<p>Nowadays, Dynamic Geometry Software (DGS) is widely accepted as a tool for creating and presenting visually rich interactive teaching and learning materials, called dynamic drawings. Dynamic drawings are specified by writing expressions in functional domain-specific languages. Due to wide acceptance of DGS, there has arisen a need for their extensibility, by adding new semantics and visual objects (visuals). We have developed a programming framework for the Dynamic Geometry Software, SLGeometry, with a genericized functional language and corresponding expression evaluator that act as a framework into which specific semantics is embedded in the form of code annotated with metadata. The framework transforms an ordinary expression tree evaluator into an object-oriented one, and provide guidelines and examples for creation of interactive objects with dynamic properties, which participate in evaluation optimization at run-time. Whereas other DGS are based on purely functional expression evaluators, our solution has advantages of being more general, easy to implement, and providing a natural way of specifying object properties in the user interface, minimizing typing and syntax errors.LGeometry is implemented in C# on the .NET Framework. Although attributes are a preferred mechanism to provide association of declarative information with C# code, they have certain restrictions which limit their application to representing complex structured metadata. By developing a metadata infrastructure which is independent of attributes, we were able to overcome these limitations. Our solution, presented in this&nbsp; dissertation, provides extensibility to simple and complex data types, unary and binary operations, type conversions, functions and visuals, thus enabling developers to seamlessly add new features to SLGeometry by implementing them as C# classes annotated with metadata. It also provides insight into the way a domain specific functional language of dynamic geometry software can be genericized and customized for specific needs by extending or restricting the set of types, operations, type conversions, functions and visuals.Furthermore, we have conducted&nbsp; experiments with several groups of students of mathematics and high school pupils, in order to test how our approach compares to the existing practice. The experimental subjects tested mathematical games using interactive visual controls (UI controls) and sequential behavior controllers. Finally, we present a new evaluation algorithm, which was compared to the usual approach employed in DGS and found to perform well, introducing advantages while maintaining the same level of performance.</p> / <p>U dana&scaron;nje vreme softver za dinamičku geometriju (DGS) je &scaron;iroko prihvaćen kao alat za kreiranje i prezentovanje vizuelno bogatih interaktivnih nastavnih materijala i materijala za samostalno učenje, nazvanih dinamičkim crtežima. Kako je raslo prihvatanje softvera za dinamičku geometriju, tako je i rasla potreba da se oni pro&scaron;iruju, dodajući im novu semantiku i vizualne objekte. Razvili smo programsko okruženje za softver za dinamičku geometriju, SLGeometry, sa generičkim&nbsp; funkcionalnim jezikom i odgovarajućim evaluatorom izraza koji čini okruženje u kom su ugrađene specifične semantike u obliku koda označenog metapodacima. Ovo okruženje pretvara uobičajen evaluator stabla izraza u objektno orijentiran, te daje uputstva i primere za stvaranje interaktivnih objekata sa dinamičkim osobinama, koji sudeluju u optimizaciji izvr&scaron;enja tokom izvođenja. Dok se drugi DGS-ovi temelje na čisto funkcionalnim evaluatorima izraza, na&scaron;e rje&scaron;enje ima prednosti jer je uop&scaron;tenije, lako za implementaciju i pruža prirodan način navođenja osobina objekta u korisničkom interfejsu, minimizirajući kucanje i sintaksne gre&scaron;ke. SLGeometry je implementirana u jeziku C# .NET Framework-a. Iako su atributi preferiran mehanizam, koji povezuje C# k&ocirc;d sa deklarativnim informacijama, oni imaju određena ograničenja koja limitiraju njihovu primenu za predstavljanje složenih strukturiranih metapodataka. Razvijanjem infrastrukture metapodataka koja je nezavisna od atributa, uspeli smo prevladati ta ograničenja. Na&scaron;e re&scaron;enje, predstavljeno u ovoj disertaciji, pruža pro&scaron;irivost: jednostavnim i složenim vrstama podataka, unarnim i binarnim operacijama, konverzijama tipova, funkcijama i vizuelnim objektima, omogućavajući&nbsp; time programerima da neprimetno dodaju nove osobine u SLGeometry&nbsp; implementirajući ih kao C# klase označene metapodacima.</p>

The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.

In our presentation we will show how technology enables us to improve the teaching and learning of linear functions at the middle school level. Through various classroom activities that involve technology such as dynamic geometry software, graphing calculators and Excel, students explore functions and discover basic facts about them on their own. Students then work with real life data and on real life problems to draw graphs and form linear models that correspond to given situations as well as draw inferences based on their models. Participants will receive complete classroom materials for the unit on linear functions.

The aim of the workshop is to present and discuss several ideas which relate to technology as well as to creative teaching. Educational experience, common sense and educational research have all proven how important for comprehensive understanding different cognitive representations are. We will present and discuss several elementary mathematical ideas of which mechanical realisations mean ingenius technological inventions (for example: ‘car differential’ and ‘digital sound technology’). Technological insights can provide deep intuitive understanding of otherwise abstract mathematical concepts and therefore yield also better comprehension of mathematics. Besides that we will use and present the technology in the form of dynamic geometry programs to show, provoke and motivate rethinking and deeper understanding of several elementary mathematical concepts.

The aim of the workshop is to present and discuss several ideas which relate to technology as well as to creative teaching. Educational experience, common sense and educational research have all proven how important for comprehensive understanding different cognitive representations are. We will present and discuss several elementary mathematical ideas of which mechanical realisations mean ingenius technological inventions (for example: ‘car differential’ and ‘digital sound technology’). Technological insights can provide deep intuitive understanding of otherwise abstract mathematical concepts and therefore yield also better comprehension of mathematics. Besides that we will use and present the technology in the form of dynamic geometry programs to show, provoke and motivate rethinking and deeper understanding of several elementary mathematical concepts.

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.

Active Approaches provide chances for individual, comprehension-oriented learning and can facilitate the acquirement of general mathematical competencies. Using the example of pinboards, which were developed for different areas of the secondary level, workshop participants experience, discuss and further develop learning tasks, which can be used for free activities, for material based concept formation, for coping with heterogeneity, for intelligent exercises, as tool for the presentation of students’ work and as basis for games. The material also allows some continuous movements and can thus prepare an insightful usage of dynamic geometry programs. Central Part of the workshop is a work-sharing group work with learning tasks for grades 5 to 8. The workshop will close with a discussion of general aspects of material-based learning.

Active Approaches provide chances for individual, comprehension-oriented learning and can facilitate the acquirement of general mathematical competencies. Using the example of pinboards, which were developed for different areas of the secondary level, workshop participants experience, discuss and further develop learning tasks, which can be used for free activities, for material based concept formation, for coping with heterogeneity, for intelligent exercises, as tool for the presentation of students’ work and as basis for games. The material also allows some continuous movements and can thus prepare an insightful usage of dynamic geometry programs. Central Part of the workshop is a work-sharing group work with learning tasks for grades 5 to 8. The workshop will close with a discussion of general aspects of material-based learning.