Es ist eine weit verbreitete Wahrnehmung, dass der Übergang zwischen der Mathematik der gymnasialen Oberstufe und der Mathematik an der Universität für Studierende problematisch sein kann. Besondere Verständnisschwierigkeiten in Bereich der lineare Algebra (lA) bereiten den Studierenden die verschiedenen Herangehensweisen auf diesen beiden Ebenen. Dies lässt sich auf die strukturell-axiomatischer Herangehensweisen an die lA an der Universität, im Gegensatz zu ihrer arithmetisch-geometrischen Darstellung in der Schule, zurückführen. Dies bedingt ebenfalls Unterschiede im prozeduralen und konzeptuellen Verständnis. Ziel dieser Arbeit ist es, zu untersuchen, wie Schüler konzeptuelles Verständnis, Bezug nehmend auf die Theorien von concept definition/image in Verbindung mit multiplen Modi der Beschreibung und des Denkens von Konzepten wie Bilinearität z.B. Skalarprodukt und Multilinearität z.B. Determinanten gewinnen können. Um dies zu erreichen wurde eine substanzielle Lehr-Lernumgebung unter Verwendung einer dynamischen Geometriesoftware (DGS) entwickelt. Die Lerneinheit wurde an einem Berliner Gymnasium eingesetzt und dabei ein vollständiger design-based research Zyklus durchlaufen und eine multiple-level Datenanalyse durchgeführt. Die Ergebnisse der Untersuchung zeigen nicht nur, dass eine Erweiterung der Vorstellungen der Schüler, eine Entwicklung multipler Denkmodi und ein Gewinn tieferen konzeptuellen Verständnisses in der lA erfolgreich vermittelt werden können, sondern geben auch Einblicke in ein mögliches theoretisches Modell, mit dessen Hilfe sich diese Prozesse weiter untersuchen lassen. Weiterhin werden die interaktiven Lehr-Lernmaterialien für die weitere Verwendung im Rahmen von Lehre und Forschung zur Verfügung gestellt. Es öffnen sich neue Forschungsfragen hinsichtlich lokalen Axiomatisierens in der lA der gymnasialen Oberstufe, welches auf einer Integration geometrischer, algebraischer und axiomatischer Denkmodi, unterstützt durch DGS, basieren könnte. / A common perception among researchers in mathematics education is that the transition between secondary- and tertiary level of mathematics may be problematic for the students. In particular, the exact and abstract nature of the theory of Linear algebra versus its arithmetic-geometric presentation in school appears to be difficult for the novice students. The application of properties for defining concepts at university in contrast to their usage for describing concepts in school points out a possible occurrence of obstacles for learning and discrepancies in procedural and conceptual understanding. The aim of this study is to examine how could upper-high school students develop a conceptual understanding based on concept definition and concept image in connection to multiple modes of description and thinking about concepts such as bi-linearity exemplified by the dot product of vectors and multi-linearity exemplified by determinants. In order to achieve this, I have created a specific teaching/ learning sequence in a dynamic geometry environment (DGE), then implemented it and evaluated it in a high school in Berlin, following a complete cycle of design-based research and conducting a multiple-level data analysis. The findings of the study show not only that widening students' concept images, developing multiple modes of thinking and gaining deeper conceptual understanding can successfully be mediated by dynamic geometries, but also give insights into an eventual theoretical model of how can they be further examined. Moreover, the study promotes authorized open-source interactive teaching/ learning materials for further sustainable practice and research. It opens new research questions about revisiting axiomatic approaches on local levels in upper high-school Linear algebra which may base on the integration of all three modes of description and thinking geometric, algebraic and abstract possibly facilitated by DGE. / Честа перцепција кај многумина истражувачи во областа на математичкото образование е дека транзицијата помеѓу средното и високото образование по математика може да биде проблематична за студентите. Егзакноста и апстрактноста на теоријата по Линеарна алгебра наспроти нејзината аритметичко-геометриска презентација во средното гимназиско образование се покажува како особено тешка за студентите. Примена на својствата на математичките поими за нивно дефинирање на универзитетско ниво наспроти нивното употреба за опишување на претходно дефинирани поими на училишно ниво, укажува на можна појава на тешкотии при нивното изучување и несовпаѓање на процедуралното и концептуалното разбирање на истите. Целта на оваа студија е да истражи како средношколците би можеле да развијат концептуално разбирање на поимите врз основа на концепт дефиниција и концепт слика во врска со мулти-моди на мислење, конкретно за поими како билинеарност, пр. скаларен производ на вектори, и мултилинеарност, пр. детерминанти. За да ја постигнам оваа цел, креирав наставна содржина поддржана од еден динамичен геометриски систем (ДГС) и следејќи целосен циклус на т.н. design-based research и спрoведувајќи мулти-анализа на податоци, истата ја имплементирав и евалуирав во едно средно училиште во Берлин. Резултатите од студијата укажуваат не само на фактот дека проширувањето на концепт сликите на учениците, развојот на мулти-моди на мислење и стекнувањето на длабоко концептуално разбирање на поимите можат да бидат успешно посредувани од ДГС туку овозможија и увид во еден теоретски модел за тоа коко тие можат понатаму да се истражуваат. Уште повеќе, студијата промовира авторизирани open-source интерактивни материјали за предавање и учење на содржините кои може да служат за понатамошни одржливи истражувања и развој. Студијата отвора нови истражувачки прашања за средношколската Линеарна алгебра која може да се базира на интеграција на сите три моди на мислење, геометриски, алгебарски и апстрактен, поддржан од ДГС.

Abstract in English, Tswana and Northern Sotho / The study aimed to explore teachers’ mathematical knowledge in respect of teaching the concept of fractions to Grade 6 learners. To that end a qualitative study was done, using a case study design. Data were collected through the observation of, and interviews with, three teachers at three schools in the Capricorn South district. Rooted in the theory of constructivism, the study was supplemented by the conceptual framework of mathematical knowledge for teaching (MKT) (Ball et al., 2008) and Shulman’s (1986) notion of pedagogical knowledge for teaching (PCK). The key finding of this investigation revealed that, of the three teachers, two did not develop the concept of fractions for their learners, but merely followed the traditional method of teaching the concept by encouraging their learners to memorise rules without understanding. Only one teacher emphasised an understanding of mathematical concepts. The main observation which the researcher made, was that teachers require a great deal of knowledge and expertise, in carrying out the work of teaching subject matter related to fractions. / Maikaelelo a thutopatlisiso e ne e le go tlhotlhomisa kitso ya dipalo ya barutabana malebana le go ruta barutwana ba Mophato wa 6 mogopolo wa dikarolwana. Go fitlhelela seo, go dirilwe thutopatlisiso e e lebelelang mabaka, go dirisiwa thadiso ya thutopatlisiso ya dikgetsi. Go kokoantswe data ka go ela tlhoko le go nna le dipotsolotso le barutabana ba le bararo kwa dikolong tsa kgaolo ya Capricorn Borwa. Thutopatlisiso eno e e theilweng mo tioring ya kago ya kitso e ne e tshegeditswe ke letlhomeso la sediriswa sa tokololo sa kitso ya dipalo ya go ruta (MKT) (Ball et al. 2008) le mogopolo wa ga Shulman (1986) wa kitso e e kgethegileng ya go ruta (PCK). Phitlhelelo ya botlhokwa ya patlisiso eno e senotse gore mo barutabaneng ba le bararo, ba le babedi ga ba a tlhamela barutwana ba bona mogopolo wa dikarolwana, mme ba latetse fela mokgwa wa tlwaelo wa go ruta mogopolo ka go rotloetsa barutwana go tshwarelela melawana kwa ntle ga go tlhaloganya. Ke morutabana a le mongwe fela yo o gateletseng go tlhaloganngwa ga megopolo ya dipalo. Temogo e kgolo e e dirilweng ke mmatlisisi ke gore barutabana ba tlhoka kitso le boitseanape jo bogolo go tsweletsa tiro ya go ruta dithuto tse di amanang le dikarolwana. / Dinyakišišo di ikemišeditše go utolla tsebo ya dipalo ya baithuti mabapi le go ruta kgopolo ya dipalophatlo go baithuti ba Kreiti ya 6. Ka lebaka la se go dirilwe dinyakišišo tša boleng, go šomišwa tlhamo ya dinyakišišo tša seemo. Tshedimošo e kgobokeditšwe ka go lekodišiša, le go dira dipoledišano le, barutiši ba bararo ka dikolong tše tharo ka seleteng sa Borwa bja Capricorn. Ka ge di theilwe go teori ya gore baithuti ba itlhamela tsebo, dinyakišišo di tlaleleditšwe ke tlhako ya boikgopolelo ya tsebo ya dipalo go ruteng (MKT) (Ball le ba bangwe, 2008) le kgopolo ya Shulman (1986) ya tsebo ya diteng tša thuto (PCK). Kutollo ye bohlokwa ya dinyakišišo tše e utollotše gore, go barutiši ba bararo, ba babedi ga se ba ba le kgopolo ya dipalophatlo go baithuti ba bona, eupša fela ba no latela mokgwa wa setlwaedi wa go ruta kgopolo ye ya dipalophatlo ka go hlohleletša baithuti ba bona go tsenya melawana ye ka hlogong ka ntle le go e kwešiša. Ke fela morutiši o tee yo a gateletšego gore go swanetše go ba le kwešišo ya dikgopolo tša dipalo. Temogo e tee yeo monyakišiši a bilego le yona, ebile gore barutiši ba hloka tsebo ye kgolo le botsebi, go phethagatša mošomo wa go ruta diteng tša thuto tšeo di amanago le dipalophatlo. / Mathematics Education / M. Ed. (Mathematics Education)