Developing Key Competences

by Mathematics Education

Carolin Gehring, Volker Ulm (Ed.)

Carolin Gehring, Volker Ulm (Ed.)

Developing Key Competences

by Mathematics Education

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Online Material Allworksheets forpupilsprinted in thisbookand furthermaterial formathematicseducationareavailabledigitallyat:www.KeyCoMath.eu ISBN978‐3‐00‐051067‐0©2015UniversityofBayreuth,GermanyDesignofthebookcover:CarstenMiller,BayreuthPhotoonthebookcover:©MaksimŠmeljov–Fotolia.comTranslationsandediting:AnnikaMüller,BayreuthTheproject“DevelopingKeyCompetencesbyMathematicsEducation”hasbeenfundedwiththesup‐portoftheLifelongLearningProgrammeoftheEuropeanUnion.Thiscommunicationreflectsonlytheauthor’sview.TheEuropeanCommissionandtheEACEAcannotbeheldresponsibleforanyusewhichmaybemadeoftheinformationcontainedtherein.

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Contents Introduction:TheEuropeanProjectKeyCoMath(Carolin Gehring,Volker Ulm) ...................................................................................................................................................... 5The Competence-oriented Approach to Mathematics Education KeyCompetencesforLifelongLearningandMathematicsEducation:ConceptsforTeachingand

LearningMathematicsinClass (Demetra Pitta-Pantazi, Constantinos Christou, Maria Kattou, Marios Pittalis,

Paraskevi Sophocleous) .......................................................................................................................................................... 7KeyCompetencesforLifelongLearning:ConceptsforInitialTeacherEducation (Mette Andresen, Helena Binterová, Carolin Gehring, Marta Herbst, Pavel Pech, Volker Ulm) ........ 18ConceptsforIn‐ServiceMathematicsTeacherEducation:ExamplesfromEurope (Stefan Zehetmeier, Manfred Piok, Karin Höller, Petar Kenderov, Toni Chehlarova, Evgenia

Sendova, Carolin Gehring, Volker Ulm) ........................................................................................................................ 23Learning Environments for Mathematics Education LearningEnvironments:AKeytoTeachers'ProfessionalDevelopment (Carolin Gehring, Volker Ulm) .......................................................................................................................................... 34LengthsofDaysovertheCourseoftheYear ModellingwiththeSineFunction (Julian Eichbichler) ............................................................................................................................................................... 36WhichShapeisbestforaHoneycombCell? OptimisationTasks (Marion Zöggeler, Hubert Brugger, Karin Höller) .................................................................................................. 42FunctionsandtheirDerivatives CoursesofFunctions,Monotony,andCurvature (Johann Rubatscher) ............................................................................................................................................................. 49ImportantLinesinTriangles AComputer‐AssistedLesson (Walther Unterleitner, Manfred Piok, Maximilian Gartner) .............................................................................. 51MeasuringDistanceandHeight FunctioningofApps (Marion Zöggeler, Hubert Brugger, Karin Höller) .................................................................................................. 53

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ThinkinginFunctions IntroductiontotheTopicofFunctions (Dorothea Huber) ................................................................................................................................................................... 56WoodStack–WhereistheLimit? DiscoveringtheHarmonicSeries (Aron Brunner, Johann Rubatscher) .............................................................................................................................. 66MovingtoaNewHouse ReflectingonProblemSolving (Panayiota Irakleous) ....................................................................................................................................................... 69AttheZoo PlanningaRoute (Eleni Constantinou) ......................................................................................................................................................... 73TheEnergyProblem WorkinginComplexReal‐LifeSituations (Panayiota Michael, Elena Sazeidou, Stella Shiakka) ................................................................................. 75

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Introduction:

The European Project KeyCoMath Carolin Gehring, Volker Ulm

Thisbookisaresultoftheproject“KeyCoMath–Developing Key Competences by MathematicsEducation”.IthasbeenfundedwiththesupportoftheLifelongLearningProgrammeoftheEuro‐peanUnionbetween2013and2015.1 General Aims Theproject KeyCoMath aims to developpupils'key competences in primary and secondaryschools.Didacticconcepts,teachingandlearningmaterial as well as corresponding assessmentmethods for mathematics education are devel‐oped,tested,evaluatedanddisseminatedontheEuropeanlevel.KeyCoMathusesthepowerofin‐itialandin‐serviceteachereducationtoputinno‐vative pedagogical and didactical approachesintopractice.2 Project Consortium KeyCoMathisbasedonthecooperationofeightpartnersfromeightEuropeancountries: UniversityofBayreuth(DE) BulgarianAcademyofSciences(BG) UniversityofSouthBohemia(CZ) UniversityofBergen(NO) UniversityofCyprus(CY) UniversityofKlagenfurt(AT) German Department of Education in South

Tyrol(IT) SchoolRottenschwil(CH)

3 Key Competences Keycompetencesarenecessaryforallcitizensforpersonal fulfilment, active citizenship, social in‐clusionand employability in today’s knowledgesociety.TheprojectKeyCoMathdevelops,imple‐ments,andevaluateswaysofworkingaccordingto the “European Reference Framework of KeyCompetences for Lifelong Learning” in mathe‐maticseducation.

Theactivitiesfocusonthefollowingkeycompe‐tences(basicskillsandtransversalcompetences)inprimaryandsecondaryschools: Mathematicalcompetence Communicationinthemothertongue Digitalcompetence Learningtolearn Socialcompetences Senseofinitiative

Furtherbackgroundanddetailsaredescribedinthefollowingsections.4 Target Areas Theprojectaimstoinnovatemathematicseduca‐tionontheEuropeanlevelinfourtargetareas. KeyCoMathaimsatfundamentalchangesof

pupils' learning.Ashifttowardsmoreactive,exploratory, self‐regulated, autonomous,communicativeandcollaborativelearningisintended. This way of doing mathematicshelpstodevelopabroadvarietyofkeycom‐petences.

These changes of pupils' learning require

andarebasedonchangesofteaching.Teach‐ersdevelopexpertisetocreatelearningen‐vironmentsinorderforpupilstoworkintheintended way and to develop key compe‐tences.

Assessment methods that correspond to the

competence‐oriented approach are devel‐oped, tested and evaluated. Pupils workmathematicallywithopen"learning/assess‐ment scenarios", they write down theirthoughts and findings, present and discussresults. Teachers are sensitized to use thisinformation to diagnose pupils' compe‐tences,toadaptteachingto learners'needsand especially to support pupilswith diffi‐culties.

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KeyCoMathcontributestoovercometheEu‐ropean problem of low‐achievers. Further‐more, the partners identify and developstrategies for integrating the European di-mensioninteachingandlearningmathemat‐ics.

5 Fields of Activity KeyCoMathoperatesinfourfieldsofactivity: The partners use the power of in-service

teacher education for initiating innovationsin mathematics education. Networks ofteachersareintensivelyinvolvedinthepro‐ject. They are made familiar with didacticconcepts for teaching, learning,andassess‐menttosupportpupils'keycompetences.

Initial teacher educationisthebasisoffuturemathematicseducation.Therefore,thepart‐nersimplementthedidacticalconceptsinin‐itialteachereducation.

Dissemination and exploitation strategieswith publications of didactic concepts andconcreteteaching/learning/assessmentma‐terial in Bulgarian, Czech, English, German,GreekandNorwegiancontributetothe im‐provement of mathematics education in schoolonthelargescaleinEurope.

Toenhancetheimpactonthelevelofeduca‐tional systems, project results are distrib‐utedamongdecision‐makers in politics, ad-ministration andeducational practiceaswellasinthescientific communityofmathematicseducation.

Learning/developmentof...

Teaching/supportof...

Assessmentof...

Europeandimensionof...

KeyCompetences

TargetAreasofKeyCoMath

In‐serviceteachereducation

Initialteachereducation

Mathematicseducationinschool

Administration,politics,science

KeyCompetences

FieldsofActivityofKeyCoMath

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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) CulturalawarenessandexpressionThesecompetencesareequallyimportantandin‐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,

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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

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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

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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.

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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.

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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

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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.

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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

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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:theymainlyusedforthe

assessmentofsomeofthekeycompetences(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).

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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.References Cobb,P.,Stephan,M.,McClain,K.,&Gravemeijer,K.(2011).

Participatinginclassroommathematicalpractices.InA.Sfard, K. Gravemeijer, & E. Yackel (Eds.), A journey in mathematics education research: Insights from the work of Paul Cobb(pp.117‐163).NewYork:Springer.

Common Core State Standards Initiative. (2011).Common core state standards for mathematics.America.Retrievedfrom http://www.corestandards.org/wp‐content/up‐loads/Math_Standards.pdf

CommonwealthofAustralia.(2009).Shape of the Australian Curriculum: Mathematics. Australia, Retrieved fromhttp://www.acara.edu.au/verve/_resources/australian_curriculum_‐_maths.pdf

Ernest, P. (2010). The social outcomes of learningmathe‐matics: Standard, unintended or visionary? In C. Glas‐codine&K.‐A.Hoad(Eds.),Conference Proceedings ACER: Teaching mathematics? Make it count: What research tells us about effective teaching and learning of mathe-matics(pp.21‐25).Australia:AustralianCouncilforEd‐ucationalResearch.

EuropeanCentrefortheDevelopmentofVocationalTrain‐ing(Cedefop).(2010).Learning outcomes approaches in VET curricula: A comparative analysis of nine countries.

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Luxembourg:PublicationofficeoftheEuropeanUnion.Retrievedfromhttp://www.cedefop.europa.eu/EN/Files/5506_en.pdf

EuropeanCommission.(2012).Education and training 2020 work programme. Thematic working group “Assessment of key competences”. Brussels: European Commission.Retrievedfromhttp://ec.europa.eu/education/policy/school/doc/keyreview_en.pdf

European Commission/EACEA/Eurydice. (2012). Develop-ing Key Competences at School in Europe: Challenges and Opportunities for Policy. Eurydice Report. Luxembourg:Publications Office of the European Union. Retrievedfrom http://eacea.ec.europa.eu/education/euryd‐ice/documents/thematic_reports/145EN.pdf

EuropeanCommunities.(2007).Key competences for lifelong learning: European reference framework. Luxembourg:OfficeforOfficialPublicationsoftheEuropeanCommu‐nities.Retrievedfromhttp://www.britishcouncil.org/sites/britishcoun‐cil.uk2/files/youth‐in‐action‐keycomp‐en.pdf

European Parliament & Council of the European Union.(2006). Key competences for lifelong learning(2006/962/EC).Official Journal of the European Union, L394,pp.10‐18.

Eurydice(2009).National testing of pupils in Europe: Objec-tives, Organisation and use of results.Brussels:EuropeanCommission.

Frederiksson, U., & Hoskins, B. (2008). Learning to learn: What is it and can it be measured?Ispra:EuropeanCom‐misionJRC.

Gallin,P.,&Ruf,U. (1998).Furtheringknowledgeand lin‐guisticcompetence:LearningwithKernelideas&jour‐nal.Vinculum, 35(2),pp.4‐10.

Gallin,P.(2012).Ontheequilibriumbetweenofferanduse‐a practical example from a Swiss upper secondaryschool:Theoffer‐and‐usemodelwithinthecontextofdi‐alogic. InP.Baptist&D.Raab (Eds.), Implementing in-quiry in mathematics education (pp.125‐132).TheFibo‐nacciproject:Disseminating inquiry‐basedscienceandmathematicseducationinEurope.

Gordon,J.,Halasz,G.,Krawczyk,M.Leney,T.Michel,A.,Pep‐per,D.,etal.(2009).Key competences in Europe: Opening doors for lifelong learners across the school curriculum and teacher education.Warsaw.

Harlen,W. (2007). Criteria for evaluating systems for stu‐dent assessment. Studies in Educational Evaluation, 33(1),pp.15‐28.

Hull, T. H., Balka, D. S., & Harbin‐Miles, R. (2011). Visible thinking in the K-8 mathematics classroom. ThousandOaks,CA:CorwinPress.

Looney, J. (2011).Alignment in complex education systems.OECDEducationWorkingPapers.Paris,OECD.

National Council of Teachers of Mathematics (NCTM)(2000). Principles and standards for school mathematics. Reston,Virginia:NCTM.

Otten,H.,&Ohana,Y.(2009).The eight key competencies for lifelong learning: An appropriate framework within which to develop the competence of trainers in the field of Euro-pean youth work or just plain politics? Europe: Salto‐Youth Training and Cooperation Resource Centre &iKAB. Retrieved from https://www.salto‐youth.net/downloads/4‐17‐1881/Trainer_%20Compe‐tence_study_final.pdf

Pepper, D. (2011). Assessing key competences across thecurriculum–andEurope.European Journal of Education: Research, Development and Policy, 46(3), 335‐353.doi:10.1111/j.1465‐3435.2011.01484.x

Pepper,D.(2013).KeyCoNet 2013 Literature review: Assess-ment for key competences. Key competence network on school education. KeyCoNet. Retrieved from KeyCoNetproject:http://keyconet.eun.org/c/document_library/get_file?uuid=b1475317‐108c‐4cf5‐a650‐dae772a7d943&groupId=11028

Sadler,D.R. (1987).Specifyingandpromulgatingachieve‐ment standards.Oxford review of education, 13(2), pp.191‐209.

Sullivan,P.(2011).Teaching mathematics: Using research-in-formed strategies.(AustralianEducationReview,n.59).Australia:ACERpress.Retrievedfromhttp://research.acer.edu.au/cgi/viewcontent.cgi?arti‐cle=1022&context=aer

VictorianDepartmentofEducationandEarlyChildhoodDe‐velopment of the Government in Australia. (2009). Key characteristics of effective numeracy teaching 7-10: Differ-entiating support for all students.Melbourne,Australia.Re‐trievedfromhttp://www.education.vic.gov.au/Docu‐ments/school/teachers/teachingresources/disci‐pline/maths/keycharnum710.pdf

Wolf,A.(2001).Competence‐basedassessment.InJ.Raven&J.Stephenson(Eds.),Competence in the Learning Soci-ety.NewYork:PeterLang.

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Key Competences for Lifelong Learning:

Concepts for Initial Teacher Education Mette Andresen, Helena Binterová, Carolin Gehring, Marta Herbst, Pavel Pech, Volker Ulm

1 Introduction Initialteachereducationisthefirststepputtingateacher’s career on course (Terhart, 2004). Fu‐tureteachersacquireprofessionalknowledgeaswellastheoreticalbasicknowledge(e.g.inpeda‐gogyandpsychology)forthesubsequentprofes‐sionalpractice(Neuweg,2004).Theyarelearnerandteacheratthesametime;onemightspeakofastudent’sdoubleroleduringtheinitialteachereducation: “being supported in learninghow toteach, and supporting pupils in how to learn.”(Caena, 2014) To apply their newly acquiredknowledgeinpracticalclasseswillbeachalleng‐ingtask(Keller‐Schneider&Hericks,2011).Thiscircumstanceshouldusefullybereflectedinthestructuringofinitialteachereducation.Asthepa‐rametersaresetbytheprofessionalpractice,itismentallydemandingtoanalyse,questionandre‐viewtheoreticalconcepts.Thefocusissetontheentityofaperson,includingtheirbehaviorsandattitudes, their knowledge and competences.(Caena,2014)Thedevelopmentofcompetencesnotonlyplaysanimportantroleforthedevelop‐mentof thepupils as futureworld citizens, butalsofortheteachers.Teacherstudentsandteach‐ersarerequiredtogaincompetencesandthusactasaninspiringexamplefortheirstudents.(OECD,2011)TheprojectKeyCoMath tackles the challengeofimplementing the “European Reference Frame‐work of Key Competences” (see previous chap‐ter)inmathematicseducation.Theprojectdevel‐opsmeanstoputthisReferenceFrameworkintopractice in schooland to increase its impactontheeducationalsystem.Initialteachereducationis the basis of future mathematics education.Thus, the partners implement didactical ap‐proachesforsupportingkeycompetencesinini‐tial teacher education. Teacher students shoulddevelop corresponding professional compe‐tences.

2 Didactical Approaches Theprojectactivitiesininitialteachereducationfocusonthefollowingsixkeycompetencesanduse specific didactical approaches to supporttheminmathematicseducation:Key Competence

Didactical Approach

Mathematicalcompetence

KeyCoMathpromotespupils’active,exploratorylearninginopenandrichsituationstodeepentheirabilityofmathematicalthinking.

Communica‐tioninthemothertongue

KeyCoMath closelyintertwinesdoingmathematicsandcommunicatingwithothersorallyorinwrittenform.Pupilsareencouragedtotalkaboutmathematics,todiscussideas,towritedownthoughtsandreflections,andtopresentresults.

Digitalcompetence

InKeyCoMathpupilsworkwithlearningenvironmentsthatincludedigitalmedia(e.g.spreadsheets,dy‐namicgeometry,computeralgebra).Byworkingmathematicallytheyshoulddevelopaconfident,criticalandreflectiveattitudetowardsICT.

Learningtolearn

KeyCoMathemphasizespupils’self‐regulatedandautonomouslearning.Thus,theydevelopabilitiestoman‐agetheirlearning–bothindividuallyandingroups–,toevaluatetheirwork,andtoseekadviceandsupportwhenappropriate.

Socialcompetences

KeyCoMathfosterspupils’collabora‐tionandcommunication.Theydomathematicscooperatively,discussideas,presentfindingsandhavetounderstanddifferentviewpointstoachievemathematicalresults.

Senseofinitiative

KeyCoMath strengthensexploratory,inquiry‐basedlearninginmathemat‐ics.Pupilsareencouragedtobecrea‐tive,proactive,andtoturnideasintoaction.Theydevelopabilitiestoplan,organize,andmanagetheirwork.

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3 Strategies for Initial Teacher Education

TheuniversitiesparticipatinginKeyCoMathim‐plementthedidacticalapproachesofthetableinsection 2 in initial teacher education. They ar‐range seminars and lectures where universitystudentslearnhowtosupportpupils’keycompe‐tences.Mathematical expertise Mathematical knowledge is abasicprerequisiteforfuturemathematicsteachers.Evenifsomeel‐ements of higher mathematics play no role inteachers’ practices at work with pupils, someotherelementsstillneedtobeincludedinteach‐ers’ educatione.g.polynomials, analyticgeome‐try, derivatives, integrals, lines and surfaces inspace,etc.Nevertheless,oneshouldlookforcon‐nectionsandexamplesoftheapplicationofcon‐cepts whenever it is possible. Mathematicscoursesneedtobesetinsuchwaythatuniversitystudentsareabletogainanunderstandingoftherole of school mathematics in contemporarymathematicsscience.Furthermore,initialteachereducationshoulden‐ableteacherstudentstoobtainadeeperunder‐standingofbasicelementsofmathematics,suchasthechangebetweenformsofrepresentationortheappropriateuseofthelanguageoflogic,e.g.words"and","or","if...then","all","some",etc.Experiencing mathematical processes In initial teachereducation,attentionshouldal‐waysbepaidtoensurethatuniversitystudentsgain(more)experiencetodomathematicsintheform of observation, comparison, experiments,attempt, analysis, analogy, generalization, syn‐thesis,induction,anddeduction.Theoretical background Inseminarsandlectures,universitystudentsareacquainted with general ideas and theories ofteaching and learning, they are made familiarwith the didactic concepts for supporting keycompetencesandtheydiscussandreflectoned‐ucationalprocesses.Subject-matter didactic competences In addition to introduction tomathematics the‐ory, student teachers need to know the entiresubjectmatterofschoolmathematicsaswellasthedidactictransformationofit.Thismeansthatinaccordancetothegoalsofmathematicseduca‐tion,teachersneedtodeeplyunderstandthetop‐icsofschoolmathematics,suchassets,measures

andmeasurement,arithmetic,algebra,geometry,functions, probability etc. These topics are spi‐rallydevelopedthroughoutthegradelevels.Forthisreason,itisveryimportantforfutureteach‐erstoknowhowtotransformthecontent fromthese fields ofmathematics into forms that areappropriateforlearninginparticularpupils’agelevels.Learning/assessment scenarios as a tool for professional development Tobridge thegapbetween theoryandpractice,theuniversitystudentslearnbydoing:Theyap‐plythegeneralconceptstospecificsituationsofmathematics education and design correspond‐ing“learning/assessmentscenarios”.These“sce‐narios"shouldsupportpupilsinworkingaccord‐ingtothedidacticalconceptsdepictedinthetableinsection2andthus,todevelopavarietyofkeycompetences.Furthermore,theyshouldprovidefeedbackfortheteacherandthelearnerondefi‐cienciesandproficiencywithrespecttothekeycompetencesaimedat.Taxonomy of Educational Objectives Todevelopsuchsuitabletasksfor"learning/as‐sessment scenarios" is a challenge not only forteacherstudents,butalsoforteachers.Itisveryimportant for themtobecomeawareof thede‐greeoftaskdifficulty.OneclassificationschemeforcategorisingtypesofquestionsisprovidedbyBenjaminBloomwithhistaxonomy(seee.g.Al‐ford,Herbert,Frangenheim2006).Thesecategoriescanbedescribedinthefollow‐ingway: Knowledge: Recalling information, dis‐

covering, observing, listing, locating,naming

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Comprehension:Understanding, translat‐ing, summarising, demonstrating, dis‐cussing

Application: Using and applying know‐ledge, using problem solving methods,manipulating,designing,experimenting

Analysis: Identifying and analysing pat‐terns, organising ideas, recognizingtrends

Synthesis: Using old concepts to createnewideas,designingandinventing,com‐posing, imagining, inferring, modifying,predicting,combining

Evaluation: Assessing theories, compar‐ing ideas, evaluating outcomes, solving,judging,recommending,rating

Withtheaidof this taxonomy, teacherstudentscanreflect theirprepared“learning/assessmentscenario”:Which level of task is it? Is it the in‐tendedone?Howcanthetasksberephrasedtotargetahigherorlowerrequirement?Corresponding methodical concepts Forthepracticalrealizationof“learning/assess‐ment scenarios” in class, methodological ap‐proachesareessentialaswell.Thus,inuniversitycourses e.g. the teaching principle “I‐You‐We”,developed by Peter Gallin und Urs Ruf, is im‐parted(Gallin,Ruf2014).Itisanidealentryde‐signforcooperativelearning:Afteraphaseofin‐dividualanalysis,thepupilsexchangetheirviewwithapartner.Intheplenarytheresultswillbepresentedandreflected.Experiences in practice in school Wheneverpossible,theseactivitiesarerelatedtopracticalcoursesinschoolwheretheuniversitystudentsworkwithpupils.Reflection in teams Allparticipantspresent,discussandreflecttheir“learning/assessment scenarios” with corre‐spondingpedagogicalanddidacticalideasaswellastheirexperiencesfromtheworkinschoolco‐operativelyinuniversitycourses.Refinement of materials Finally, after all discussionsand reflections, the"learning/assessment scenarios” are refined bythestudentsonthebasisofallexperiences.Goodlearning/assessment scenarios that match thequalitystandardsofKeyCoMatharepublishedonthe project website and/or in further publica‐tions. It ismotivating for the students to know

thattheirworkisembeddedinaEuropeanpro‐jectandthattheirproductsarepublishedonaninternationallevel.4 Examples Thefollowingexamplesillustratetheideaofdis‐cussing and reflecting tasks with respect toBloomstaxonomy.4.1 Knowledge, comprehension –

understanding, discussing Decide if the geometric shape, which is shown below, is a rhomboid. Explain your answer. Inthis exercise,thepupilsareintentionallygivenavisualrepresentationofanon‐existentgeomet‐ricfigure.Theirtaskistomakeadecisionbasedonpresentedpropertieswhether thepresentedfigureisarhomboid.Thepupilsaretoapplythebasic knowledge of diagonals and lengths ofrhomboid’ssides.Usingthetransferofwhathasbeenlearnedandconsideringwhattheyalreadyknow,basedonthisalreadyexistentmentalrep‐resentation,theyshouldcomparethepresentednonverbalelementtotheincorrectdataanddis‐coverthereasonfornon‐existenceofthefigure.Thecorrectansweristhediscoverythatthefig‐urecannotbearhomboidwiththegivenlengthsofthesides,sincethediagonalsbisecteachotheratarightangle.4.2 Comprehension, application –

understanding, applying knowledge Decide and write down what kind of geometric object is in the picture below. (Square, triangle, rhombus, rhom-boid, rectangle, tetrahedron, parallelogram, trape-zium.)

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a) b) Thisexercisepresentstwovisualrepresentationsofaquadrangletothepupils.Pupilshavetode‐cidewhatkindsofquadranglesareinthepicture.Statedvisualrepresentationsarearhomboidandatrapeziumwhoseshapesaremoreevocativeofarectangleandasquare.Thepupilshavetocon‐sideralloftheimportantpropertiesofthesevis‐ual representations and compare them to theircreated mental representations of a rectangleandasquare.Thisiswhereincorrectmentalrep‐resentations–basedonlyonshapesofgeomet‐ricalfiguresandnotoncharacteristicpropertieslikesizesofinnerangles,diagonalsbisectingeachother,oppositesidesbeingparallel–canbecomeevident.4.3 Analysis, synthesis – combination,

organisation of ideas Draw and name a geometric shape with the following properties: It is a geometric shape, whose opposite sides are parallel and congruent. Consecutive angles are different, but they are supplementary. Opposite angles are congruent. The diagonals of this shape bisect each other.

Inthisexercisepupilsarepresentedwithalistofgeometrical properties, basedonwhich thepu‐pilsaretosketchavisualmodelcorrespondingtoa geometric object thatmeets given conditions.Thepupilsaretousebasicpropertiesofaquad‐rangleconcerningitsinnerangles,diagonals,andoppositesides.Inthiscaseoftransferofwhathasbeenlearned,thepupilsshouldcreateacorrectmental representation of a geometrical objectthey have inmind and then express it visually.Thecorrectsolutionisasketchofarhombusorarhomboid.5 Concrete examples for initial teacher

education Atprojectmeetings,thepartnersexchangedex‐periencesfrominitialteachereducationandgaverecommendations for mathematics courses atuniversity.Lectures Forexample,theUniversityofBergen,Norway,usesthedidacticconcept“learningbyteaching”inlectures.Ineachsession,twouniversitystu‐dentshavetoprepareacontributiontothelec‐ture. These two students present an aspect ofthelectureforabout20minutes.Thisformsanintegralpartofthelecture.Tutorials Implementing a tutor system appears to be agoodwaytodownsizebiguniversitylectures.Ingroupsofabout20people,thestudents workmathematicallyontheirleveland

gainexperienceswhenworkingmathe‐matically,theyexperiencemathematicsasaprocessofexperience,exploration,discussion, documentation, presenta‐tion,

reflect on activities, build a bridge togeneralaimsandstandards,extractgen‐eralideasofmathematicseducation,for‐mulategeneraldidacticideas,

apply these general didactic ideas tospecifictopicsonpupils’level,constructlearning environments for pupils ac‐cordingtothegeneraldidacticconcepts.

Theacademicsortutorshavethepossibilitytomakevideoanalysesandtostimulatethedevel‐opmentofstudents’ideas.

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Seminars Academics have awide creative leeway to or‐ganizetheirseminars.Fordevelopingkeycom‐petences,itishelpfulforstudentstoexperiencelearningscenariosbythemselvesalreadyatuni‐versity. Their fellow students could take apu‐pil’spositionandtesttheseassignments.Asanoption,anewmathematical topicwithadifficulty level between school and universitymathematics can be approached. In the firststep,theuniversitystudentsbringaphenome‐nonintoquestionandclarifyitandinthesecondstep, they transfer it into practical classes(“WhatdoIdointheclassroom?”).Inthisman‐ner,abridgebetweentheoryandpracticecanbebuilt.Afurtherexampleofapracticallyorientedsem‐inar can be seen at the University of Bergen,Norway.Here,videomaterialwithrealclasssit‐uationsisoftenusedtobeanalysed.Practical studiesInsights and initial professional experiencemakegoodsenseineveryrespectandthereisavarietyofimplementations.The German and Czech project partners e.g.havethesameprocedurefortheirteacherstu‐dents.Atthebeginningoftheirteachertraining,university studentsvisit classroomsas (guest)observers.Advancedteacherstudentscompleteauniversity‐relatedinternshipataschooldur‐ingoneuniversityterm.Itisveryvaluableinor‐dertocombinepracticalstudiesinschoolwithseminarsatuniversity.Intheseseminars,teach‐ingatschoolcanbeplannedandpreparedcoop‐eratively,learning/assessmentscenarioscanbedesignedinaprocessofcommondiscussionandallexperiencesfromthelessonsinschoolcanbediscussed and reflected with fellow students,the teachers at school and the university lec‐turer.Thesereflectionsandprocessesarecru‐cialfordevelopingthestudents’beliefsinmath‐ematicsandmathematicseducation.ThesituationisdifferentinSouthTyrol,Italy.Af‐tertheirprofessionalstudies,studentscontinuetostudyatuniversity(1.5daysperweek)extra‐occupationally, whilst working as teachers atschool.

Other activities TeacherstudentsoftheUniversityofSouthBohe‐mia regularly organizemathematical camps forpupilsunderthedirectionofmembersoftheFac‐ulty of Education. During their holidays, pupilscan takepart inmathematicalactivitiessuchasgeocaching, puzzling, paper folding etc. All ofthese activities are planned and supervised byuniversity students. On the one hand, this is agood possibility for students to stay in contactwith pupils and to get an impression of pupils’wayofworkingandthinking.Ontheotherhand,university students get the chance to try them‐selvesoutaslearningguidesandtotesttheirself‐preparedlearningmaterials.Summing up: All these efforts mainly aim tochange and develop the teacher students’ atti‐tudesandbeliefstowardsmathematicseducationandtheirroleinteachingandlearningprocesses.References Alford,G.,Herbert,P.,Frangenheim,E.(2006):Bloom’sTax‐

onomy Overview. Innovative Teachers Companion, pp.176‐224.ITCPublications.

Caena,F.(2014): Initial teacher education in Europe: an over-view of policy issues. http://ec.europa.eu/education/policy/strategic‐framework/expert‐groups/docu‐ments/initial‐teacher‐education_en.pdf

Gallin, P.,&Ruf,U. (2014):Dialogisches Lernen in Sprache und Mathematik.Seelze:Kallmeyer.

Keller‐Schneider,M.,&Hericks, U. (2011):Forschung zum Berufseinstieg. Übergang von der Ausbildung in den Beruf.In:E.Terhart,H.Bennewitz&M.Rothland(Ed.):Hand‐buch der Forschung zum Lehrerberuf, Münster:Waxmann,pp.296‐313.

Neuweg,G.H.(2004):Figuren der Relationierung von Lehrer-wissen und Lehrerkönnen. In: B. Hackl&H. G. Neuweg(Ed.):ZurProfessionalisierungpädagogischenHandelns.ArbeitenausderSektionLehrerbildungundLehrerbil‐dungsforschunginderÖFEB,Münster:LIT,pp.1‐26.

OECD (2011): Preparing Teachers and Developing School Learders for 21st Century – Lessons from around the world, BackgroundReportfortheInternationalSummitontheTeachingProfession.PISA:OECDPublishing.

Terhart,E.(2004):Struktur und Organisation der Lehrerbil-dung in Deutschland. In: S. Blömeke, P. Reinhold, G.Tulodziecki&J.Wildt(Hrsg.):HandbuchLehrerbildung,BadHeilbrunn:Klinkhardt,pp.37‐59.

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Concepts for In-Service Mathematics Teacher Education:

Examples from Europe Stefan Zehetmeier, Manfred Piok, Karin Höller, Petar Kenderov, Toni Chehlarova, Evgenia Sendova, Carolin Gehring, Volker Ulm

1 Introduction1 Teachers are considered to play a central rolewhenaddressingprofessionaldevelopmentpro‐grammes:“Teachersarenecessarilyatthecenterofreform,fortheymustcarryoutthedemandsofhighstandards intheclassroom”(Garet,Porter,Desimone,Birman,&Yoon,2001,p.916).Ingvar‐son,Meiers,andBeavis(2005)sumup:“Profes‐sional development for teachers is now recog‐nisedasavitalcomponentofpoliciestoenhancethe quality of teaching and learning in ourschools.Consequently,thereisincreasedinterestin research that identifies features of effectiveprofessionallearning”(p.2).Inthepast20years,newlyemergingchallengesfortheteachingprofessionhaveresultedinanin‐creaseddemandforcorresponding(new)profes‐sionalcompetencesandanadequateframework(Posch,Rauch&Mayr2009).Untilthe1990s,theroleofteacherswasmostlylimitedtoofferingpre‐determinedcontentsinamannerthatisasclearandillustrativeaspossi‐ble, toensuringdisciplineandassessingperfor‐mance. This “static” culture of school‐basedteachingand learninghascomeunderpressurein recent years, both with regard to stu‐dent/teacher interaction and how classroomwork is defined in substantive terms. Increas‐ingly,thenormsapplicableatschoolareatvari‐ancewith thewealthofextramuralexperienceschildren and adolescents gather. As a conse‐quence, thereareeffortsto furtherdevelopandinstildynamicmomentumintothestaticcultureofteachingandlearning.The conditions of work and employment havechanged:Graduatesareincreasinglyexpectedtohavemulti‐functionalskills.Theyshouldbeabletoperformconceptual,planningandsupervisorytasks.Higherdemandsarebeingplacedinterms1ThisintroductionisbasedonRauch,Zehetmeier&Erlacher(2014)andZehetmeier,Rauch&Schuster(inpreparation).

ofaself‐reliantdesignofworkprocesses.Andul‐timately–duetothegrowingcomplexityofworksettings– teamworkhasgained increasing im‐portance.Requirementssuchasthesehavecre‐ated a demand for what has been called “dy‐namic”skills,forself‐reliance,independentman‐agementanduseofknowledge,andasenseofre‐sponsibility.Thesocialisationwhichchildrenandadolescentshaveexperiencedbeforetheyenrolatschoolhasalteredsignificantlyinthepast30years.Familiestendtohavefewerchildren,whilsttheindividualchildplaysamorecentralrole.Therelationshipofparentsandchildrenhasshiftedtoonethatre‐volvesaroundpartnership.Whatisallowedandwhatisforbiddenismoreandmoretheoutcomeof a negotiating process between children andparents.Itiswiththisbackgroundofexperiencesthat children and adolescents come to school,projectingitontothisinstitution.Astheprevail‐ingworkcultureatschoolrestsonasocialmodelinwhichchildrenandadolescentsacceptauthor‐itariandecisionsofadults,theresultantpotentialforconflictisinevitablyhuge.Against this backdrop, teachers are facing chal‐lengesonaprofessionalandhumanscalewhichare novel in many aspects. KeyCoMath aims atchanges of pupils' learning. In particular, moreactive, exploratory, self‐regulated, autonomous,communicativeandcollaborative learning is in‐tended.Researchonthiskindoflearningusuallyfocuses on the students’ level. Questions liketheseareraised:Howcanstudents‘ learningbewelldefined?Howcanwedescribeandexplainstudents’ progress and difficulties? However,these changes of pupils' learning are based onchangesofteaching.Teachersdevelopexpertisetoarrangelearningenvironmentsinorderforpu‐pilstoworkintheintendedwayandtodevelopkeycompetences.Researchontheteachers’level

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includesquestionslike:Howcanteachersbesup‐ported in activities dealing with this kind oflearning?Whatdowelearnfromresearchwhensupportingteachersinimplementingtheselearn‐ingactivities?Teachersareconsideredtoplayacentral role in planning, implementing, and re‐searching professional development pro‐grammes.This section provides some insights and exam‐ples,howvariouscountriesdeal(t)withtheseis‐sues. In particular, examples fromAustria, Bul‐garia,GermanyandSouthTyrolareprovided.2 Austria: The IMST Project2 Whereasinthelastdecadesofthetwentiethcen‐turymanycountrieslaunchedreforminitiativesinmathematics and science instruction, or con‐cerning teaching and teacher education in gen‐eral,similarsystemicstepsinAustriadidnothap‐pen.Thisledtoabiggapbetweenintendedandimplementedinstruction,bothinschoolsandatteachereducationinstitutions.Althoughthepro‐motionofstudents’understanding,problemsolv‐ing, independent learning, etc. and the use ofmanifold forms of instruction and didactic ap‐proachesinmathematicsandscienceinstructionwereregardedasimportant,teacher‐centredin‐structionandapplicationofroutinesdominated.Retrospectively, it seems clear that the educa‐tional system needed an external impulse, andthis appeared in the late 1990s in the form ofTIMSS1995andlaterPISA.TheinitialimpulsefortheIMSTproject3inAus‐triacamefromtheTIMSSachievementstudy in1995. Whereas the results concerning the pri‐maryandthemiddleschoolwereratherpromis‐ing, the results of theAustrianhigh school stu‐dents(grades9to12or13),inparticularwithre‐gard to the TIMSS advanced mathematics andphysicsachievementtest,turnedouttobedisap‐pointing(andevokeddiscussionsoneducationalpractice,researchandpolicy,influencedbycriti‐cal reports in the media). The ranking listsshowedAustriaasthelast(advancedmathemat‐ics)andthelastbutone(advancedphysics)of16nations(seee.g.Mullisetal.1998,pp.129,189).Thisandotherindicatorsshowedthattheteach‐ingofmathematicsandscienceinAustrianeededashiftfrom“transmission”to“inquiry”.

2 ThisexampleisbasedonKrainer&Zehetmeier(2013)andZehetmeier&Krainer(2013).3IMST=originally,InnovationsinMathematicsandScienceTeaching(1998‐1999);later,Innovationsin

Asinmanyothercountries(seee.g.PrenzelandOstermeier 2006), the responsible ministry re‐actedtothesituation.InAustria,anationaliniti‐ativewiththeaimtofostermathematicsandsci‐enceeducationwaslaunchedin1998:theIMSTproject.Sincethen,thisinitiativehasundergoneseveraladaptionsandisstillrunning.IMSTwasimplementedinthreesteps:1.The taskof the IMST research project (1998–1999)wastoanalysethesituationofuppersec‐ondarymathematicsandscienceteachinginAus‐tria and towork out suggestions for its furtherdevelopment.Thisresearchidentifiedacomplexpictureofdiverseproblematicinfluencesonthestatus and quality of mathematics and scienceteaching: For example, mathematics educationand related research was seen as poorly an‐choredatAustrianteachereducationinstitutions.Subjectexpertsdominateduniversityteachered‐ucation, while other teacher education institu‐tionsshoweda lackof research inmathematicseducation. Also, the overall structure showed afragmentededucationalsystemconsistingoflonefighterswith a high level of (individual) auton‐omyandaction,but little evidenceof reflectionandnetworking(Krainer,2003;seesummarizedinPegg&Krainer,2008). 2. The IMST² development project (2000–2004)focusedontheuppersecondarylevelinresponsetotheproblemsandfindingsdescribed.Inaddi‐tion,itelaboratedaproposalforastrategyplanfortheministry,aimingatimprovingtheinquiry‐based learning (IBL) of STEM in secondaryschools.Thetwomajor tasksof IMST²were(a)the initiation, promotion, dissemination, net‐working and analysis of innovations in schools(andtosomeextentalsointeachereducationatuniversity);and(b)recommendationsforasup‐portsystemforthequalitydevelopmentofmath‐ematics,scienceandtechnologyteaching. Inor‐dertotakesystemicstepstoovercomethe“frag‐mentededucationalsystem”,a“learningsystem”(Krainer,2005)approachwastaken. Itadoptedenhancedreflectionandnetworkingasthebasicinterventionstrategytoinitiateandpromotein‐novations at schools. Besides stressing the di‐mensionsofreflectionandnetworking,“innova‐tion”and “workingwith teams”were twoaddi‐tional features. Teachers and schools defined

Mathematics,Science,andTechnologyTeaching(2000‐2009);since2010‐motivatedbyaddingGer‐manstudiesasonemoresubject‐InnovationsMakeSchoolsTop.

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theirownstartingpointforinnovationsandwereindividually supported by researchers and pro‐jectfacilitators.TheIMST²interventionbuiltonteachers’strengthsandaimedtomaketheirworkvisible (e.g., by publishing teachers’ reports onthewebsite).Thus,teachersandschoolsretainedownership of their innovations. Another im‐portant feature of IMST² was the emphasis onsupportingteamsofteachersfromaschool.3.TheIMST3 support system(infourstages2004–2006, 2007–2009, 2010–2012, 2013–2015, afifthstage2016–2019isinpreparation)startedto implementparts of the strategyplan, amongotherwaysbycontinuouslybroadeningthefocustoallschool levelsandtothekindergarten,andalsotothesubjectGermanlanguage(duetothepoorresultsinPISA).TheoverallgoalofIMSTisto establish a culture of innovation and thus tostrengthen the teaching of mathematics, infor‐mationtechnology,naturalsciences,technology,andrelatedsubjectsinAustrianschools(seee.g.Kraineretal.2009).Here,cultureof innovationmeansstarting fromteachers’ strengths,under‐standingteachersandschoolsasownersoftheirinnovations, and regarding innovations as con‐tinuousprocesses that lead to anatural furtherdevelopmentofpractice,asopposedtosingularevents that replace an ineffective practice (formoredetailsseee.g.AltrichterandPosch1996;Krainer2003).Forthe future, theministryexpressed its inten‐tiontocontinueIMST.Theoverallgoalissettingupandstrengtheningacultureofinnovationsinschoolsandclassrooms,andanchoring thiscul‐turewithintheAustrianeducationalsystem.3 Bulgaria: How can Teachers be

Supported in Inquiry-Based Mathematics Education with Focus on Key Competences

3.1 Background To create a class culture inwhich the teachersandthestudentscouldworkasaresearchteamusing the ICT in support of the inquiry‐basedlearning has been the goal of a long‐term re‐searchinBulgariadatingfromtheearly80’s.Thefirst attempts are related with the ResearchGrouponEducation (RGE)–havingcarriedoutaneducationalexperimentlaunchedbytheBul‐garianAcademyofSciencesand theMinistryof

Education in 1979 (Sendov 1987, Sendov, Fili‐monov, Dicheva 1987; Sendova 2011). It com‐prised29pilotschools(2%oftheBulgarianK‐12schools)anditsmaingoalwastodevelopanewcurriculumdesignedtomaketheuseofcomput‐ers one of its natural components. The guidingprinciplesofRGEwerelearningbydoing,guideddiscovery, and integrated school subjects. TheRGEexperimentranfor12years.SpreadingthepositiveexperienceofRGEonabroaderscaleatthetimeturnedouttobeverydifficultforvariousreasons–botheconomicandpolitical.However,evenwiththeseisolatedexperimentsthelessonslearnedwerevaluable–the learners’ and teach-ers’ creativity alike can be enhanced when pro-vided with appropriate environments.At a university level new courses were intro‐duced, reflecting the need to prepare teachersworkinginthestyleofprojectbasedlearningandguideddiscovery learningpromoted in theRGEschools,e.g.attheFacultyofMathematicsandIn‐formatics at SofiaUniversity the curriculum forfuturemathematicsteacherswasenrichedbythecourseTeaching Mathematics in Laboratory Type Environment (Nikolova, Sendova, 1995). Afteryearsofstudyingandreproducingverysophisti‐catedmathematicalfactstheseteachers‐to‐beex‐perienced situations in which they could say:“Look atmy construction!”, “Can you provemytheorem?” (Denchev, Kovatcheva, Sendova2012).Suchaspiritofdiscoverywasexpectedtobe transferred lateron to their students.Still, anegative tendency (noticednotonly inBulgariabutalsointhemostofEasternEurope)wastheslowdeclineoftheeducationalsysteminthe90’sand the early 2000’s (Denchev, Kovatcheva,Sendova2012).3.2 New Life for Inquiry-Based Learning WiththeadventofpowerfulICTandspeciallyde‐signededucationalsoftwareformathematicsex‐plorations awaywas opened for inquiry‐basedlearning (IBL) in a number of European coun‐tries. Mathematics was an especially importantdomainofIBLthankstothedevelopmentandthedisseminationofdynamiclearningenvironmentsinwhichvariousexperimentswithmathematicalobjectscouldbeperformedleadingtotheformu‐lationandverificationofhypothesesofthelearn‐ersthemselves.Activitiesofthiskindwerethefo‐calpointoftheinvolvementoftheBulgarianKey-CoMath team members in previous European

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projects,includingInnoMathEd, Fibonacci, Dyna-Math, Math2Earth, Mascil and Scientix(Kenderov2010;Kenderov,Sendova,Chehlarova,2012).Ourworkwithin‐serviceteachersisbasedontheunderstanding that for the teachers tobemoti‐vated theyshouldexperience thesame intellec‐tual pleasure we expect their students to un‐dergo.Theinquiry‐basedmathematicseducation(IBME)ispromotedbyourteamattwolevels–nationallyandlocally,inmajorregionalcentres.On a national level the promotion instrumentsareworkshops,seminarsandspecialsectionsofthenationalconferencesorganisedbytheUnionof Bulgarian Mathematicians. On a local levelIBME is promoted and supported by multipletraining andpresentation sessionsorganised infifteen Bulgarian regions with the help of localboards (Chehlarova, Sendova 2012; Sendova,Chehlarova2012).3.3 Activities and Resources in Support of

Inquiry Based Mathematics Education The activities of the Bulgarian KeyCoMath re‐searchteamintermsoforganisingeventsandde‐velopingeducationalresourcesembracethefourlevelsoftheIBL:Level1‐ConfirmationInquiry,inwhichstudentsconfirmaprinciplethroughanactivitywherebytheresultsareknowninadvance;Level2‐Structuredinquiry,inwhichstudentsin‐vestigateateacher‐presentedquestionthroughaprescribedprocedure;Level3‐Guidedinquiry,inwhichstudentsinves‐tigate a teacher‐presented question through aproceduretheydesigned/selectedthemselves;Level4‐Openinquiry,inwhichstudentsinvesti‐gateaquestiontheyhaveformulatedthemselvesthrough a procedure they designed themselves(Banchi,Bell2008;Sendova2014).Hereisashortdescriptionofsuchactivitiesandresources:PD courses (from 2 to 128 hours) Thesecoursesarebeingorganizedby the Insti‐tuteofMathematicsand Informaticsof theBul‐garian Academy of Sciences (IMI‐BAS) in theframework of European projects (InnoMathEd,Fibonacci, Mascil, KeyCoMath and Scientix), aswell as by sections of the Union of BulgarianMathematicians(UBM),bytheMinistryofEduca‐tionandScience,bypublishinghousesforeduca‐tional literature, and by PD centers. The maingoalofthecoursesisinharmonywiththemostrecent educational strategies for updating the

mathandscienceeducationintheECcountries:the development of key competences by imple‐mentingtheinquirybasedlearninginintegrationwith the world of work. These PD courses arebasedona teamwork(of the lecturersand theparticipants alike) and implement educationalmodelsadaptabletovariousschoolsettings.Thecrucialpartofthecoursesisfortheparticipantsto experience different stages and levels of IBLwithemphasisonkeycompetences.Each summer in the years 2014 and 2015 theKeyCoMathteamhasofferedtrainingcoursesforIBLforteachersinBulgariawhoarenotinvolveddirectlyintheprojectbutwanttolearnandusethemethodsofIBL.Theteachersworkonpeda‐gogicalproblemsrelatedwith: reformulatingmathproblemsinIBLstyleso

as to enhance the development of specifickeycompetences;

formulating their own math problems re‐flectingreal‐lifesituations,notsolvablewiththecurrentmathknowledgeofthestudentsbut allowing for explorations by means ofdynamicgeometrymodelsleadingtoagoodenoughapproximationofthesolution;

studyingandproposingmethodsfortacklingproblemswhichareunstructured,orwhosesolutionsareinsufficientorredundant;

solving “traditional problems” with “non‐traditional”data,forwhichtheuseofacom‐putingdeviceisnecessary;

applying game‐design thinking soas to en‐gagebetterthestudentsintheproblemsolv‐ing;

formulatingmorerelevantevaluationcrite‐riaforthestudents’achievements;

project‐basedworkwithpresentationoftheresults;

assessmentoflearningresourcesintermsofformationanddevelopmentofIBLskillsandkeycompetences(Kenderov,Sendova,Cheh‐larova2014;Chehlarova,Sendova2014).

Thecourseshavetwophases.Inthefirstphase(3daysinthebeginningofthesummer)theteach‐ersbecomeacquaintedwiththedynamicgeome‐trysoftwareGeoGebraaswellaswithplentyofexamplesofhowtousedynamicscenariosintheIBLstyle.Theparticipantsofthecoursesareas‐signedprojectsasa“home‐work”.Inthesecond“follow‐up” phase (with duration 1‐2 days andconductedattheendofthesummer)thepartici‐pants present the advancement in thework ontheirprojectinfrontoftheotherparticipantsinthecourse.

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PD events (seminars and workshops) in the frames of conferences Thekeyfeatureofthese eventsisthattheteach‐ersplayanactiveroleandactaspartnersinare‐searchteam–theysharetheirgoodpracticesinoral or poster presentations (sometimes jointlywiththeirstudents),work ingroupsonspecifictasks and present their ideas to the rest of theparticipants.TypicalexamplesincludetheScien-tix National ConferencewithintheNationalsemi‐narInquiry Based Mathematics Education,theDy-namic Mathematics in Educationconference(Fig.1),theseminarswithintheSpringconferencesofUBM, theregionalconferencesorganizedbyUBMsections, the International UNESCO workshopQED(Chehlarova2012;Sendova2015). Fig 1. The Scientix National Conference demon-strated good practices of teachers implementing IBL with emphasis on specific key competences The inquiry based learning, its connectionwiththeworldofwork,goodpracticesandproblemsdirectedatthedevelopmentofspecifickeycom‐petences,havebeen the focusofourworkwithin‐serviceteachers:Using specific learning scenarios in support of IBL Agood repositoryofsuchresourcesistheVirtual School Mathematics Laboratory (Fig. 2,http://www.math.bas.bg/omi/cabinet/) beingdevelopedbyIMI‐BAS(Chehlarova,Gachev,Ken‐derov,Sendova,2014),whichcontainsover800scenarioswithdynamicfilestransparentfortheusers. Theway teachers are encouraged to usetheseresourcesistostimulatestudentstobehavelike working mathematicians: to make experi‐ments,tolookforpatterns,tomakeconjectures,toverifythemexperimentally,toapply“what‐if”strategies so as tomodify/generalize the prob‐lem,andeventousethemasapreparationforarigorousproof.Todo thiswithout leaving theircomfortzone,theteachersentertheroleoftheir

studentsandexperiencethesametypeofactivi‐ties during our courses, and when working ontheirown.They firstuse thedynamic files sup‐porting the scenarios as a ground for explora‐tions.Thenextstepforthemistoproposeappro‐priatemodificationof the files forsimilarprob‐lems,ortousethemasamodelforcreatingoneoftheirownfromscratch.Thus,itwouldbequitenatural for them to tell the students: “I don’tknowtheanswer,but Ihope to find it togetherwithyou,thankstoYOURefforts,toourjointef‐forts…” Fig. 2. Virtual School Mathematics Laboratory: dy-namic files for the open problem on finding the lo-cus of a regular m-gon inscribed in a regular n-gon Building and developing competences necessary for the students to participate in new types of mathematics contests Examples are “Mathematics with a computer”and“Themeofthemonth”(Fig.3,seeKenderov,Chehlarova2014;Chehlarova,Kenderov2015). Fig 3. “Theme of the month”: an invitation for a long-term activity on a chain of math problems modeling real-life situations

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Mathematics performances Eventsraisetheawarenessofthegeneralpublicabouttheroleofmathematicsforenhancingchil‐dren’sscientificcuriosityandendeavortolearn.The examples include: Performance at the His‐toryMuseum in Stara Zagora, organizedby theUBM section in the town, performances duringthe Researchers’ Nights (2011‐2015), Sciencefestivals (in Italy, Romania, Greece). It is im‐portanttonotethattheteachersactasmultipli‐ersoftheIBLideasduringtheseeventsaswell–they participate with their students, and occa‐sionallyleadtheperformance. Fig. 4. Posters for math performances within Sci-ence Fairs and Researchers’ nights Individual work with teachers Itincludessupportforthedevelopmentofales‐son, educational materials, mathematical festi‐vals,courseprojects,peerreviews,andprepara‐tionofapedagogicalexperiment.Activities in support of the Open Inquiry The fourth level of IBL (the Open Inquiry) hasbeen promoted to reach teachers and studentsfromalloverthecountrywithpotentialtodore‐searchinmathematicsandinformatics.ThishasbeendoneintheframesoftheHigh‐SchoolStu‐dents’ Institute ofMathematics and Informatics(HSSI) where secondary school students work

(underthesupervisionofamentor)ontheirownprojects,focusedonthestudyofagivenproblemfromMathematics, Informatics and/or IT (Ken‐derov, Mushkarov, Parakozova 2015). The stu‐dentsdelivertheirresultsandfindings,inwrittenandoralform,infrontofajuryandpeersattwosessions – in January and April. A three weekSummer School is organized for the involvedteachersandthebestachievingstudents.Duringthe SummerResearch Schools special seminarsfor teachers are organized. During these semi‐narsexperiencedteacherssharetheirgoodprac‐ticesinmentoringstudentswithhighpotentialindoing research, andprofessional researchers inmathematicsandinformaticstogetherwithPhDstudents (HSSI alumni)deliver lectureson con‐temporarytopicsinthesefieldssuggestingpossi‐bletopicsforfutureresearchprojects.3.4 The Main Achievements A community of teachers who implement andspreadtheinquirybasedlearningofmathematicsand informatics has been created. Theypartici‐pateinpedagogicalexperimentsnotonlyasare‐ality‐proofofresearchersbutasmembersofare‐search team.These teachers implement,modifyanddevelopfromscratcheducationalresourcesin support of IBL, share their good practices atseminars,nationalandinternationalconferencesandinprofessionaljournals.Someofthemorgan‐izepubliceventsataschoolandregionallevelforpopularizingtheinquirybasedmathematicsedu‐cation.TeachersarealsokeyfiguresinorganizingthenewmathematicscontestsMathematics with a computer andTheme of the month, inmakingthemknowntoabroaderaudience.Thefurthergoalistoreachfasterandmoreeffec‐tivelylargergroupsofotherteachersandschoolstudentsinacquiringtheIBLapproachwithfocusonthedevelopmentofkeycompetences.TheactivitiesoftheHighSchoolInstituteofMath‐ematicsandInformaticsandtheteachersresultsareencouraging.For its15yearsofexistence ithasprovedthatresearchpotentialofthestudentsshouldbesupportedanddevelopedstartingataveryearlyagetogetherwiththedevelopmentofother important competences including teamworkandpresenting(orallyandinwrittenform)tovariousaudiences.Someofthestudents’find‐ingswereonsuchahighlevelthattheywerepub‐lished(andinsomecasesquotedbyspecialists)in mathematics research journals. The HighSchoolStudent’sInstituteplaysanimportantrole

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withrespecttothedisseminationofIBLbecausethe participating students (and their mentors)arecomingfromdifferenttownsinthecountry.AnotherpositiveeffectisthatanumberofHSSIalumni act as mentors (virtually and face‐to‐face),thuspassingthetorchontothenextgener‐ationofyoungresearchers.4 Germany: Multipliers Concept for

the Urban Network of Primary Schools

TheUniversityofBayreuth,Germany,developedapractice‐basedmultipliersconceptforanurbannetworkofprimaryschoolswith theaimtode‐velopmathematicseducationandtosupportpu‐pils’keycompetences.The idea is implementedon a local scale and addresses primary schoolteachersofthecityAugsburg.Theconceptisre‐alisedwithintheframeworkoftheEuropeanpro‐ject KeyCoMath. 28 primary schools are takingpartinaface‐to‐faceprofessionaldevelopment.4.1 Involved Parties Ontheurbanscale,theUniversityofBayreuthco‐ordinates,organisesandsupervisesactivitiesfortheprofessionaltrainingofprimaryschoolteach‐ers. It works directly with a group of qualifiedmaths teachers as coaches for teachers' profes‐sional development. These teachers with theiradvisoryfunctionformalinkbetweenuniversityandschoolandbetweenscienceandpractice,be‐causetheyareresponsibleforappointedmathe‐maticaltutorsfromprimaryschoolsintheentirecity.4.2 Organisation TheChairofMathematicsandDidacticofMathe‐maticsisregularlyincontactwiththelocaledu‐cationauthorityandwithninededicated teach‐erswhowereassignedtobecomeadvisorsduetotheirhighprofessionalexpertiseandtheirinno‐vationcapacity.Groupsoftwoorthreeteacher‐advisorsguide28primaryschools in theentirecity. Therefore, a classification into four schoolgroupsaccording to location (north, south, eastand west) has been assigned. Every primaryschoolappointsatleasttwomathematicaltutors–oneresponsible forthefirstandsecondform,the other for the third and fourth form. Hence,oneschoolgroupconsistsofaminimumoftwelve

mathematical tutors who will further intro‐duce/present newly gained acquaintances totheirlocalcolleagues.4.3 Functions TheteamattheUniversityofBayreuthhassev‐eralduties: theykeep in touchandmakeallar‐rangements with the local education authority.Teacher‐advisors are appointed and meetingsare summoned.Theyprovide an academic sup‐portforschoolsandmakefundsaswellaslearn‐ing and teachingmaterials available. Above all,they operate a hosting platform for the infor‐mationandmaterialexchange.Theninemathe‐matical teacher‐advisors meet the universityteam regularly and they organise and guideschool group meetings. There, they assist themathematical tutors in planning teaching units.Thetutorstakeanactivepartinthemeetingsintheirassignedschoolgroup.Theyprepareandre‐alise planned mathematical lessons and docu‐mentteachingapproaches.4.4 Activities Once a year, amajor event for all involvedpri‐mary school teachers takesplaceat theuniver‐sity.Suchameetingconsistsofalecturetoasu‐perordinate topic like developing key compe‐tencesbymathematicseducation.Theattendingteachersreceivetheoreticalinputandadoptitaf‐terwards inworkshops.The inclusionofpupils’utterancesandsolutionprocessesconditionedbythestudyofpupils’writtenexercises,calllogsorvideoshotsdemonstrateshowchildrenoperateinaspecificlesson’ssequence.In addition, every local primary school has thepossibility to book in‐house advanced trainingcoursesforitsteachingstaffaccordingtothere‐quirements. Normally, four school groupmeet‐ingsperschoolyearareorganisedbytheteacher‐advisors. For thesemeetings, themathematicaltutorshavetopreparethefollowing:Theyprevi‐ously realise a teaching unit in their own classand collect pupils' approaches. Above all, themembersofaschoolgrouporganisejointvisitstoclasses, compare their varying approaches ofteachingonthesamespecificmathematicaltopic,togetherreflectonone’sownworkandexchangetheir ready‐made experiences, wherebymathe‐maticseducationcanberefined.

30

Thefocusisnotsolelysetonspreadingteachingmaterialsandongivingrecommendationsforles‐sons.Theobjectiveof theproject lies toagreatdegree on developing the level of inner convic‐tionsandbeliefsteachershaveoftheirsubject,onteachingandlearningprocesses,andtheirroleinthelessons.Theseplayamajorrolefortheplan‐ningandimplementationoflessons(Blömekeetal.2010;Kunteretal.2011).4.5 Experience Thismultipliersconceptfortheurbannetworkofprimaryschoolshasprovenitselfoveryearsandwillbecontinuedinfuture. Acoordinatorsays:“It is encouraging to witness how mathematics education progresses at many primary schools in Augsburg, how teachers set out together to develop good mathematical assign-ments and how they bring teaching concepts of the common retrospect into question and thus improve them. By observing the children, I recognized their growing interest for mathematical issues, their creative dealing with numbers, patterns and struc-tures, a greater openness for mathematical obser-vations of everyday life and an increasingly safer handling with basic mathematical skills and abili-ties. The cooperation with school administration (education authority, administration) is successful because of good personal contacts.”Theinvolvedprimaryteachersworkasmultipli‐ersduring their leisure time, includinganaddi‐tionaleffortintheirdailyworkload,whichisun‐fortunately neither compensated nor squaredwith class hour reduction, but which demon‐strates the commitment of the participatingteachers–since"Staying still means taking a step back"(ateacheradvisor).Concerningtheobservationoflessons,itisnotal‐wayseasytofindavolunteerwillingtopresenttheown teaching approach to colleagues.How‐ever,thisriskiseffectivelypreventedbytheofferof a joint preparation of classes. "I can work through exciting new ideas and discuss problems with motivated colleagues. Instead of sitting alone in my room I have many like-minded people around me."(aparticipatingteacher)and"At the moment it's the only opportunity to get a direct look at the work of my colleagues. I can challenge myself with the implementation of the curriculum and at the same time receive inspiration through the teachers and their contact with students. Eve-

ryone learns from each other in a relaxed atmos-phere without the pressure of being judged." (ateacherwhoisonparentalleave)5 South Tyrol: Cooperative Design of

Learning Environments IntheframeworkofKeyCoMaththeGermanDe‐partmentofEducationinSouthTyrolorganisedateacher training for thedevelopmentof compe‐tence‐focusedtasksformathematicseducationatsecondary schools. In a series of meetings theteachersweremadeacquaintedwithpedagogicalanddidacticaltheoriesconcerning openquestions, networking, individuallearning, dialogiclearningand assessment.

The participants, working in teams, developedtasksfortheirownlessons.Theseweretestedintheclassesandthen,afteracommonreflection,optimised.Thetasksfocusedontheacquisitionofmathematicalcompetences,butalsooncommu‐nicative,socialanddigitalcompetences.Thebestpracticeexamplesofthetaskswerepublishedina printed book „Tasks for competence‐focusedteachingofmathematics“(Höller,Ulm2015)andonthewebsitewww.KeyCoMath.eu.Strategies for Teachers’ Professional Development Particularly valuablewas the commonwork ontasksintheteamofteachers–fromtheconcep‐tiontothereflectionandoptimisation.Therefore,theplanningoffurthertrainingsessionswillnotonlyincludecontentsandmethods,butalsowaysof cooperative learning. The cooperation of theparticipantswilltakeplaceas: teachingdevelopmentinpeergroups, commonlearningwithmutualobservation, exchanging views through e‐learning plat‐

forms.Thefocusliesonthedevelopmentofteachingbythecommondesignoflearningenvironmentsforstudents.Moreover,theissueofevaluationisin‐cluded.Thesebasicstrategiesforteachers’professionaldevelopment are shown by the following dia‐gram:

31

Fig. 5. Teachers’ professional development by working with learning environments 6 Conclusion IntheprojectKeyCoMathpartnersfromeightEu‐ropean countries developed concepts for inno‐vatingmathematicseducation.The focus layonsupportingstudents’keycompetences.Althoughtheapproachesinthedifferentcountrieshadtoconsidernationalframeworks,acommonpatternofthestrategiescanbeidentified.Aiming at Teachers Thekeypeopleforinnovationsinschoolaretheteachers. Their beliefs, motivation and profes‐sionalexpertisearecrucialforeverydayteachingand learning. Thus, KeyCoMath focused on theteachers’professionaldevelopment.Networks of Schools Since learning is a social process, regional net‐worksofschoolswereestablished.Theyofferedframeworksforteachers’exchangeofexperienceandfortheircooperativelearning.Coaches for Teachers’ Professional Development Theregionalschoolnetworkswereledbyacoachorateamofcoacheswhocouldbee.g.veryexpe‐riencedteachers,teachereducatorsorscientists.Theteachersweremadefamiliarwithgeneraldi‐dactical andpedagogical concepts.They relatedtheseideastotheirdailyworkatschool;theyde‐signedlearningenvironmentsfortheirstudents,andusedthemintheirclasses.Theteacherspre‐sented,discussedandreflectedtheirexperiences

cooperativelyintheirnetworkofschoolsguidedbytheircoach.Aiming at the Meta-Level Initiativesforsubstantialinnovationsoftheedu‐cational systemshouldaimat themeta‐level ofteachers’attitudesandbeliefs.Thisconcernse.g.theroleof the teacher, theroleof thestudents,thenatureofthesubjectsandgeneralaimsofed‐ucation.Development of Learning Environments Tobridge thegapbetween theoryandpractice,teachers individually and cooperatively devel‐oped learning environments for their students,workedwiththeminclass,optimizedthemonthebasisofallexperiences,andexchangedanddis‐cussedthemintheirschoolnetwork.Thus,byde‐signingandworkingwithspecificlearningenvi‐ronmentsteachersbecameacquaintedwithgen‐eral pedagogical ideas. Learning environmentsfunction as tools for systemic teachers’ profes‐sionaldevelopment.Areas of Activity Participating schools and teachers should con‐centrateononeorafewareasofactivity,e.g.ex‐ploratorylearning,promotingstudents’coopera‐tion,cumulativelearningorfosteringkeycompe‐tences. Such bounded fields of activity enableteacherstobeginwithsubstantialchangeswith‐out the risk of losing their professional compe‐tenceinclass.Universities as Innovation Centres Intheseprocessesteachersandcoachesreceivedguidance and advice from universities. Theyservedas innovationcentres for teachereduca‐tion. They provided regular and systematic in‐serviceteachereducationoffersandcoachedthecoaches.(Inter-)National Teacher Education Teachers and teacher educatorsweregiven theopportunity to exchange experiences with col‐leaguesandtoparticipateinprofessionaldevel‐opment offers on a regional or anational level.Theyunderstood thatproblems andnecessitiesofinnovationshavesystemiccharacterandcon‐cern the fundaments of education far beyondtheirownprofessionalsphere.Moreover,theyre‐ceivedideasforinnovationactivitiesfromalargecommunity.

Input

Cooperation

DevelopmentofLearningEnvironments

TeachinginClass

Reflection

32

References Altrichter,H., Posch, P. (1996):Mikropolitikder Schulent‐

wicklung.Innsbruck,Austria:Studienverlag.

Banchi,H.,Bell,R.(2008):The‐Many‐Levels‐of‐Inquiry,Sci‐enceandChildren,46(2),26‐29,2008.https://engage.intel.com/docs/DOC‐30979

Blömeke, S.,Kaiser,G., LehmannR. (2010):TEDS‐M2008.ProfessionelleKompetenzundLerngelegenheitenange‐henderMathematiklehrkräftefürdieSekundarstufeIiminternationalenVergleich.Münster:Waxmann.

Chehlarova,T.(2012):IfonlyIhadsuchamathteacher….In:Sendova,E.(Ed.):Re‐DesigningInstitutionalPoliciesand Practices to Enhance the Quality of Educationthrough Innovative Use of Digital Technologies –UNESCOInternationalWorkshopSofia,StateUniversityofLibrarystudiesandIT,2012,pp.135‐140.

Chehlarova,T.,Sendova,E. (2012): IBME in thesecondaryschool:Overviewandexamples inaBulgariancontext.In:Baptist,P.,Raab,D.(Eds.): ImplementingInquiry inMathematics Education, Bayreuth 2012. рр. 114‐124,ISBN978‐3‐00‐040752‐9

Chehlarova,T.,Gachev,G.,Kenderov,P.,Sendova,E.(2014):AVirtual SchoolMathematics Laboratory. In:Proceed‐ingsofthe5thNationalConferenceone‐learning,May17,2014,Rousse,Bulgaria,pp.146‐151.

Chehlarova, T., Kenderov, P. (2015): Mathematics with acomputer–acontestenhancingthedigitalandmathe‐maticalcompetencesofthestudents.In:Kovatcheva,E.,Sendova, E. (Eds.): UNESCO International WorkshopQualityofEducationandChallenges inaDigitallyNet‐workedWorld,Faleza2000,pp.61‐72.

Chehlarova, T., Kenderov, P., Sendova, E. (2015): Achieve‐mentsandProblemsintheIn‐serviceTeacherEducationinInquiryBasedStyle,ProceedingsoftheNationalCon‐ferenceon“EducationandResearchintheInformationSociety”,Plovdiv,May,2015,pp.21‐30.

Chehlarova,T.,Sendova,E.(2014):DevelopingCommunica‐tionCompetencesintheContextofMathematicsEduca‐tion,Proc.VCongres2014,Скопје,Македонија,рр.127‐235.

Denchev,S.,Kovatcheva,E.,Sendova,E.(2012):Educationintheknowledge&creativity‐basedsociety.InternationalConference “ICT in Education: Pedagogy, EducationalResourcesandQualityAssurance”(pp.22‐26).Moscow,RF: UNESCO Institute for Information Technologies inEducation.

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Garet, M., Porter, A., Desimone, L., Birman, B., Yoon, K.(2001): What makes professional development effec‐tive?Resultsfromanationalsampleofteachers.Ameri‐canEducationalResearchJournal,38(4),915‐945.

HöllerK.,UlmV.(Ed.,2014):Aufgabenfürkompetenzorien‐tierten Mathematikunterricht, Bozen: Deutsches Bil‐dungsressort.

Ingvarson,L.,Meiers,M.,Beavis,A.(2005):Factorsaffectingthe impact of professional development programs onteachers’knowledge,practice,studentoutcomesandef‐ficacy.EducationPolicyAnalysisArchives,13(10),1‐28.

Kenderov,P.(2010):InnovationsinMathematicsEducation:EuropeanProjectsInnoMathEdandFibonacci,In:Math‐ematicsEducation:EuropeanProjectsInnoMathEdandFibonacci.Union of Bulgarian Mathematicians39.1(2010):63‐72.http://eudml.org/doc/250965

Kenderov,P.Mushkarov,O.,Parakozova,B.(2015):15yearsHighSchoolInstituteofMathematicsandInformatics.In:ProceedingsoftheFortyFourthSpringConferenceoftheUnionofBulgarianMathematicians,SOK“Kamchia”.

Kenderov,P.,Chehlarova,T.(2014):Thecontest“VivaMath‐ematicswithacomputer”and its role for thedevelop‐ment of digital competence of the students Shumen,МАТТЕХ.2014.pp.3‐10(inBulgarian).

Kenderov,P.,Sendova,E.,Chehlarova,T.(2012):IBMEandICT–theexperienceinBulgaria.In:Baptist,P.,Raab,D.(Eds.):ImplementingInquiryinMathematicsEducation,Bayreuth2012.рр.47‐54,ISBN978‐3‐00‐040752‐9

Kenderov,P., Sendova,E., Chehlarova,T. (2014):Develop‐ment of key competences by mathematics education:The KeyCoMath European project (in Bulgarian), Pro‐ceedingsofthe43rdConferenceoftheUnionoftheBul‐garianMathematicians,Borovets,2‐6April,2014.pp.99‐105.

Krainer, K., Zehetmeier, S. (2013): Inquiry‐based learningforstudents,teachers,researchers,andrepresentativesofeducationaladministrationandpolicy:reflectionsona nation‐wide initiative fostering educational innova‐tions.ZDM‐The International JournalonMathematicsEducation,45(6),875‐886.

Krainer,K.(2003):Innovationsinmathematics,scienceandtechnology teaching (IMST2): Initial outcome of a na‐tion‐wideinitiativeforuppersecondaryschoolsinAus‐tria.MathematicsEducationReview,16,49‐60.

Krainer,K. (2005):Pupils, teachersandschoolsasmathe‐maticslearners.InC.Kynigos(Ed.),Mathematicseduca‐tionasafieldofresearchintheknowledgesociety.Pro‐ceedingsoftheFirstGARMEConference,pp.34‐51.Ath‐ens,Greece:HellenicLetters(Pubs).

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Pegg,J.,Krainer,K.(2008):Studiesonregionalandnationalreform initiatives as ameans to improvemathematicsteaching and learning at scale. In:K.Krainer, T.Wood(Eds.): Internationalhandbookofmathematics teachereducation, Vol. 3: Participants inmathematics teachereducation: Individuals, teams, communities and net‐works,pp.255‐280.Rotterdam,TheNetherlands:SensePublishers.

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34

Learning Environments:

A Key to Teachers' Professional Development Carolin Gehring, Volker Ulm

1 Introduction As itwas described in the previous chapters, acore element of the project KeyCoMath wasworking with learning environments. Teachersand teacher students were made familiar withgeneral didactical and pedagogical concepts forcompetence‐oriented mathematics education.Theyrelatedtheseideastopracticebydevelop‐inglearningenvironmentsinteams.Theteachersusedthemin theirclasses, the teacherstudentsworkedwiththeminpracticalcoursesinschool.Allexperienceswerecooperativelyreflectedanddiscussedafterwards.Thisstrategyforteachers’professional development should closely inter‐twine pedagogical and didactical theories withpractice in school. It should help teachers andteacherstudentstodevelopattitudesandbeliefsfor competence‐orientated mathematics educa‐tion.2 A Model of Teaching and Learning Teaching and learning are very complex pro‐cesses. The model of “learning environments”showninFig.1simplifiesreality–asanymodel

does.However,basicstructuresof teachingandlearningprocessesarerevealed.According to constructivist points of view, ateacher cannot put knowledge directly into thelearners’heads.The learning environment istheessentiallinkbetweenteacherandlearner.Thisnotionincludesfivecomponents:thetasksforthelearner toworkwith thecontent, themethodofteachingandlearning,thearrangementofmedia,and the social situation with the teacher andotherlearnersaspartnersforlearning.Itispartoftheteacher’sprofessionalexpertisetodesign learningenvironments.Thus,heoffersabasis for the learner’sworking. This allows theteacher to receive feedback about both thelearnerandthelearningenvironment.Thismodelisbasedonandextendsthedidacticalconceptsof“substantial learningenvironments”byWittmann (1995, 2001) or “strong learningenvironments”byDubs(1995).Sometimes“learningenvironments”aresynony‐mouslycalled“learningscenarios”.Ontheonehand,thismodeloflearningenviron‐ments shows that a teacher cannot enforce orsteerstudents’learningdirectly.Limitationsofateacher’s influence on students’ learning are

Feedback

Design

Work

Offer Teacher

TasksMedia

MethodPartnersContent

Learner Learning Environ-

ment

Fig. 1: Model of learning environments

35

shown.Thismightbedisappointingorevenfrus‐tratingforteachers.On the other hand, interpreted positively, themodelpointsoutthatitisateacher’stasktode‐sign learning environments in order to initiateand foster thestudents’ learningandtouse thefeedbackforfurtherdiagnosisandsupportingac‐tivities.3 Examples from Practice In the project KeyCoMath teachers and teacherstudents designed learning environments, usedtheminclassandreflectedexperiencescoopera‐tively. These activities should initiate and sup‐portprocessesofteachers’professionaldevelop‐mentasitisdescribedinsection1.Inthefollowingchaptersofthisbook,tenlearn‐ing environmentswhich result from these pro‐cessesarepresented.Theygiveanimpressionofmathematicseducationthatfocusesonthedevel‐opment of key competences.However,with re‐specttothemodelinFig.1itmustbekeptinmindthatinthestrictsenseonly“tasks”arepublishedinthefollowingsections.

Theycanunfoldtheirpotentialforcompetence‐orientedmathematicseducationonlywhentheyare embedded in appropriate learning environ‐mentswith suitablemethodical concepts. To il‐lustrate this, the authorsof the following chap‐tersgivesomeinsightintothewaytheyworkedwiththetasks.Theydescribethe learningenvi‐ronmentswhichtheycreatedfortheirstudents,andtheyreportonresultsfrompractice.Further examples from the project work areavailableonthewebsite:www.KeyCoMath.euReferences Dubs,R.(1995).Konstruktivismus:EinigeÜberlegungenaus

derSichtderUnterrichtsgestaltung.ZeitschriftfürPäda‐gogik,41(6),pp.889‐903.

Wittmann, E. (1995).Mathematics Education as a ‘DesignScience’. Educational Studies in Mathematics, 29, pp.355‐374.

Wittmann,E.(2001).DevelopingMathematicsEducationisaSystemicProcess.EducationalStudiesinMathematics,48,pp.1‐20.

36

Lengths of Days over the Course of the Year

Modelling with the Sine Function Julian Eichbichler

Intention Inthislearningenvironment,thepupilsshouldnotonlybecomeacquaintedwiththecourseofthetrigonometricfunctionsbutalsolearnabouttheinfluenceofparametersonamplitude,perioddura‐tionandphaseshift.Thisisachievedthroughtheuseofexplicitexamples.Afterthis,themotionsofthemooninrelationtotheearthcanbetakenalookat:Howaretheseasonscreated?HowdoesthelengthofdaychangeoverthecourseoftheyearwhenIamatNorthorSouthPole?Whichdayisthelongest/shortest?Why?Background of Subject Matter Thechangeinthelengthofdayofthehometownshouldbepresentedgraphicallyandbeapproxi‐matedwithanadequatecurve(regressionmodel).Thisfunctionwillbeanalyzedindifferenttasks.Throughthis,thepupilsareacquaintedwiththegeneralsinefunctionandlearnabouttheinfluenceofthedifferentparameters.Furthermore,theyshouldbeenabledtounderstandandquestionphe‐nomenaindailylife.Forthistopic,aboutfourlessonsarerecommended.Thepupilsshouldalreadybefamiliarwiththetrigonometricfunctionsontheunitcircle.Thus,theyshouldhaveknowledgeofthefunction sin anditsco‐domain,graphandperiodicity.Thetask“Identifythechangeinthelengthofdayoverthecourseoftheyear“isnotimmediatelycon‐nectedtotrigonometricfunctionsbythepupil.Thisisonlypossiblethroughtheregressionmodel.Asaresult,thepupilsbecomefamiliarwiththesignificanceoftheparametersinthefunctiontermofthegeneralsinefunction ∙ sin .

37

Methodical Advice Onthewebsite

http://aa.usno.navy.mil/data/docs/RS_OneYear.phpitispossibletocalculatetheexacttimeofsunriseandsunsetforthehometownsofthepupils.

Firstly,thelongitudeandlatitudeofthehometownhavetobediscovered.Furthermore,thetimezonehastobeidentifiedanditsinfluenceexplained.Onthewebsite,thereareinstructionsabouttheimportofcollecteddatatoExcel.Inclass,thefollowinggraphswerefound:

38

0

100

200

300

400

500

600

0 50 100 150 200 250 300 350 400

Sun

rise

[m

in]

Days

Sunrise

900

950

1000

1050

1100

1150

1200

1250

0 50 100 150 200 250 300 350 400

Sun

set

[min

]

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Sunset

8,00

9,00

10,00

11,00

12,00

13,00

14,00

15,00

16,00

0 50 100 150 200 250 300 350 400

Len

gth

of

Day

[h

]

Days

Length of Day

39

ItisonlypossibleinGeoGebratoapproximatethediscoveredcurvewiththesinefunctionasthenec‐essarytoolsdonotexistinExcel. Withthehelpoftheapproximatedfunction ∙ sin ,thedatacanbeanalyzed:Averagelengthofday: 728,7min 12h9minDifferencelongest–shortestday: 2 429,8min 7h10minPeriodduration: 374Days(Problem:roundingerrors)Phaseshift: 78,4DaysThus,the18thofDecemberistheminimumorshortestday.

40

Comparisonwithactualresults:

Itisnecessarytoexplicitlynamethewebsiteonwhichthedatashouldbecollectedassomecitieshavealreadyputgraphsonthelengthofdayonline.However,especiallytheanalysisofdataisdifficultforthepupils.Performance Rating Especiallyimportantarethecompletenessoftherequiredreportandthelogicoftheargumentation.Inthenextexamitispossibletoaskaquestionabouttheproject(e.g.whichfunctiondescribesthechangeofthelengthofdayoverthecourseofayear?Explain).Further Considerations Aftertheendoftheproject,thegeneralsinefunctionshouldbeanalyzedsystematically.a) Investigatetheinfluenceoftheparameters , , ,and onthegraphofthefunction

∙ sin .

Example:Takeacloselookatthegraphsofsin ,sin 2 ,sin 3 ,sin 0,5 ,sin 0,2 .Whichchangesdoestheparametergenerateinthegraphofthefunction?

b) Writeaclearreportaboutyourobservations.c) Findadequatetermsforthesignificanceoftheparameters.

41

Identify the Lengths of Days over the Course of the Year in Your Home Town! In the following, you can look at the change in the lengths of days over the course of the year in your home town. After that, you will be able to present your results graphically and approximate them with a suitable curve (regression model). As a basis, you use data collected on the internet. It is your decision, which programs you would like to use in order to process and analyze the data. 1 Collect Data Find information about the exact time of sunrise and sunset in your home town on this site:

http://aa.usno.navy.mil/data/docs/RS_OneYear.php 2 Analyse Data Analyse the collected data: How long is an average day?

By how many minutes differ the longest and the shortest day?

Calculate the change in the lengths of days per day in minutes. Why does the change in

some periods of time happen more rapidly than in others? How do the lengths of days change over the course of the year at the North Pole?

If a day is longer at the North Pole than it is here, the temperature there should in com-

parison be higher! Argue. Are your data and results realistic?

3 Appendix The sun is not punctiform, so why should the solar day be defined? Does the sun really rise in the east? … Think of different places on earth. 4 Basis for Evaluation Write a well-structured report about your observations and find logical terms for the different quantities.

42

Which Shape is best for a Honeycomb Cell?

Optimisation Tasks Marion Zöggeler, Hubert Brugger, Karin Höller

Intention Thislearningenvironmentisfocusedontheoptimalshapeofahoneycombcell.Thisisanillustrative,interdisciplinaryexampleofanoptimizationtask.Ifthetaskisusedinlowerclasslevels,thefocusisplacedonthequalitativedescriptionoffunctionsandtheexperimentalapproachatfindingsolutions.Inhigherclasslevelsthislearningenvironmentcanbeusedtoworkwithdifferentialequationsofoneormultiplevariables.Aminimumoffourlessonsarerecommendedforthislearningenvironment. Background of Subject Matter Thetasksaresupposedtoguidethepupilstotheoptimalhoneycombcell.Thefirsttasksaremoresimpletasksintheplanewhichhavetobetranslatedintothethreedimensionalspace.Hereareafewsuggestedsolutions: For 1 Maximal Area InGeoGebra,thedifferentpolygonswithgivenperime‐terareconstructedandtheirareasarecalculated.Hereitcanbeseenthattheareaconvergeswithanincreaseinvertices.Thelimitistheareaofacircle.TheGeoGebrafilecanbedownloadedatwww.KeyCoMath.eu.

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Areas of Regular Polygons in a Circle InExcel,atabularoverviewoftheareasofdifferentpolygonscanbecreated.Exampleofacircleradiusof10cm:

Number of vertices n

Central angle in degree measure

Central angle in radian measure

Area of the n-gon in cm²

3 120,00 2,094395102 129,9038106

4 90,00 1,570796327 200,0000000

5 72,00 1,256637061 237,7641291

6 60,00 1,047197551 259,8076211

7 51,43 0,897597901 273,6410189

8 45,00 0,785398163 282,8427125

9 40,00 0,698131701 289,2544244

10 36,00 0,628318531 293,8926261

20 18,00 0,314159265 309,0169944

30 12,00 0,209439510 311,8675362

40 9,00 0,157079633 312,8689301

50 7,20 0,125663706 313,3330839

60 6,00 0,104719755 313,5853898

70 5,14 0,089759790 313,7375812

80 4,50 0,078539816 313,8363829

90 4,00 0,069813170 313,9041318

100 3,60 0,062831853 313,9525976

150 2,40 0,041887902 314,0674030

200 1,80 0,031415927 314,1075908

250 1,44 0,025132741 314,1261930

300 1,20 0,020943951 314,1362983

350 1,03 0,017951958 314,1423915

400 0,90 0,015707963 314,1463462

450 0,80 0,013962634 314,1490576

500 0,72 0,012566371 314,1509971 Result:Theareaofthepolygonapproximatestheareaofacircle.TheExcelfilecanbedownloadedatwww.KeyCoMath.eu.Forhigherclasslevels,acalculationofthelimitmaybeadequate:

lim→

∙2∙ sin

2 lim

→2 ∙

²2∙ ²

Here,thesubstitution andthelimitlim→

1havebeenused.

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For 2 Special Case: Rectangle Here,arectanglewithgivenperimeteristobeconsidered:Inwhichmannerdothesides and havetobechoseninordertogenerateamaximalarea?Thisproblemleadstotheminimumofaquadraticfunction.Apupil’ssuggestedsolution:

For 3 The Other Way Round: Minimal Perimeter

Inthistasktheperimeterofarectangle 2 withagivenarea shouldbeexaminedin

relationtoasidelength .Again,aminimumoughttobefound.Inthiscase,agraphicalrepresentationcanbehelpful.Inhigherclasslevels,calculuswillbeused.Apupil’ssuggestedsolution:

For 4 Special Case: Triangle

WithHeron’sformula ,theareaofatrianglewithgivenperimetercan

beexpressedwithtwovariables.Here .ExperimentalsolutionscanbefoundwithExcelorGeoGebra.Exactsolutionsrequiretheuseofmeth‐odsfrommathematicalanalysisintwovariables.

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For 5 From Plane to Space Inthistasktermshavetobetranslatedfromtheplanetothethreedimensionalspace. Perimeter equals surfacearea Area equals volume Rectangle equals cuboid Square equals cube

Allpupilsshouldbeabletodothis.Findingoptimalsolutionsinthreedimensionalspace,however,willbeespeciallyinterestingformotivatedpupils. Example:Consideracuboidwithgivenvolumeanddeterminethesides , and inthemannerthattheresultisasurfaceareawhichisassmallaspossible.Apupil’ssuggestedsolution:

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For 6 Optimal Shape of a Honeycomb Cell – Hexagonal Base Pupilsshouldshowthat theplanecanonlybecompletelycoveredbytriangles,squaresorregularhexagons.Ateachvertexofapolygonintheplaneatleastthreeotherpolygonsmeet.Thesumoftheneighbouringinterioranglesis360°.Foraninteriorangleofaregular ‐gon,itisvalidthat:

2 ∙ 180°

Thishastobeadivisorof360°,whichisonlyvalidfor 3,4and6.Inordertofindtheminimalperimeterofthepolygon,resultsfromtask1canbeused. For 7 Optimal Shape of a Honeycomb Cell – Optimal Inclination Angle Thepupilscanbuildamodelofahoneycombcelloutofpaper.Thisincreasestheunderstandingofthesituation.Thistaskcanonlybeusedinhigherclasslevelsascalculusisneeded. Methodical Advice Itmakessensetosolvethetaskstogetherwithapartneroringroups.Afterthis,thesuggestedsolu‐tionscanbediscussedinclass. Performance Rating Theperformanceratingcanbebasedonthedrawingupofthemodels,theworkonthecomputer,thepresentationoftheresults,andthewayofworking. Bibliography Schlieker,V.,Weyers,W.(2002):Bienenbauenbesser,in:mathematiklehren,Heft111,FriedrichVerlagDietrich,V.,Winter,M,Hrsg.(2003):ArchitekturdesLebens,MathematischeAnwendungeninBiolo‐

gie,Chemie,Physik,CornelsenVolkundWissenVerlagSteiner,G.,Wilharter,J.(2007):MathematikundihreAnwendungeninderWirtschaft3,RenietsVerlag

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Optimization task: Which is the Best Shape for a Honeycomb Cell? In nature, building styles have improved through evolution and are now almost optimal – for in-stance the building of a honeycomb cell. But what is optimal in this context? Solve these tasks in order to answer the question: 1 Maximal Area Given is a piece of rope of a certain length. How can you enclose an area which is as large as possible? Try different geometric shapes. Document your results. Which is the most adequate shape? Give reasons. Use Excel or GeoGebra to support your claim.

Hint: Formula for the area of a regular polygon: ∙ ∙ sin

2 Special Case: Rectangle Choose a rectangle with given perimeter. Determine the side lengths that generate the largest area. Argue your suggested solution. Are there different approaches? 3 The Other Way Round: Minimal Perimeter Consider a rectangle with given area. Which side lengths should it have to generate a minimal perimeter? Argue your suggested solution. 4 Special Case: Triangle Examine any triangle with a given perimeter. How long should the side lengths be in order to create a maximal area? Argue your suggested solution by considering the area as a function with two variables.

Hint: Heron’s formula for the area of triangles: , with

5 From Plane to Space Translate tasks 2 and 3 to the three dimensional space. Find optimal solutions for this as well.

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6 Optimal Shape of a Honeycomb Cell – Hexagonal Base Firstly the problem is considered in a plane. The honeycomb cells completely cover the plane. Which polygons are adequate for this? Hint: formula for the sum of the interior angles of a regular n-gon:

2 ∙ 180°

Which of these polygons has the minimal perimeter? Hint: Use the solution of task 1. 7 Optimal Shape of a Honeycomb Cell – Optimal Inclination Angle

The figure shows the model of a honeycomb cell. You can see that the top is no plane hexagon. The biologist D’Arcy discovered that the surface area is just dependent of the inclination angle of the three areas forming the top of the cell. He even found a formula for this:

632

√3 cossin

Here, is a side of the hexagon and is the longer side edge. Determine angle in the manner that it generates a minimal surface area.

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Functions and their Derivatives

Courses of Functions, Monotony, and Curvature Johann Rubatscher

Intention Theaimofthisworksheetisomakepupilsfamiliarwiththeconnectionbetweenafunctionanditsderivatives.Nocalculationsareneededinordertosolvethetasks.Practicalexperienceshowsthatabout50minareneededtocompletetheworksheet. Background of Subject Matter Multipletasksareaimedatanunderstandingoftherelationbetweenmonotonyandcurvature. Methodical Advice Thepupilsshouldsolvethetasksaloneortogetherwithapartner.Thetasks’levelofdifficultyisav‐erage. Performance Rating Inanexam,itiseasilypossibletotestifthestudentshaveunderstoodtherelationsandareabletoarguetheirclaimsthroughvaryingthetasks.

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Functions and their Derivatives May function be continuous and differentiable. Task 1 May function be strictly monotonically decreasing in the interval I. What happens to functional value in the interval if the -value of the function is decreasing? Argue. Task 2 In a right curve, may the function have the slope 4 in . Which slope will have in in this right curve, if is located to the right of ? Argue. Task 3 May ′′ possess the value – 5 in . What does this tell you about ′ and ′ in a neighbourhood of ? What can be said about ? Argue. Task 4 May ′ intersect the -axis in from above. Which special point is ? Argue. Task 5 If necessary, correct the following argumentation:

In a right curve of a function the slope of the tangent decreases if moves from left to right. As the derivate of a function describes the slope of the tangent in a particular point, ′ and ′′ are strictly monotonically decreasing in the right curve.

Argue if you have changed the argumentation. Task 6 If possible, draw a sketch of a section of a function which has a left curve and is strictly mono-tonically decreasing. What can be said about ′′ in this part of the curve? Argue. Task 7 May function change from a right curve into a left curve in . Additionally, may be strictly monotonically increasing close to . Is this possible? Argue with the help of a sketch. Task 8 May function change from a right curve into a left curve in . Additionally, may be strictly monotonically decreasing close to . Is this possible? Argue with the help of a sketch. Task 9 May ′′ be monotonically decreasing in an interval and possess an -intercept in this interval. What can you conclude about in this interval? Argue. Task 10 May ′ be strictly monotonically increasing in an interval I but may ′ possess only negative func-tional values in this interval. Sketch the course of in the interval and argue your solution.

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Important Lines in Triangles

A Computer-Assisted Lesson Walther Unterleitner, Manfred Piok, Maximilian Gartner

Intention ThepupilsconstructthemostimportantfeaturesoftriangleswiththehelpofGeoGebraandstudytheircharacteristicsbychangingtheshapeofthetriangles.Fourtosixlessonsarerecommendedforthistasksdependingonthepupils‘knowledgeofGeoGebra. Background of Subject Matter Themostimportantpointsintriangles(Pointofintersectionofperpendicularbisectorsascircumcen‐tre,pointofintersectionofanglebisectorsasincentre,centroid(centreofmass),andpointofinter‐sectionofaltitudesasorthocentre)arebeingconstructed.Thelocationofthesepointsvariesdepend‐ingontheshapeofthetriangle;threeoutofthefourarelocatedontheEulerline.GeoGebraenablesthepupilstotestvariouscasesandthroughthisdrawconclusions.Additionally,fasterpupilscantrytofindtheFermatpoint. Methodical Advice Inordertoworkwiththislearningenvironment,pupilsrequireknowledgeofthefollowingterms: Perpendicularbisector Anglebisector Median Altitude Distance

The first taskcanbeusedfordemonstrationpurposes inordertoexplainGeoGebra’s tools.Pupilsoughttoworkaloneortogetherwithapartner. Performance Rating Thepupilssolvethetasksandwritedowntheirthoughtsandresultsonthecomputer.TheproducedgraphsfromGeoGebrawillbeaddedtothedocument.Thisdocumentwillthenbegradedaccordingtocontent,clarity,andtheproficiencyinconstructionandargumentation.

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The Most Important Lines in Triangles For each task, create an own window in GeoGebra which you can access at any time. 1 Altitudes Construct a random triangle and draw in the three altitudes. Construct their point of intersection O. Move the vertices of the triangle so that the orthocentre O is relocated as a result. Note down how the location of the orthocentre moves in relation to the shape of the triangle. 2 Perpendicular Bisectors Construct the perpendicular bisectors of a triangle. Label their point of intersection as C. Move the vertices of the triangle so that the point of intersection C is relocated as a result. Note down how the location of C changes in relation to the shape of the triangle. 3 Angle Bisectors Construct the angle bisectors of a triangle. Label their point of intersection as I. Is it possible that I is located outside of the triangle? Argue. 4 Medians Construct the line segments from the midpoints of the sides to the opposite vertices. Label their point of intersection as S and study its location. 5 Circles and Relations Which one of the points has the same distance from all three vertices of the triangle? Which one of the four points has the same distance from all three sides of the triangle? Measure the distance between the vertices and the four points of intersection and the distance between the four points of intersection and the three sides of the triangle. What do you notice? Additional Task: Is there a point for which the sum of the distances to the vertices is minimal? 6 Connection between the Different Points of Intersection Construct a triangle that covers most of the screen. Draw the now familiar special lines and their points of intersection into the triangle. Fade out the special lines so that a clear picture remains. Move the triangle and track the motion of the four points of intersection. What do you notice? Are you able to move the vertices in the manner that all four points of intersection are situated on a line?

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Measuring Distance and Height

Functioning of Apps Marion Zöggeler, Hubert Brugger, Karin Höller

Intention Smartphonesandtabletcomputersofferavarietyofappswhichareoftenbasedonmathematicalcal‐culations.Thisisthecasewiththemeasuringofdistanceandheight.Thefollowingtaskissupposedtoencouragethepupilstoquestionthesetechnicalgadgets.Forthistopicapproximately1‐2lessonsarerecommended. Background of Subject Matter Inordertofulfillthetask,basicknowledgeintrigonometryisrequired. Methodical Advice Thistaskisdesignedforgroupsconsistingoftwotothreepupils.Greatimportanceisattachedtothedocumentationofthepupils’worksandthedescriptionoftheirapproaches.Thefollowingquestionsaresupposedtosteerthisdocumentation.Forthisexperimentaltask,asufficientamountofsmartphonesortabletcomputershavetobeathand.Additionally,tapesformeasuring,rulers,protractors,andcordsshouldbewithinreach. Performance Rating Eachpupil’ssolutioncontainsasketchthatshowswhichquantitiesaregivenorarebeingmeasured.Themissingquantitiesarecalculatedthroughtheapplicationoftrigonometricrelations.Theperfor‐manceratingtakesthesetwoaspectsintoconsideration.Theprocessofmeasuringitselforthesearchforadditionalinformation(e.g.intheapp’susermanual)shouldalsocontributetothefinalrating. Pupil’s Example

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Measuring Distance and Height Functioning of Apps Smartphones and tablet computers offer a variety of apps which make use of mathematical calcu-lations. An example is the measuring of distance and height. Small Distances (1 - 50 m) Taken from the “Smart Measure Lite“-Manual:

The measured length is for reference.

Before using this App, calibrate your devices with known distances.

If the measured distance is longer

than it, reduce by 5% at manual calibration. If shorter, increase by 5%. Do it several times. You can find your own best calibration.

© Smart Tool co

55

Tasks Shown above are the instructions for the app “Smart Measure“. Measure some distances and heights with your smartphone or tablet PC and check the results with a tape measure. Compare. Now we want to take a closer look at the app: Which quantities are necessary for the app?

The device can measure angles of inclination. Which angles of inclination are necessary

for calculating the distance and height of an object?

By which means are distance and height calculated?

Draw a sketch, measure the required quantities with a tape measure and calculate the distance and height yourself using trigonometry.

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Thinking in Functions

Introduction to the Topic of Functions Dorothea Huber

Intention In this learning environment, pupils should work extensively with functional relations and theirgraphs.AccordingtoKröpfl(2007),thetwomainideasoffunctionsarethedependenceofavariableofoneormorevariablesandthecourseofthefunction.Thecourseofthefunctionisexpressedinthechangeswhicharenoticeableinthevaluetableandthegraphofthefunction.Indailylife,ratherfewfunctionsarefundamentaltypesoffunctions.Thepupilsareprovidedwithageneralintroductiontothefieldoffunctions.Atthispointtheywillnotbeacquaintedwithlinearandquadraticfunctionsasspecialtypesoffunctions. Background of Subject Matter Inthislearningenvironment,thepupilsbecomeacquaintedwiththedifferenttypesoffunctions,es‐peciallythosewhichcannotbedescribedinafunctionalequation.Theylearnthatsomegraphsdonotportrayafunction.Additionally,theyhearaboutcontinuousanddiscretefunctions.Furthermore,theyrepeatpercentagecalculationandtheformulafortheareaofarectangleandstudygraphsoffunctionsaccordingtotheirrateofchange.Thefollowingthematicaspectswillbeconsidered: Time‐Distance‐Functions(linearandquadraticfunctions) Functionsdescribingtheprocessoffillingacontainer(VolumeWaterlevel) Powerfunctions Exponentialfunctions

Methodical Advice Inordertoguideandhelpthepupilsmoreeasily(e.g.withtheultrasoundscanner),mostofthelessonisorganisedaccordingthemarketplacemethod.Hereitisveryimportanttoassistespeciallyweakerpupils.Apossiblecourseofthelessonscouldbe: 1st lesson

Atfirst,apersonalgraphofthepupils‘well‐beingonthedaybeforeisdrawnbythepupilsthemselves.Afewinterestinggraphsarepresentedinclass.Thentheclassisseparated.Onepartisworkingingroupsoftwoonstall2“walkingalonggraphs”.Theotherhalfoftheclassisworkingwiththeultrasoundscanner.Thisscannermeasuresthedistanceofanobjectinrelationtothetime.Theteacherassiststhestudentswiththefirstmeasurements.Thisstallconsistsoftasksofvaryingdifficultyandthesolutionscanbecheckedbythepupilsthemselves.Atstall2“walkingalonggraphs“,studentswalkalongTime‐Distance‐Graphsingroupsoftwo.Thereisalsoanimpossiblegraph.Thepupilsshouldrecognizethatthiscannotbethegraphofafunction.Thehomeworkdealswiththe“hoistingofaflag”.

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2nd lesson

Thepupilscontinuetheworkinthemarketplace.Atstall3theymeasurethewaterlevelofanErlen‐meyerflaskinrelationtothevolume.Attheendofthistask,thepupilsshouldbeabletomatchgraphandcontainer.Stall4“rectangles”dealswithindirectproportionalfunctions.Theformulafortheareaofarectangleisrepeated.Stall5isanexampleforadiscreteexponentialfunction.

3rd lesson Theworkinthemarketplaceiscompleted.Theideasandresultsofindividualstallscanbediscussedinclassifnecessary. Practical Experience Thefollowingobservationshavebeenmadeinclass: Insteadoftheexpectedargument“itisnotpossibletobeintwoplacesatonce“,“timecannot

standstill“isusedtoarguetheimpossiblegraph. Especiallythecourseoffunctionsisregardedinthisapproach. Regularityinthelossofdrawingpinsisnotrecognized.

Performance Rating Theacquiredcontent,knowledge,andskillscanbetestedinanexaminthefollowingmanner: Describinggraphsofmovement Identifying/describingthewaterlevelofagraph Similarfunctionalrelationships:e.g.timespeed,timepagesread Experiment:foldingasheetofpaperrepeatedlyinthemiddle.Recognizingthenumberofpa‐

perlayersinrelationtothenumberoffoldingsasafunctionalrelationandbeingabletodepictthis.

Bibiliography BüchterA.,HennH.‐W.(Hrsg.)(2008):DerMathe‐Koffer,FriedrichVerlagKröpfl,B.(2007):HöheremathematischeAllgemeinbildungamBeispielvonFunktionen,ProfilVerlag,

MünchenWienMathematik5bis10,Heft8|2009,Friedrich‐Verlaghttp://www.nawi‐aktiv.de/umaterial/labor/laborgeraetequiz_zuordnung.htmhttp://www.individualisierung.org/_neu/download/funktionen_pool1.pdf

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Introduction to Thinking in Functions Personal graph of my well-being Draw yesterday’s graph of your well-being.

(see: Mathematik 5 bis 10, Heft 8|2009, Friedrich-Verlag)

6AM 8AM 10AM 12PM 2PM 4PM 6PM 8PM

1

2

3

4

5

‐5

‐4

‐3

‐2

‐1

0

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Working in the Market Place 1 Market Stall “Ultrasound Scanner“ Move your hand away from the ultrasound scanner and draw the graph. Move your hand closer to the ultrasound scanner. Draw the graph and think about which quantity should be measured on the x-axis and which quantity on the y-axis. In which way do you have to move your hand in order to generate the following graph? Try.

In which way do you have to move your hand in order to generate the following graph? Try.

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2 Market Stall “Walking along Graphs“ Choose a wall and walk along the graphs. The graphs should describe your distance to the wall in relation to the time. Additionally, you should describe your movements in words. If it is not possible to put some graphs into practice, give reasons for this.

(see: Mathematik 5 bis 10, Heft 8|2009, Friedrich-Verlag)

Distance

Time

Distance

Time

Distance

Time

Distance

Time

Distance

Time

Distance

Time

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3 Market Stall “Filling Containers“ For this experiment, an Erlenmeyer flask, a ruler, and a graduated cylinder are necessary. Check how the water level changes when the Erlenmeyer flask is evenly filled with water. Repre-sent your findings graphically. Label the x-axis volume and the y-axis water level.

(see: http://www.nawi-aktiv.de/umaterial/labor/laborgeraetequiz_zuordnung.htm)

Find the graph which describes the process of filling the containers.

(see: Mathematik 5 bis 10, Heft 8|2009, Friedrich Verlag)

Describe the relation between container and graph. Find examples of your own.

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4 Market Stall “Rectangles“ Take a sheet of paper and cut out four different rectangles with areas of 9 cm² !

Draw a coordinate system in which you can compare the connection between the side lengths of possible rectangles. The axes should be labelled with the two side lengths a and b of the rectangle. (Scale: 1 square ≜1 cm)

With which formula can you calculate the second side length if one side is given? Glue the rectangles on the chart in an appropriate manner.

Does the graph intersect the axes? Give reasons.

If you have completed stall 5 already: Which differences and similarities can you discover be-tween the two graphs?

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5 Market Stall “Loss of Drawing Pins“ When you let drawing pins fall to the floor, some will land on their backs while others will land on their sides. Now your task is to investigate according to which rule this happens. You will need 100 drawing pins and a paper cup for throwing the pins.

Throw the drawing pins to the floor and sort out the ones lying on their sides. Note down the number of the rest of the pins. Repeat the experiment with the remaining pins four times.

Represent your findings graphically. Label the x-axis with “number of throws” and the y-axis with “number of remaining drawing pins“.

Are there regularities? Which ones? If you have completed stall 4 already: Which differences and similarities can you discover between the graphs?

(see: Mathekoffer, Friedrich-Verlag)

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Homework 1 Hoisting a Flag The nine figures partially describe the hoisting of a flag. You may have noticed that this process can be done quite differently. Some people quickly hoist the flag, others take small breaks, etc. Describe, if possible, how the hoisting of the flag happens in each figure.

(see: http://www.individualisierung.org/_neu/download/funktionen_pool1.pdf)

2 Bikerace The diagram describes a bike race. Comment on the race like a radio reporter.

Distance

Time

Height

Time

Height

Time

Height

Time

Height

Time

Height

Time

Height

Time

Height

Time

Height

Time

Height

Time

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Test about Functions Task 1 Peter, Paula und Maria are classmates and live on the same street. Their school is located at the end of the street. Every morning they walk to school, which starts at 7:50 AM. The graph shows where they were yesterday morning at different times. What happened? (It does not have to be an exciting story but it has to be clear that you are able to read graphs.)

Task 2 Draw the graph which describes the process of filling the container. Task 3 What do the graphs of linear functions look like? What can be seen in their value tables?

Draw the graph of: 3 Task 4 Determine the functional equations of the graphs.

Time 7:50 AM

Distance from School

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Wood Stack – Where is the Limit?

Discovering the Harmonic Series Aron Brunner, Johann Rubatscher

Intention Thequestiononhowfarequalwoodencuboids(e.g.dominoes,CD‐covers)with the dimensions l, b, and h in a stack can bemovedwithout the stack collapsing has been asked for centuries. Themaximaloverhangsofthecuboidontopshouldbediscoveredhere.Newdevelopmentscanbefoundinthebibliography.

https://de.wikipedia.org/wiki/Harmonische_Reihe

Background of Subject Matter Astackofcuboidsdoesnotcollapseifthefootofthecommoncentreofmassofthecuboidsdoesnotleavethecuboidbelow.Themaximalshiftdistances canbediscoveredbyfindingthecentreofmass.Throughaddingnewcuboids,scanbedescribedthroughtheseries

/2 /4 /6 /8 /10 /12 /14 ⋯,

whichisequalto

2⋅ 1

12

13

14

15

16

17

⋯ .

Thebracketscontaintheharmonicseries,whichdoesnotconverge.Thus,byaddingcuboidsanyshiftdistancecanbereached. Working out the Results Thefirstcuboidontopcanbeshiftedby /2asthenitscentreofmassrestsovertheedgeofthecuboidbelow.Thecentreofmassoftwocuboidsislocatedinthemiddleofthecentresofmassofthetwocuboidsontheirown(symmetry).Thus,thetwocuboidstogethercanbeshiftedby /4. Thecentreofmassofthreecuboids,however,isnotlocatedinthemiddleofthecentreofmassofthetwocuboidsandthethirdbutisshiftedinthedirectionofthecentreofmassofthetwocuboidsasthetwopossessmoremass.Duetothis,thecentreofmassisshiftedintheratio2:1,whichisequivalenttoathirdofl/2.Thus,itisvalidthat: 1/3 ⋅ /2 /6Thecentreofmassoffourcuboidsisshiftedaccordingtotheratio3:1,whichleadsto: 1/4 ⋅ /2 /8Thecentreofmassoffivecuboidsisshiftedaccordingtotheratio4:1,whichleadsto: 1/5 ⋅ /2 /10

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Continuationfor , , ,…generates:

⋯ /2 /4 /6 /8 /10 /12 ⋯ Thecontinuationoftheharmonicseriesshouldbepointedouttothepupils.Aspreadsheetprogramcancalculatetheharmonicseriesuptoabout 200.000.Bibliography Pöppe, C. (2010):TürmeausBauklötzen–MathematischeUnterhaltungen, SpektrumderWissen‐

schaft,September2010,S.64Paterson, M., Peres, Y., Thorup, M., Winkler, P., Zwick, U. (2008): Maximum Overhang, arXiv.org,

CornellUniversityLibrary,http://arxiv.org/pdf/0707.0093v1.pdf

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Wood Stack – Where is the Limit?

https://de.wikipedia.org/wiki/Harmonische_Reihe

1 Description of the Task and First Experiments Taketwoequalwoodencuboids(dominoesorCD‐covers)oflengthl,widthb,andheighth(withhalot smaller than l and b), stack them on top of each other andmove the cuboid on top so far inlongitudinaldirectionthatitsticksoutasfaraspossible.Putathirdequalcuboidunderneaththetwofirstonesandmovethetwouppercuboidsagainasfaraspossible.Continuethisprocess.2 Calculation Considerwithpen,paper,pocketcalculator,etc.,howfaryoucanmovethecuboidssuccessivelyifthetotalnumberofblocksis 2, 3, 4, 5, …?Howfaristhetotalshiftdistancesineachcase? 3 How far? Whichisthenumber ofwoodenblocks,withwhichastackofthismannercanbebuiltthatdoesnotcollapse?4 With Help from the Computer Calculate withaspreadsheetprogram. 5 Further Considerations Howbigiss,ifeachcuboidismovedbyl/3orl/4?Whatwouldhappenwithwoodendisks?Whatwouldchangeifyouwereallowedtousecounterweightsoranyotheradditions? Readthefollowingarticle:

Paterson, M., Peres, Y., Thorup, M., Winkler, P., Zwick, U. (2008): Maximum Overhang,arXiv.org,CornellUniversityLibrary,http://arxiv.org/pdf/0707.0093v1.pdf

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Moving to a New House

Reflecting on Problem Solving Panayiota Irakleous

Goals (Mathematical Competence) Thestudentsshouldacquirethefollowingcompetences:Numbers Usenaturalnumbersuntil1000anddecimalstosolveproblems. Compareandordernaturalnumbersuntil10000. Usetheconceptofratioandsolveproportionalproblems. Judgethereasonablenessofcalculatedresults.

Algebra Useverbalandalgebraicexpressionstorepresentadditiveandmultiplicativerelations. Understandtheconceptofvariable,interpretandexplainrelationsbetweenvariables.

Geometry Describethepositionofobjects,usingconceptssuchasup‐down,behind‐infrontof,nextto,

between,right‐left.

Measurement Convertunitswithinthemetricsystem. Usestandardunitsofmeasurementforlength. Makeestimationsofdistances. Interpretscaledrawings.

Statistics – Probabilities Interpretanddesignfrequencycharts(barchart,piechart,lineargraph,plotsandtables). Performtheproceduresofdataclassification,quantification,weightingandgrouping.

Key Competences for Lifelong Learning DigitalCompetence SocialCompetences Communicationinthemothertongue LearningtoLearn SenseofInitiative

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Moving to a New House A Warm up activity Family stories … Leonidas and Ioanna have rented a furnished apartment at the center of Nicosia. They pay 620 euros per month in rent. They have two children, George and Marilia, who are twins. They are 7 years old and go to the A’ Primary School in Latsia. Both Leonidas and Ioanna have worked as accountants in the same office for 8 years, but last month Ioanna was fired. Thus, they want to move to a new house. The two parents believe that their children should still go to the same school, in order to protect their psychological balance. They seek to find an affordable apartment that meets their basic needs. To this end, they have conducted a market research and have collected information about two-bedroom apartments, as shown in the table below.

Below you can find a map that shows the position of each apartment as well as the office where Leonidas works.

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Comprehension questions 1. What problem does the family face? 2. How far is the apartment B from the office? 3. Which apartment is closest to Leonidas’ office? 4. Which apartment has the highest rent? Which apartment has the highest rent in relation to

its area? Are these apartments the same? Why? Β Problem You should classify the apartments into three groups: very suitable for the family, moderately suit-able and unsuitable. Write a letter to describe the method you applied in order to classify the apart-ments. The parents are going to use this method in the case of a future moving to a new house. C Reflection

1. Step Diagram Draw a step diagram to represent “changes in thinking” that your group went through during the solution of the problem as well as your level of engagement. Start of session End of session

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2. Which mathematical concepts did you use during the solution of the problem?

3. Which mathematical procedures did you use during the solution of the problem?

4. In your opinion, how well did you understand the concepts you have used? Explain why.

Not at all To a small extent To a moderate extent To a large extent Absolutely

5. How difficult was the problem for you? Explain why. Very easy Easy Neutral Difficult Very difficult

6. In your opinion, which of the methods you have thought so far is the most appropriate?

Ratio

Standard Deviation

Mean

FrequenciesProbabilities

Percentages

Fractions

Length

Volume

Mass

DensityDivisibility

Symmetry

Area

Data classification

Data quantification

Mass measurement

Length measurementGraph design

Length estimation

Data processing

Geometrical transfor-mations

Performing operations

Interpreting scale

drawings

Areameasure-

ment

Volume measure-

ment

Designing mapping diagrams

Velocity caclulation

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At the Zoo

Planning a Route Eleni Constantinou

Goals (Mathematical Competence) Thestudentsshouldacquirethefollowingcompetences:Numbers Usenaturalnumbersanddecimalstosolveproblems. Performadditionandsubtractionofnaturalnumbers. Usetheconceptofratioandsolveproportionalproblems. Judgethereasonablenessofcalculatedresults.

Geometry Describethepositionofobjects,usingconceptssuchasup‐down,behind‐infrontof,nextto,

between,right‐left.

Measurement Usestandardunitsofmeasurementforlength. Convertunitswithinthemetricsystem. Interpretscaledrawings. Recognizerelationsbetweenunitsoftime. Estimatetimeduration.

Statistics – Probabilities Organizeandpresentdataintables. Performtheproceduresofdataclassification,quantification,weightingandgrouping.

Key Competences for Lifelong Learning DigitalCompetence SocialCompetences Communicationinthemothertongue LearningtoLearn SenseofInitiative

Tools available Thestudentsshouldhavethefollowingtoolsfortheirwork: Mapofthezoo(digitalandhardcopy) Familypreferences Adviceoffriendswhohavevisitedthezoo PC Stopwatch Measuringtape SkitchTouchapplication

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At the Zoo Problem Erodotos and his family are in Barcelona for holidays. One of Barce-lona's top touristic attractions is the Zoo. They are planning to visit the zoo on Friday (26/03), but they will have only 2 ½ hours available. You should help the family to plan the most appropriate route in order to see as many animals as possible and also to decide the best time to visit the zoo. Additionally, you have to present this route, explaining the reasons why the particular route is ap-propriate for the family and for any other visitor who has less than 3 hours to visit the zoo. Family preferences The family members have special interests: “I would like to see …

… the lions and the gorillas.” (Mr. Nikos) … the seals and the flamingos.” (Mr. Danae) … the elephants and the kangaroos.” (Aphrodite, 6 years old) … the dragons, the crocodiles and the camels.” (Achilleas, 10 years old)

Advice of friends who have visited the zoo “The dolphins’ show is incredible! You should definitely watch it!” (Georgia) “The lions sleep from 12 o'clock until 5 o’ clock, so it is difficult to see them.” (Alexander)

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The Energy Problem

Working in Complex Real-Life Situations Panayiota Michael, Elena Sazeidou, Stella Shiakka

Goals (Mathematical Competence) Thestudentsshouldacquirethefollowingcompetences:Numbers Usenaturalnumbersuntil10000anddecimalnumberstosolveproblems. Performaddition,subtraction,multiplicationanddivisionofnaturalnumbersanddecimals. Compareandordernaturalnumbersuntil10000. Usetheconceptofratioandsolveproportionalproblems. Judgethereasonablenessofcalculatedresults.

Statistics – Probabilities Describeandcomparedatasetsbyusingtheconceptofarithmeticmean. Performtheproceduresofdataclassification,quantification,weightingandgrouping.

Key Competences for Lifelong Learning SocialCompetences Communicationinthemothertongue LearningtoLearn SenseofInitiative

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The Energy Problem A Problem The University of Cyprus senate has decided to give the opportunity to a group of students to choose the most appropriate type of lamp for a specific lecture hall. The dimensions of the partic-ular lecture hall are 24 m x 20 m. The maximum amount of money that can be spent on the lighting of this hall for 12.5 years is € 800. B Problem You are a member of the Central Committee of Electricity Authority of Cyprus. It is your duty to describe a procedure by which the customers will be able to choose the lamp that meets their needs, based on the following table. C Data

* 3 hours daily

Lamp Electric power (Watt)

Lumino-sity (Lm)

Lm/Watt Number of days in use*

Number of replace-ments over

25 years

Cost per lamp

(EUR)

Electric power

(kW) for 25 years

Cost (kWh) for 25 years

Led 7 256 36,5 9000 1 35 252 0,07

Halogen 17 256 15,07 500 19 2,50 612 0,07

High-pres-sure sodium

273 256 0,93 5000 2 14,96 9.828 0,07

Light bulbs 24,6 256 10,4 250 25 0,75 885,6 0,07

Fluorescent 95,7 256 2,67 2.500 4 1,30 3.445 0,07

Mercury 347,2 256 0,73 4000 3 8,30 12.499 0,07

Key competences are necessary or all citi ens or personal ul lment, active citi enship, social inclusion, and employability in a knowledge society. As a result, it is a fundamental task of the school system to assist children and adolescents in their development of key competences. In a European project, eight partners have developed strategies for changing mathematics education in order to contribute to this objective and have put these concepts into practice.

This book illustrates results of the project „KeyCoMath – Developing Key Competences by Mathematics Education”.

> Concepts for improving mathematics education with the aim of developing key compe-tences are suggested.

> Proved strategies for initial and in-service teacher education, which foster competence- oriented attitudes and beliefs of university students and teachers, are presented.

> As examples, ten learning environments for competence-oriented mathematics educa-tion, which were developed by teachers or university students and tested in school, are given.

www.KeyCoMath.eu ISBN 978-3-00-051067-0

With support of the “Lifelong Learning Programme” of the European Union