunity to present their mathematicalapproaches, solutions,projects toothers inthe form of a presentation or a poster. Ofcourse,suchactivitieswillalsoenhancestu‐dents’ social and communication skills aswellastheirmathematicalskills.

Engage students to activities that requiremathematizationofeverydaylife:Suchactiv‐itiesaimtoconnectmathematical languagetomothertongueandeverydaylife.Throughtheprocessofmathematizationstudentsareasked to represent a real‐world situationsymbolically (inmathematics language) bydescribing,identifying,formulatingandvis‐ualizing themathematical problem in theirownway;andthenmovingbackbymakingsenseofthemathematicalsolutionintermsoftherealsolution,includingthelimitationsofthesolution.

Askstudentstoreformulateamathematicalidea:Studentsareaskedtoexpressthesamemathematicalideaindifferentways.

Offerstudentsthepossibilitytoworkonpro‐jects:Studentsmaydevelopcommunicationintheirmothertongueiftheyareworkingontheirownprojects.

Ask students to communicate and defendmathematicalideaswiththeuseoflanguage,symbols,diagrams,actions.

Ask students to listen and read carefullyotherpeople’sarguments.

Askquestionstoclarify issuespresentedtostudents.

Askstudentstodecidewhethercertainargu‐mentsmakesensetothem.

Onemay think that communication ina foreignlanguagemaybesomethingthatdoesnothaveaplace in the mathematics classroom. However,this is far fromthetruth.Foreign languagemaybeawayofdescribingthecommunicationinan‐other languagewhich students are not familiarwith.Thiskeycompetenceisnecessarytobede‐velopedwhen students are asked to communi‐catewithaprogramminglanguage.

Approaches Learningtocommunicatewiththeprogram‐

ming language: Students may be asked towrite or read something in programming

language.Ofcoursethismaybedoneinvar‐iouslevelsofeducationandinvariouslevelsofdifficulty.

Socialcompetencemaybedevelopedthroughtheinteraction and communication that studentshaveintheclassroombothwiththeirpeersandtheirteacher.

Approaches Ask students to listen, talk, write, under‐

stand,andcommunicate:Studentsshouldbegiven the opportunity to communicate inverbalandwrittenform.Theyshould learnhowtolistencarefullytoother,totalkwithprecisionandmakethemselvesunderstoodbyothers.

Requirefromstudentstoacceptdifferentso‐lutions and respect arguments: Studentsmustlearntorespecttheopinionsandargu‐mentsprovidedbyotherpeopleandalsoac‐ceptdifferentsolutions.

Engage students in co‐actions when com‐municating.

Require from students accuracy in theircommunication: Insist on accuracy in vari‐ousactivities.

Apply dialogic learning in classrooms:Teachingshouldinvolvetheapproachofdi‐alogiclearning.

Model with mathematics, contextualizing and de-contextualizing everyday situations. Students should be able to usemathematics todealwithproblemsthatariseineverydaylife,so‐ciety and workplace. They should be able tomodel,contextualizeanddecontextualizeevery‐day situations. For instance, in early years thismaystartwithsimplystatinganequationtosolveamathematicalproblemwhilelaterstudentsmayhavetoconstructmorecomplexmodelstosolveaproblemwhichmayinvolvesituationssuchasdecision making, system analysis and troubleshooting.

Approaches Askstudentstoapplytheirknowledgeinreal

worldproblems. Require from students to make sense of

quantitiesandrelationshipsinproblems. Askstudentstousedifferenttypesofmod‐

els, representations, graphs, diagrams torepresent various relationships amongquantities,concepts,shapes.Createcoherentrepresentations that facilitate to solve aproblem.

Offerstudentsthepossibilitytocontextual‐izeanddecontextualizemathematicalideas.Be able to represent concrete situationssymbolicallyandbeabletounderstandanddealwith themand the reverse, be able tounderstandsymbolsandtranslatethemintoreallifeconcretescenarios.

Offer activities to students in order to be‐come able to understand and translate ab‐stractideasintosymbolsandbecomeabletousethem.

Use appropriate tools strategically. Students with mathematical proficiency shouldbeable tomakedecisions regardingwhich tooltheyneedorisappropriatetocarryoutaspecifictask.Toolsmaybesimplypaperandpencil,rul‐ers, protractors, calculators, mathematics soft‐ware. Nowadays, the ability to use digital toolsstrategically is one of the central competencesthatindividualsneedtodevelop.Digitalcompetenceallowsstudentstodevelopin‐tuitions about various mathematical concepts.Students need to know and also use variousmathematicalprograms.Furthermore,whenre‐ferring to digital competence it is important toraise the issue that studentsneed to learnhowvariousmachineswork.

Approaches Offerstudentsthepossibilitytodecidewhich

toolstouseandhowtousethem,forexam‐ple,tools,suchaspaperandpencil,calcula‐tor, ruler, digital technologies, protractor,compass.

Train students to use any kind of softwaresystem: Asking students to work with anykindofsoftwaresystemoftenincreasesandimproves students’ mathematical under‐standing.Thismayallowstudentstobettervisualize,compareandpredictresults.

Offerstudentsthepossibilitytousemodernsoftware systems: Students must be giventheopportunity toworkwithmodernsoft‐waresystems.

Engage studentswith virtual reality activi‐ties inmathematics: Studentsmust also begiventheopportunitytoworkindevelopingmathematical ideas with the use of virtualreality.

Look for and make use of structure and generali-zations, attend to precision. Studentsshouldbeable to identifyapatternorstructure in what they see inmathematics and

alsobeabletofindcommonalitiesandreachgen‐eralizations or find shortcuts in procedures. Intheprocessofdoingtheseaswellasintheprod‐ucts of their mathematical activities studentsshould be able to communicate with precisionandaccuracy.

Approaches Ask students to look for patterns or struc‐

ture. Require from students to recognize that

quantities can be represented in differentways.

Give students the opportunity to use pat‐ternsorstructurestosolveproblems.

Offerstudentsthepossibilitytoviewcompli‐cated concepts either as single objects orpartofcompositionsofseveralobjects.

Givestudentstaskswheretheyneedtono‐ticerepeatedpatterns,calculationsormeth‐ods and look for general methods orshortcuts.

Askstudentstoreflectandevaluatetherea‐sonablenessofresultsandmakegeneraliza‐tions.

Requirefromstudentstocommunicatepre‐ciselyandwithaccuracy.

Ask students to state the meaning of con‐cepts,symbols,andcarefullyspecifyunitsofmeasure.

Require from students to calculate accu‐ratelyandefficiently,tocarefullystateexpla‐nationsandprovideaccuratelabels.

5 Key Competences and Assessment

In recent years great emphasis is given on thewaysusedtoassesskeycompetences.Gordonetal.(2009)foundthatthemostoftheEUmemberstates implemented curricula that include com‐petences and identified assessment as one oftheirmostimportantcomponents.Inparticular,assessmentmightgive informationabout learn‐ingprocess,can leadtothedevelopmentofkeycompetencesandmaysupportconsequentlyef‐fective changes (European Commission, 2012).Indeed, ‘Theassessmentofkey competencesorsimilar learning outcomes that emphasize notonlyknowledgebutalsoskillsandattitudesinre‐lationtocontextsintendedaspreparationforlife‐longlearning’(EuropeanCommission,2012p.4).Despitetheimportanceoftheassessment,theEu‐ropean Commission/EACEA/Eurydice report(2012),declaresthatmostcountriesusenational

teststoassessonlythreeoutoftheeightkeycom‐petences(communicationinthemothertongue,communicationinforeignlanguagesandmathe‐matical competence and basic competences inscienceand technology). Inaddition to this, thenationalassessmentsinthemostcountriesfocusonthebasicskills(mothertongueandmathemat‐ics)andinsomeofthemfocusonthescience,for‐eignlanguagesandsocialandciviccompetences(European Commission/EACEA/ Eurydice,2012).Forthis,thereisaneedtorevealassess‐mentmethodsforthekeycompetences.

The first step towardsassessment is theopera‐tionalization of the key competences, as it hasbeen done in previous sections (Gordon et al.,2009).Bydefiningthekeycompetencesandthescopeoftheassessment,aswellbydiscussingthelearningoutcomesmightgivean insight for theassessment criteria (Sadler, 1987;Wolf, 2001).Furthermore,bydescribingthecharacteristicsoftheassessmenttasksandtheanalysisoftheirre‐sultsinthelearningprocess,studentsandteach‐erscanidentifyandappraiseperformances.

KeyCoMathprojectproposedfourcharacteristicsthat tasks for measuring students key compe‐tencesshouldhave:(1) include important mathematical aspects

withoutcomplexity,(2) allowtheemergenceofdifferentsolutionsor

processes,(3) encourage and request pupils’ productions

(drawings,reasons),(4) demandreflectionssuchasdescriptions,ex‐

planationsorreasons.

First,astheaimoftheassessmenttasksistore‐veal solver’s cognitive processes and results, asuitabletaskshouldbeclearlyspecified,easytounderstand,iscomprehensiblyformulatedanditfitstothecurrentlearners’stateofknowledge.Asimilar conception is mentioned by Harlen(2007)‘Acleardefinitionofthedomainbeingas‐sessedisrequired,asisadherencetoit’(p.18).Secondly, although the task is based on knownfactsitmightofferopportunitiestothesolvertolinkvariouscompetencies,tothinkcriticallyandtodealactivelyinasituation.Theexplorationofopensituationsoffersopportunitiestothesolvertoprovideavarietyofsolutionsordifferentwaysof approaches and problem‐solving strategies.Thesetypeoftasksallowspupilstoworkontheirownpaceandabilities,todevelopmathematicalcompetences on their individual level, forcingthemtoactivatetheirinitiativeandautonomy.

Thirdly,theuseofappropriatescenariossuchasoccupationalandsocialcontextsaskslearnerstooperatewith everyday life problems and situa‐tionsandadapttheirknowledgeindifferentsetofcircumstances(EuropeanCentrefortheDevel‐opmentofVocationalTraining,2010).Suchcon‐textsdemandsfromthesolvertoexploittheirso‐cial,civicandculturalawareness.Finally, appropriateassessment task shoulden‐couragetheactualuseandleveloflanguage,en‐hancing the competence of communication. AsGallinandRuf(1998)mentioned,theuseoflan‐guageenablesthoughtstobeclarifiedandhelpsaresponsetobeelicited.

Otherassessmentmethodswiththepotentialtoassesskeycompetencesareamongothersstand‐ardised tests, attitudinalquestionnaires,perfor‐mance‐based assessment, and portfolio assess‐ment,teacher,peerandself‐assessmentpractices(Looney,2011;Pepper,2013),asfollow:

Standardised tests:theymainlyusedfortheassessmentofsomeofthekeycompetences(communication,mathematicalcompetence,competencesinscienceandtechnology),duetothefactthatthesethreecompetencescanbedirectlylinkedtoindividualsubjects(Eu‐rydice,2009).Astheothercompetencesaremoregeneralandarerelatedtoseveralsub‐jects,difficultappropriatestandardisedtestmightbedeveloped(Pepper,2013).

Attitudinal questionnaires: these question‐naires aimed at capture learners’ attitudesfor learning to learn and generally for as‐sessing affective andmetaffective domains.However there is a difficulty of separatingcognitive and affective aspects of learning(Frederiksson & Hoskins, 2008). Addition‐ally a discrepancy between what studentsanswer andwhat they actually do appears(EuropeanCommission,2012).

Performance-based assessment: this type ofassessmentincludesportfolios,reflectivedi‐aries, experiments, group work, play,presentations, interviews and role plays(Looney,2011).Theobservationofsuchbe‐haviorsmightprovidereliableresults.‘Vari‐ationinteachers’judgementswithinandbe‐tweenschoolsisnonethelessarisk…’(Pep‐per,2013,p.18).

Classroom observation and dialogue is consid‐ered as an appropriate method for the assess‐mentofkeycompetences,incontrasttoquestion‐nairesandtests(Pepper,2013).Aninquirybasedlearningenvironmentmightoffersgreatoppor‐tunitiestothisdirection(Gallin,2012).Throughstudents experimentation in combination withteachers’ involvement several key competencesmayberevealedandconsequentlymightbeas‐sessed.Forinstance,asstudentsattempttofindoneormoresolutionstoagiventask,mathemat‐icalcompetence is inevitablyengaged.Acarefulselectionofataskmightalsoengagetheculturalawareness and expression competence as thegiven information or even the required onesmightbeconnectedwiththesocialandeconomicopportunitiesinculturalactivity.Additionally,ifateacherprovidesstudentswithdigital and electronic means as facilitators ofproblemsolvingprocessdigitalcompetencemayalso be developed. The use of digitalmeans, incombinationwithcooperativework,forceslearn‐erstodepictthecommunicationandsocialcom‐petences.

Exchanging thoughts among the learners, dis‐cussingideasandsolutionsbetweenteacherandstudents, keeping track of their thoughts, prob‐lems and findings, judging others solutions, ad‐vising theirpeersmightdevelopandassess thecompetencesof communication and learning tolearn.Moreover, the role of teachers is important toprovideappropriatefeedbackonremarkablein‐sights. Reflection and redirection of students’thoughtsmayenable them tobeawareof theirstrengths andweaknesses and consequently toidentifyfruitfuluseofthem(Gallin&Ruf,1998).

6 Conclusions

Summingup,theimportanceofkeycompetencesforlifelonglearninghasbeenwidelydocumented(e.g.Eurydice,2012;Otten&Ohana,2009).Basedontheprevioussectionsofthischapter,isobvi‐ousthatmathematicseducationmayachievethegoalofdevelopingkeycompetenceinasufficientleveland thusbeable tobuildamorequalifiedcitizen for the needs of today’s world. Citizenswhoareabletoexploreeverydayproblemswithperseverance,whoareabletocommunicateandcollaborate effectively, construct viable argu‐mentsandjudgethereasoningofothers,whode‐velopmodels,whoareabletousetoolsappropri‐ately and who look for and use structures and

generalizations,aresufficientlyequippedtocon‐frontcontemporarychallenges.

Besides the development of key competence,theirassessmentisofequalimportance.Thede‐velopment and assessment of key competencesshouldbedirectedlinkedinordertogainbetterresults in education. In particular, by revealingthe ways that lead to the enhancement of keycompetences,insightsaregivenforthedevelop‐mentofappropriateassessmentcriteria(Sadler,1987;Wolf,2001).Additionally, theassessmentprocessmightgiveinformationaboutthelearn‐ingprocess,andthusleadtothedevelopmentofkeycompetences(EuropeanCommission,2012).

Hence,theresearchprogramKeyCoMathcontrib‐uted in two strands: first, by defining the keycompetences and by proposing appropriate di‐dactical approaches for their development; sec‐ondly, by proposing assessment methods withthepotentialtoassesskeycompetences.Inordertoensurethat“keycompetences”areinthefocuspointofbothresearchersandeducators,itisim‐portanttoeducateteachersonthistopicandes‐peciallytrainthemaboutthewaysinwhichkeycompetences canbe interwovenwith their cur‐riculum.Therefore,furtherstudiescouldfocusondevelopingdidacticalconceptsthataimtomax‐imizetheimpactofkeycompetencesinthefieldofteachereducationand/orwaystomeasuretheeffectivenessofsuchprograms.

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