Title: Construction of elementary pupils' geometric schemas via motivating learning environments Author: Jaroslava Kloboučková Department: Department of mathematics and mathematics education Supervisor: doc. RNDr. Darina Jirotková, Ph.D. Abstract: The aim of the dissertation is to discuss teaching geometry as integral part of mathematics education at the primary school level. The thesis also documents a longitudinal teaching study which was initiated in 2010 and which gives us a base for discussion of some fundamental questions regarding the process of learning geometry for pupils in their early school years. The main objective here is to attempt to answer the following four didactic questions: In which way do pupils learn about geometrical objects? How do they share their geometrical knowledge, experience and discoveries with one another? How much (at what level) are they able to understand mathematical concepts that the official curricular documents (the Czech Framework for Education Program) place in later years of schooling? What phenomena are they able to grasp and describe using their mother tongue? The theoretical framework focuses on the learning process and the typology of mathematical problems in geometry. Four specific engaging environments (Cube Buildings, Origami, Wooden Sticks, and Tiles) and...

Der Grundbaustein des Lebens, die Desoxyribonukleinsäure (DNS), ist aufgrund ihrer spezifischen Basenpaarung ein geeignetes Molekül, um stabile und vielfältige nano- beziehungsweise mikrometergroße Strukturen herzustellen. Diese selbstorganisierten DNS-Strukturen eignen sich als Grundeinheiten für die Ausrichtung anorganischer und organischer Materialien. Für die Synthese solcher DNS-Strukturen werden insbesondere die Kachel-basierte Assemblierung (engl. tile-based assembly, im Folgenden als Tile-basierte Assemblierung bezeichnet) oder die DNS-Origami-Methode verwendet. Die Tile-basierte Assemblierung beinhaltet die Verbindung einzelner DNS-Bausteine, den sogenannten Kacheln (engl. tiles), zu komplexeren DNS-Strukturen. Hingegen entspricht die DNS-Origami-Methode der Faltung eines langen einzelsträngigen DNS-Moleküls, dem sogenannten scaffold, anhand von hunderten kurzen Oligonukleotiden (Heftklammer-Oligomeren, engl. staple strands) hin zu einer entsprechenden Form. Hinsichtlich einer zukünftigen Erstellung von DNS-basierten, nanoelektronischen Systemen war das Ziel dieser Arbeit einheitliche zwei- (2D) und dreidimensionale (3D) DNS-Nanostrukturen herzustellen, Methoden für deren kontrollierte Vernetzung zu entwickeln sowie deren chemische Funktionalisierung mit Nanomaterialien und einer beispielhaften Integration in lithographisch gefertigten Mikrokontaktstrukturen durchzuführen. Hierfür war es notwendig, einen weiten Bogen zu spannen, welcher einerseits verschiedene Konstruktionsprinzipien der DNS-Nanotechnologie vorteilhaft miteinander vereint und der andererseits die weitreichenden Möglichkeiten der chemischen Funktionalisierung der sogenannten DNS-Templatstrukturen auslotet. Konkret wurden zur Erstellung von einheitlichen DNS-Strukturen Assemblierungskonzepte verwendet bzw. entwickelt, welche auf die Ausrichtung einzelner kurzer Oligonukleotide anhand eines langen einzelsträngigen DNS-Templates beruhen. Im ersten Teil der Arbeit ist anhand eines selbstkomplementären Einzelstranges aufgezeigt, wie sich prinzipiell die Wachstumsrichtung einer Tile-basierten Struktur durch die Verwendung eines einzelsträngigen DNS-Templates beeinflussen lässt. Bei diesem Ansatz bildet sich entlang des DNS-Templates eine 2D-Gitterstruktur aus einheitlichen und abschnittsweise selbstkomplementären hexagonalen oder tetragonalen Oligonukleotideinheiten aus. Diese gerichtete Selbstassemblierung führt schließlich zum Aufrollen und Zusammenschluss der 2D-DNS-Struktur zu einer tubulären Struktur. Die Größe und Geometrie der Oligonukleotideinheiten bestimmen dabei maßgeblich den Durchmesser dieser DNS-Nanoröhren. Zur Erklärung von experimentellen Beobachtungen wurde ein Modell entwickelt, welches die Templat-gestützte Assemblierung theoretisch beschreibt. Die erstellten, strukturellen Anforderungen genügenden Nanoröhren eignen sich für eine gleichmäßige Funktionalisierung mit Nanomaterialien, wie anhand der Ausrichtung von Gold-Nanopartikeln gezeigt wurde. In einem weiteren Teil der Arbeit wurde eine ca. 400 nm lange DNS-Nanoröhre anhand der DNS-Origami-Methode erstellt. Diese Nanoröhre diente als Modellsystem zur Untersuchung der Integration von tubulären DNS-Strukturen in Mikrokontaktstrukturen mittels der Dielektrophorese. Eine positive dielektrophoretische Antwort der 3D-DNS-Strukturen konnte im MHz-Bereich festgestellt werden. Des Weiteren wurde für mit Gold-Nanopartikeln funktionalisierte DNS-Nanoröhren eine verstärkte dielektrophoretisch Antwort beobachtet. Neben der Manipulation bzw. Ausrichtung von DNS-Nanostrukturen wurden Konzepte entwickelt, welche zusätzlich zum Aufbau komplexer DNS-Netzwerke innerhalb einer Mikrokontaktstruktur erforderlich sind. Konkret konnte eine Verbindung der 3D-Nanoröhren (i) untereinander über eine 200 nm lange kreuzartige DNS-Zwischenstruktur und (ii) endständig mit einer Goldoberfläche ermöglicht werden. Der dritte Teil dieser Arbeit befasste sich mit der Entwicklung einer modularen 2D-DNS-Struktur, welche unter anderem für eine vergleichbare Untersuchung zur Immobilisierung von Nanomaterialien auf DNS-Strukturen dienen kann. Anhand der DNS-Origami-Methode wurde eine spezifische DNS-Gerüststruktur entworfen, welche die Ausstattung mit einer funktionalisierbaren Tile-basierten Einheit erlaubt. Um die Modularität der DNS-Gerüststruktur zu verdeutlichen, wurden zwei unterschiedliche, drei-beinige Tiles entworfen und anhand eines Ein- oder Zwei-Schritt-Verfahrens in die DNS-Gerüststruktur integriert. Die Anbindung eines Gold-Nanopartikels an jedes Bein des eingebundenen Tiles demonstriert die spezifische Funktionialisierbarkeit dieses Modellsystems. Zudem wurden Methoden, welche zur Aufreinigung der funktionalisierten DNS-Gerüststrukturen dienen, wie auch Effekte der Vernetzung von DNS-Origami-Strukturen anhand unspezifischer Wechselwirkungen untersucht. Die Ermittlung der Struktureigenschaften beziehungsweise der Assemblierungsqualität der in dieser Arbeit gezeigten DNS-Strukturen erfolgte mittels elektrophoretischer und bildgebender Untersuchungsverfahren (Rasterkraftmikroskopie, Transmissionselektronenmikroskopie, Rasterelektronenmikroskopie).

This paper addresses folding the circle in half and discussing some of over one hundred different mathematical terms and functions generated in that one fold. The simplicity of process in understanding fundamentals of mathematics by folding circles and observing what is generated is unknown because we only draw pictures of circles. Examples are given about observing and exploring relationships in the circle that are appropriate for first, second, third grade level and beyond. The traditional educational ‘parts-towhole’ approach can only be fully realized through the comprehensive frame of Whole-to-parts by folding the circle. Wholemovement of the circle is not only direct; it is the only context inclusive to progressively understanding parts within unity of the Whole.

This thesis forms part of an ongoing project in the DNA Group to build and operate a synthetic ribosome. We present two synthetic ribosome designs that can be combined with DNA-templated chemistry to generate libraries of functional synthetic small molecules. In Chapter 2 we use the DNA strand displacement technique to construct a mechanism that is capable of moving along a DNA track. We explore ways to control the speed and the driving force of the mechanism, and present a mathematical model of the system. We discuss the ability of the design to incorporate chemically-functionalised DNA strands. In Chapter 3 we use a 2D DNA origami tile as the basis of the synthetic ribosome mechanism. Functionalised DNA strands are arranged on the surface of the tile, and we demonstrate the ability to template reactions between the strands, and discuss the possibility of creating a library of distinct chemical products from a single origami tile.

This thesis considered the design of new and improved foldcore sandwich panels suitable for high-performance energy absorption applications. This was achieved by utilising origami geometry design techniques to alter foldcore structures such that they possessed different mechanical behaviours and failure modes. The major findings of this thesis were in three areas as follows. First, a modified planar foldcore geometry was developed by introducing sub-folds into a standard foldcore pattern. The new geometry, deemed the indented foldcore, successfully triggered a high-order failure mode known as a travelling hinge line failure mode. This was found to have a much higher energy absorption than the plate buckling failure mode seen in an unmodified foldcore structure. A comprehensive numerical, theoretical, and experimental analysis was conducted on the indented core, which included the development of a new foldcore prototyping method that utilised 3D printed moulds. It was shown that compared to available commercial honeycomb cores, the indented foldcore had an improved uniformity of energy absorption, but weaker overall peak and crushing stresses. Second, rigid origami design principles were used to develop extended foldcore geometries. New parametrisations were presented for three patterns, to complete a set of Miura-derivative geometries termed first-level derivatives. The first-level derivative parametrisations were then combined to create complex, piecewise geometries, with compatible faceted sandwich face geometry also developed. Finally, a method to generate rigid-foldable, curved-crease geometry from Miura-derivative straight-crease geometry was presented. All geometry was validated with physical prototypes and was compiled into a MATLAB Toolbox. Third, the performance of these extended foldcore geometries under impact loadings was investigated. An investigation of curved-crease foldcores showed that they were stronger than straight-crease foldcores, and at certain configurations can potentially match the strength, energy-absorption under quasi-static impact loads, and out-of-plane stiffness of a honeycomb core. A brief investigation of foldcores under low-velocity impact loadings showed that curved-crease foldcores, unlike straight-crease foldcores, strengthened under dynamic loadings, however not to the same extent as honeycomb. Finally, an investigation of single-curved foldcore sandwich shells was conducted. It was seen that foldcore shells could not match the energy-absorption capability of an over-expanded honeycomb shell, but certain core types did exhibit other attributes that might be exploitable with future research, including superior initial strength and superior uniformity of response.

The thesis titled /Origami in primary lessons/ focuses on the practical use of Origami, especially in arts and mathematics classes. It contains lesson plans on various topics and a detailed reflection of the lessons taught at a primary school.

In the past decade, DNA origami self-assembly has been widely applied for creating customised nanostructures with base-pair precision. In this technique, the unique chemical addressability of DNA can be harnessed to create programmable architectures, using components ranging from dye or protein molecules to metallic nanoparticles. In this thesis, we apply DNA nanotechnology for developing novel light-harvesting and optical voltage sensing nano-devices. We use the programmable positioning of dye molecules on a DNA origami plate as a mimic of a light-harvesting antenna complex required for photosynthesis. Such a structure allows us to systematically analyse optimal design concepts using different dye arrangements. Complementary to this, we use the resistive-pulse sensing technique in a range of electrolytes to characterise the mechanical responses of DNA origami structures to the electric field applied. Based on this knowledge, we assemble voltage responsive DNA origami structures labelled with a FRET pair. These undergo controlled structural changes upon application of an electric field that can be detected through a change in FRET efficiency. Such a DNA-based device could ultimately be used as a sensitive voltage sensor for live-cell imaging of transmembrane potentials.

On étudie les surfaces plates à singularités coniques, leur géométrie, leur espaces des déformations et leur paramétrisation. La surface de base est la sphère à trois trous (pantalon). On trouve trois ensembles de paramètres pour le pantalon plat à un point singulier conique et on décrit son espace des déformations. On introduit un flot que l'on appelle flot de Fenchel-Nielsen sur un espace des déformations. On étudie l'injectivité de ce flot en examinant la variation des fonctions longueur de segments géodésiques ou de géodésiques simples fermées le long de ce flot. On étudie également la paramétrisation d'une surface plate à singularités coniques utilisant des longueurs des segments géodésiques joignant des points singuliers ou un point singulier à une composante du bord. A la fin du texte, trois annexes apportent des discussions supplémentaires.

The purpose of this study is developing valid and reliable scales in order to measure beliefs and perceived self-efficacy beliefs towards using origami in mathematics education and then, investigating beliefs and perceived self-efficacy beliefs of Turkish prospective elementary mathematics teachers in using origami in mathematics education. Furthermore, gender differences in prospective teachers&#039 / beliefs and perceived self-efficacy beliefs in using origami in mathematics education were investigated. Data for the current study was collected in the spring term of 2011-2012 academic year from 299 prospective elementary mathematics teachers. These teacher candidates are from three universities located in three different regions of Turkey and all the participants have elective origami course experience. Origami in Mathematics Education Belief Scale (OMEBS) and Origami in Mathematics Education Self-Efficacy Scale (OMESS) were used as data collection instruments. Exploratory and confirmatory factor analysis results showed that OMEBS and OMESS are valid and reliable instruments in order to measure beliefs and perceived self-efficacy beliefs in using origami in mathematics education. Descriptive analysis results indicated that, Turkish prospective elementary mathematics teachers strongly believe that origami is beneficial and suitable to be used in mathematics education. However, their perceived self-efficacy belief level is at little higher than moderate level. Lastly, independent sample t-test results revealed that female teacher candidates have significantly higher belief and perceived self-efficacy beliefs in using origami in mathematics education when compared with male teacher candidates.

The main purpose of the study was to investigate the effect of origami-based instruction on fourth, fifth, and sixth grade students&rsquo / spatial ability in mathematics. More specifically, how origami-based instruction affected students&rsquo / spatial visualization and spatial orientation ability was examined. In addition, elementary students&rsquo / self-reported perceptions related to origami-based instruction was investigated. In other words, students&rsquo / attitude towards origami-based instruction, their views about the benefits of origami-based instruction and its connection to mathematics, and the difficulties that students encountered while making origami as well as by whom they overcame these difficulties were investigated. The data was collected from 38 fourth, fifth, and sixth grade students in a private school in Eryaman neighborhood in Ankara. The participants were given a Spatial Ability Test (SAT) as pretest and posttest in order to assess the effect of origami-based instruction on their spatial ability in terms of spatial visualization and spatial orientation. In addition, the participants were asked to write reflection papers related to origami-based instruction in order to examine their perceptions. The results indicated that there was a significant positive effect of origami-based instruction on elementary students&rsquo / both spatial visualization and spatial orientation ability. Moreover, the findings showed that students had positive attitude toward origami-based instruction where they wanted to continue origami-based instruction. Students also thought that origami-based instruction was beneficial for them especially in geometry topics in mathematics, and they have common views that origami-based instruction was directly related with mathematics. Furthermore, results revealed that students were generally encountered with folding and assembling difficulties, and they overcame these difficulties by themselves, by the help of the teacher, and by the help of their friends.