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Josip Juraj Strossmayer University of Osijek 2017 Faculty of Education Department of Mathematics M A T H E E M dito AT r s I : Z C d S e n E k a D K U o C la r A - B T e I g O o v N ić , R AS u žic A a S K C o la IE r -Š N u C p e E r , L A MATHEMATICS EDUCATION AS je N r k D a J A A SCIENCE AND A PROFESSION u k P ić R M O a tić F E S Editors: S I O Zdenka Kolar-Begović N Ružica Kolar-Šuper Ljerka Jukić Matić 2017 ISBN 978-953-197-592-6 JosipJurajStrossmayerUniversityofOsijek FacultyofEducationandDepartmentofMathematics Sveucˇilisˇte JosipaJurjaStrossmayera uOsijeku Fakultetzaodgojne iobrazovne znanostiiOdjelzamatematiku MATHEMATICS EDUCATION AS A SCIENCE AND A PROFESSION ˇ MATEMATICKO OBRAZOVANJE KAO ZNANOST I PROFESIJA monograph Editors: ZdenkaKolar-Begovic´ RuzˇicaKolar-Sˇuper LjerkaJukic´ Matic´ Osijek,2017 InternationalScientiﬁcCommittee JosipJurajStrossmayerUniversityofOsijek RudolfScitovski MirtaBensˇic´ ZdenkaKolar-Begovic´ RuzˇicaKolar-Sˇuper LjerkaJukic´Matic´ IvanMatic´ NenadSˇuvak AnaMirkovic´Mogusˇ UniversityofZagreb ZˇeljkaMilinSˇipusˇ AleksandraCˇizˇmesˇija VladimirVolenec Foreignuniversities AndersHast(Sweden) EmilMolna´r(Hungary) NatasˇaMacura(USA) EdithDebrenti(Romania) LjiljankaKvesic´ (BosniaandHerzegovina) TheEditorialBoard EmilMolna´r(Hungary) BettinaDahl(Denmark) AndersHast(Sweden) NatasˇaMacura(USA) ZdenkaKolar-Begovic´(Croatia) RuzˇicaKolar-Sˇuper(Croatia) LjerkaJukic´Matic´ (Croatia) EdithDebrenti(Romania) LjiljankaKvesic´ (BosniaandHerzegovina) Monographreferees: AttilaBo¨lcskei(Hungary) SˇimeUngar(Croatia) SanjaRukavina(Croatia) Supportedby: FacultyofEducation,UniversityofOsijek DamirMatanovic´,Dean DepartmentofMathematics,UniversityofOsijek MirtaBensˇic´,HeadofDepartment A word from the Editorial Board Mathematics education as a research field experienced significant growth in the last decades. Nonetheless, we still experience the same problems over and over. Thecontinuousproblemwithtransitionfromprimarytosecondaryeducation and from secondary to tertiary education is ever present. (Corriveau & Bednarz, 2017; Clark & Lovric, 2008; Thomas, 2008). It seems that mathematics teachers ofeacheducationalcyclehaveratherdifferentexpectationsoftheknowledgeastu- dentshould bring from onecycle toanother. Themathematics learning process is connected withdoingmathematics andprocesses suchasinvestigating, reflecting, reasoning,givingarguments,findingconnections,etc. Theresearchinmathematics educationhasbeenemphasizingthisaspectoflearningmathematicsformorethan twentyyears. However,manyteachersandlearnersfocusonimitatingprocedures. What causes this discrepancy between the research and practice? The findings in mathematics education showed that the teacher had a significant role on students’ achievements. Moreover, the way they conceived mathematics reflected on the learners. If the teachers saw it as a set of procedures and algorithms, they would transfer this conception of mathematics onto their students. On the other hand, if they saw it as collections of ideas, their students would have the same conception ofmathematics. Thestudent’s attitudestowardsmathematicsalsoinfluenced their learning process. This shows that attitudes and beliefs in teaching and learning mathematics have significant impact on the outcomes in the classroom. That is why they are considered to be hidden variables in mathematics education (Leder, Pehkonen, To¨rner, 2002), which cannot be omitted from the didactical triangle mathematics–teacher–students. Intheprimaryandsecondaryeducation, mathematicsisacompulsorysubject in schools, but moving to the tertiary education, we find mathematics as the key component of many natural and technical sciences. Mathematics is the heart of STEM.Itisimportantforthosewhowillcontinuetheirprofessioninthatdirection. Moreover, a strong interaction between mathematics and technology is present: developments in technology stimulate mathematics and developments in mathe- maticsoftenenhance innovations intechnology (Ho¨ft,2016). Butmathematics in theformofstatisticsappearsinhumanitiesandsocialsciences. Thereforeitseems thatmathematicsisthecorescienceandthekeyfeatureofasuccessful individual. This is also supported by the definition of mathematical literacy in PISA, where theroleofmathematics isdescribed asapart ofanindividual’s current andfuture privatelife,professionallifeandsociallifewithpeersandrelatives(OECD;2013). Considering the rapid progress of technology in the world of today, we face new challenges. Living inthe age of digital technologies, wefeel obligated toin- corporatethemintoourlivesbutalsointheotherareassuchastheschoolsubjects. Thisissueraises manystillunanswered questions. Couldtechnology improveour teaching and learning of mathematics? Should we use it constantly or occasion- ally? Howcanthistechnologybeusedefficiently? Dowedesigntasksinasimilar manner with this technology as it was the case with pencil and paper technique? Modern technology and its use in mathematics education has been examined in various studies forseveral decades. Thefindings ofthose studies showedbenefits oftheproperuseoftechnology. Spreadsheetsanddynamicsoftwareenvironments enablefastercollectingofthespecificcasesofmathematicalphenomena –values, variables,functions,shapes,locationsalongagraph,orthepropertiesofgeometric constructions(Aldon,Hitt,Bazzini&Gellert,2017). Thisprovidestheopportuni- tiesforlearnerstoconsidercollectionofcasesratherthanindividualinstances. But Applebaum (2017) warns against jumping to conclusion that technology forces automatic generalization in mathematics; this depends on the learner, similarly as it is the case with the classic environments. Digital technology can speed up the learning process, but sometimes educators want to slow this process down. They notice that the amount of information is overwhelming for the learner. At other times, technology narrows our focus too much, or not enough (Appelbaum, 2007). Therefore, this brings us to the conclusion that the use of technology has itsadvantages anddisadvantages whichteachersshould beawareof. We addressed these issues and some other in this monograph. Some papers opened new questions for research, some showed examples of good practice and others provided more information for the earlier findings. All papers portray the complex role of mathematics education: mathematics education is a science but alsoaprofession. Researchinmathematicseducationisneededtoinvestigatevari- ousphenomena, buttheresearchhasalsotheresponsibility toinformpractitioners of its findings. Mathematics teachers are a critical component in this process. Their collaboration with the researchers is necessary to achieve the shift from the traditionaltocontemporaryteachingmathematics. Thisprocessisneithereasynor rapid. Butwemustbepersistentinourefforts,becausemathematicsisinseparable fromoureverydaylife. References [1] ALDON, G., HITT, F., BAZZINI,L., & GELLERT,U. (Eds.) (2017), Mathematics and Technology. A C.I.E.A.E.M. Sourcebook, Springer, doi. 10.1007/978-3-319-51380- 5 15. [2] APPELBAUM,P.(2007),Children’sbooksforgrown-upteachers: Readingandwriting curriculumtheory,NewYork: Routledge. [3] APPELBAUM, P. (2017), in G. Aldon, F. Hitt, L. Bazzini, & U. Gellert (eds), MathematicsandTechnology.AC.I.E.A.E.M.Sourcebook,(pp.335–345),Springer, doi.10.1007/978-3-319-51380-515. [4] CLARK,M.&LOVRIC,M.(2008),Suggestionforatheoreticalmodelforsecondary- tertiarytransitioninmathematics,MathematicsEducationResearchJournal,20(1), 25–37. [5] CORRIVEAU,C.&BEDNARZ,N.(2017),Thesecondary-tertiarytransitionviewedas achangeinmathematicalcultures: anexplorationconcerningsymbolismanditsuse, EducationalStudiesinMathematics,95(1),1–19,doi:10.1007/s10649-016-9738-z. [6] HO¨FT, M. (2016), IT/Mathematics: Statistical Science, in M. Duran, M. Ho¨ft, B. Medjahed, D. Lawson & E. A. Orady (Eds.), STEM Learning, (pp. 121–152), Springer,doi.10.1007/978-3-319-26179-96. [7] LEDER, G. C., PEHKONEN, E., & TO¨RNER, G. (Eds.) (2002), Beliefs: A hidden variableinmathematicseducation?,Dordrecht,TheNetherlands: KluwerAcademic Publishers. [8] OECD (2013), PISA 2012 Assessment and Analytical Framework: Mathematics, Reading, Science, Problem Solvingand FinancialLiteracy, OECD Publishing, doi. 10.1787/9789264190511-en. [9] THOMAS, M. O. J. (2008), The transition from school to university and beyond, MathematicsEducationResearchJournal,20(1),1–4. Osijek,May10,2017 TheEditorialBoard Contents Preface ......................................................... 3 1. Students’strategies forproblemsolving BettinaDahl Thestudyapproaches ofuniversity students inacalculus class .......... 7 ZˇeljkaMilinSˇipusˇ,MajaPlaninic´,AnaSusˇac,LanaIvanjek Searchingforacommongroundinmathematics andphysics education: thecaseofintegral ...................................... 18 MateaGusic´ Functionsinthe2015and2016CroatianStateMatura inhigherlevelMathematics ........................................ 28 2. Fosteringgeometric thinking NikolinaKovacˇevic´ Spatialreasoning inmathematics ................................... 45 EmilMolna´r,Istva´nProk,Jeno˝ Szirmai Thefootball{5,6,6}anditsgeometries: fromasporttool tofullerens andfurther ............................................ 66 A.S.Leeds,NatasˇaMacura,ZacharyMoring Holesinalienquilts ............................................... 88 3. Theroleandimportanceofmathematicstextbooks LjerkaJukic´ Matic´,DubravkaGlasnovic´ Gracin Teachers’beliefsonmathematicsasabackground fortheir teachingpractice .................................................. 109 AleksandraCˇizˇmesˇija,AnaKatalenic´,ZˇeljkaMilinSˇipusˇ Asymptoteasabodyofknowledgetobetaughtintextbooks forCroatiansecondary education ................................... 127 GoranTrupcˇevic´,AndaValent LanguageofCroatianmathematical textbooks ........................ 147 4. Integration ofICTintheteachingmathematics ZˇeljkaDijanic´,GoranTrupcˇevic´ TheimpactofusingGeoGebrainteractive appletsonconceptual andprocedural knowledge ......................................... 161 BojanKovacˇic´,MirelaKatic´ Zˇlepalo Theuseofthecomputerprogram Graphinteaching application ofdifferential calculus ............................................. 175 LukaMarohnic´,MandiOrlic´ Bachler Applications offreecomputational softwareinmathcourses atZagrebUniversityofAppliedSciences ............................ 192 5. Attitudestowardandbeliefsaboutmathematicsteaching JosipaMatotek Mathematicsattitudes amongstudents ofCivilEngineering ............ 209 DusˇanMundar,DamiraKecˇek Targetingadditional effortforstudents’ success improvement: Thehighesteffectgroupselection method ............................ 223 Ivana Durdevic´ Babic´,AnaKozic´,TomislavMilic´ Discoveringstudentprofiles withregardtotheuseofmathematics tutoringservices atuniversity level .................................. 237 Ivana Durdevic´ Babic´,TomislavMilic´,AnaKozic´ Identifying mathematical anxietywithMLPandRBFneuralnetworks ... 250 ZoranHorvat Standardization oflearning outcomesinteachingmathematics .......... 258 KsenijaRomstein,LjiljanaPintaric´ Mlinar Teachingandlearning mathematicsininclusivesettings: Analysis ofcurriculum ofcompulsory education infiveEuropeancountries ....... 269 6. Someapproachestoteachingmathematics AmandaGlavasˇ,AzraStasˇcˇik Enhancingpositiveattitude towardsmathematicsthrough introducing EscapeRoomgames .................................... 281 RuzˇicaKolar-Sˇuper,AndreaSadric´,ZdenkaKolar-Begovic´,PetraAbicˇic´ Thepresenceofmathematicalgamesinprimaryschool ................ 295 EdithDebrenti Problemsolving inelementary mathematicseducation ................. 308 Index ........................................................... 325 MATHEMATICS EDUCATION AS A SCIENCE AND A PROFESSION monograph