Table Of ContentInternational Perspectives on the Teaching and
Learning of Mathematical Modelling
Raphael Wess
Heiner Klock
Hans-Stefan Siller
Gilbert Greefrath
Measuring
Professional
Competence
for the Teaching
of Mathematical
Modelling
A Test Instrument
International Perspectives on the Teaching
and Learning of Mathematical Modelling
SeriesEditors
GabrieleKaiser,FacultyofEducation,UniversityofHamburg,Hamburg,Germany
GloriaAnnStillman,FacultyofEducationandtheArts,NationalSchoolof
Education(Victoria),AustralianCatholicUniversity,Ballarat,VIC,Australia
EditorialBoard
MariaSalettBiembengut,UniversidadeRegionaldeBlumenau,Blumenau,
SantaCatarina,Brazil
WernerBlum,UniversityofKassel,Kassel,Hessen,Germany
HelenDoerr,SyracuseUniversity,Knoxville,TN,USA
PeterGalbraith,UniversityofQueensland,Brisbane,QLD,Australia
ToshikazuIkeda,YokohamaNationalUniversity,Yokohama,Japan
MogensNiss,RoskildeUniversity,Roskilde,Denmark
JinxingXie,TsinghuaUniversity,Beijing,China
Thisbookserieswillpublishvariousbooksfromdifferenttheoreticalperspectives
aroundtheworldfocusingonTeachingandLearningofMathematicalModellingat
Secondary and Tertiary level. The proceedings of the biennial conference called
ICTMA, organised by the ICMI affiliated Study Group ICTMA (International
Community of Teachers of Mathematical Modelling and Applications) will also
bepublishedinthisseries.Theseproceedingsdisplaytheworldwidestate-of-the-art
inthisfieldandwillbeofinterestforawideraudiencethantheconferencepartici-
pants.ICTMAisaworldwideuniquegroup,inwhichnotonlymathematicseducators
aiming for education at school level are included but also applied mathematicians
interestedinteachingandlearningmodellingattertiarylevelarerepresented.ICTMA
discussesallaspectsrelatedtoTeachingandLearningofMathematicalModellingat
SecondaryandTertiaryLevel,e.g.usageoftechnologyinmodelling,psychological
aspectsofmodellinganditsteaching,modellingcompetencies,modellingexamples
andcourses,teachereducationandteachereducationcourses.
Moreinformationaboutthisseriesathttp://www.springer.com/series/10093
· · ·
Raphael Wess Heiner Klock Hans-Stefan Siller
Gilbert Greefrath
Measuring Professional
Competence for the Teaching
of Mathematical Modelling
A Test Instrument
RaphaelWess HeinerKlock
MathematicsandComputerScience MathematicsandComputerScience
UniversityofMünster UniversityofWürzburg
Münster,Germany Würzburg,Germany
Hans-StefanSiller GilbertGreefrath
MathematicsandComputerScience MathematicsandComputerScience
UniversityofWürzburg UniversityofMünster
Würzburg,Germany Münster,Germany
ISSN2211-4920 ISSN2211-4939 (electronic)
InternationalPerspectivesontheTeachingandLearningofMathematicalModelling
ISBN978-3-030-78070-8 ISBN978-3-030-78071-5 (eBook)
https://doi.org/10.1007/978-3-030-78071-5
©TheEditor(s)(ifapplicable)andTheAuthor(s)2021.Thisbookisanopenaccesspublication.
OpenAccessThisbookislicensedunderthetermsoftheCreativeCommonsAttribution4.0International
License(http://creativecommons.org/licenses/by/4.0/),whichpermitsuse,sharing,adaptation,distribu-
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Series Preface
Applicationsandmodellingandtheirlearningandteachinginschoolanduniversity
havebecomeaprominenttopicformanydecadesnowinviewofthegrowingworld-
wide relevance of the usage of mathematics in science, technology and everyday
life. There is consensus that modelling should play an important role in mathe-
maticseducation,andthesituationinschoolsanduniversityisslowlychangingto
includereal-worldaspects,frequentlywithmodellingasrealworldproblemsolving,
in several educational jurisdictions. Given the worldwide continuing shortage of
students who are interested in mathematics and science, it is essential to discuss
changesofmathematicseducationinschoolandtertiaryeducationtowardstheinclu-
sionofrealworldexamplesandthecompetenciestousemathematicstosolvereal
worldproblems.
ThisinnovativebookseriesestablishedbySpringerInternationalPerspectiveson
theTeachingandLearningofMathematicalModelling,aimsatpromotingacademic
discussionontheteachingandlearningofmathematicalmodellingatvariouseduca-
tionallevelsallovertheworld.Theserieswillpublishbooksfromdifferenttheoretical
perspectives from around the world dealing with Teaching and Learning of Math-
ematicalModelling inSchooling and atTertiarylevel.Thisserieswillalsoenable
theInternationalCommunityofTeachersofMathematicalModellingandApplica-
tions(ICTMA),anInternationalCommissiononMathematicalInstructionaffiliated
StudyGroup,topublishbooksarisingfromitsbiennialconferenceseries.ICTMAisa
uniqueworldwideeducationalresearchgroupwherenotonlymathematicseducators
dealingwitheducationatschoollevelareincludedbutalsoappliedmathematicians
interestedinteachingandlearningmodellingattertiarylevelarerepresentedaswell.
SixofthesebookspublishedbySpringerhavealreadyappeared.
Theplannedbooksdisplaytheworldwidestate-of-the-artinthisfield,mostrecent
educationalresearchresultsandnewtheoreticaldevelopmentsandwillbeofinterest
forawideaudience.Themesdealtwithinthebooksfocusontheteachingandlearning
of mathematical modelling in schooling from the early years and at tertiary level
includingtheusageoftechnologyinmodelling,psychological,social,historicaland
culturalaspectsofmodellinganditsteaching,learningandassessment,modelling
competencies,curricularaspects,teachereducationandteachereducationcourses.
v
vi SeriesPreface
Thebookseriesaimstosupportthediscussiononmathematicalmodellingandits
teachinginternationallyandwillpromotetheteachingandlearningofmathematical
modellingandresearchofthisfieldallovertheworldinschoolsanduniversities.
The series is supported by an editorial board of internationally well-known
scholars, who bring in their long experience in the field as well as their expertise
to this series. The members of the editorial board are: Maria Salett Biembengut
(Brazil),WernerBlum(Germany),HelenDoerr(USA),PeterGalbraith(Australia),
ToshikazuIkeda(Japan),MogensNiss(Denmark),andJinxingXie(China).
We hope this book series will inspire readers in the present and the future to
promotetheteachingandlearningofmathematicalmodellingallovertheworld.
SeriesEditors
Hamburg,Germany GabrieleKaiser
Ballarat,Australia GloriaAnnStillman
Introduction
From a mathematics education perspective, initiating and evaluating modelling
processes offers a great potential for the acquisition of competences by students.
Throughthechoiceofappropriatetasksandinterventions,allow,interalia,aclose
connection to reality as well as a high degree of independence and openness, thus
allowing individual access to mathematics and multiple solution approaches at
differentlevels(SchukajlowandKrug,2013).Atthesametime,thegreatpotentialof
thesemathematicalmodellingprocessesposesachallengefor(pre-service)teachers
(Kuntze, Siller and Vogel, 2013). For this reason, a close examination of possible
support measures to develop professional competence for teaching mathematical
modelling in the sense of quality development in teacher education is necessary
(Blum,2015).
Inthiscontext,theoperationalisationofmodelling-specificprofessionalcompe-
tence is an important and relatively new field of research (Borromeo Ferri, 2019)
towhichwewishtocontributewiththisbook.Indoingso,weunderstandcompe-
tencesascontext-specific,cognitivedispositionsforachievementthatarefunction-
allyrelatedtospecificsituationsandrequirements(Klieme,HartigandRauch,2008),
sothatnotonlydeclarativebutalsoprocedural,situation-relatedknowledgefacets
arefocused.
Modelsdescribingtherequirementsforpre-serviceteachersareneededtomeasure
the above competences. A structural model of professional competence to teach
mathematical modelling was developed as part of a cooperation1 between several
Germanuniversitiesandwasempiricallyconfirmedtoagreatextent(Klock,Wess,
GreefrathandSiller,2019;Wess,Klock,GreefrathandSiller,2021).Amongother
things,thismodelformsthebasisfortheconceptualisationandoperationalisationof
specificdidacticknowledgefacetsaswellasotheraffective-motivationalcomponents
ofmodelling-specificteacherprofessionalism.
1This project is part of the “Qualitätsoffensive Lehrerbildung”, a joint initiative of the Federal
GovernmentandtheLänderwhichaimstoimprovethequalityofteachertraining.Theprogramme
isfundedbytheFederalMinistryofEducationandResearch.Theauthorsareresponsibleforthe
contentofthispublication.
vii
viii Introduction
Inordertopresentthisstructureandtherelatedtestinstrumentinacomprehensive
manner,thefirstpartofthebookgivesanoverviewofselectedconceptsandtheo-
retical backgrounds of mathematical modelling. This is followed by explanations
ontheconceptofcompetenceusedhereandofdifferentcompetencemodelsbefore
specific competence dimensions for teaching mathematical modelling are consid-
ered. Finally, these are used to interpret a structural model of modelling-specific
professionalcompetenceandserveasabasisfortestconstruction.
Thesecondpartpresentsoperationalisationandthetestinstrumentastheproduct
ofthisprocess.Inthiscontext,theprimaryfocusisontheanalysisofthequalitywith
which the quantitative test instrument measures the aspects of modelling-specific
professionalcompetence.
In the final part, the possibilities of use, but also limitations of the instrument,
are discussed and further implications for research and practical applications are
highlighted.
Contents
PartI TheoreticalBackground
1 MathematicalModelling ........................................ 3
1.1 TermsandDefinitionsUsedinMathematicalModelling .......... 3
1.1.1 MathematicalModellingandMathematicalModel ........ 4
1.1.2 ModellingProcessesandModellingCycles .............. 5
1.1.3 ModellingCompetencies .............................. 7
1.2 AimsandPerspectivesofMathematicalModelling .............. 8
1.3 ModellingTasks ............................................ 11
1.3.1 GeneralCategoriesofTasks ........................... 11
1.3.2 TaskCategoriesforRealisticTasks ..................... 12
1.3.3 SelectedCriteriaforModellingTasks ................... 14
1.4 DifficultiesintheModellingProcess .......................... 15
1.5 InterventionsintheModellingProcess ........................ 17
2 ProfessionalCompetenceforTeachingMathematicalModelling .... 21
2.1 TheConceptofCompetence ................................. 22
2.2 ProfessionalCompetenceofTeachers ......................... 22
2.2.1 ProfessionalCompetence ............................. 23
2.2.2 Conceptualisations of Professional Competence
ofMathematicsTeachers .............................. 24
2.3 Competence Dimensions for Teaching Mathematical
Modelling ................................................. 26
2.4 ACompetenceModelforTeachingMathematicalModelling ..... 28
2.4.1 Modelling-SpecificPedagogicalContentKnowledge ...... 30
2.4.2 BeliefsRegardingMathematicalModelling .............. 31
2.4.3 Self-Efficacy Expectations for Mathematical
Modelling ........................................... 33
2.4.4 EmpiricalValidationoftheStructuralModel ............. 34
ix
