One of the main focuses of this thesis is a special kind of symmetrical arrangement called a "Steiner system." Imagine trying to organize a group of objects in such a way that every possible subset of them fits together in a very specific and symmetrical pattern. These patterns are not just mathematical coincidences; they have real-world applications, such as in coding theory.

More precisely, among these Steiner systems, there are some particularly interesting ones known as "Witt designs." These designs are important because their symmetrical properties are used in constructing certain types of error-correcting codes. These codes can correct a certain amount of errors within information that is sent or stored, making it more reliable.

To better understand these Witt designs, the thesis explores two main ways to create them. One method uses something called the Golay code, which is a specific type of error-correcting code. The other method involves a mathematical group called the projective special linear group. Both of these constructions are explained in detail, with proofs provided to show that they indeed form the desired symmetrical patterns.

Sixty years after John Bell published his paper in which he showed that local hidden variables with statistical independence are incompatible with quantum mechanics, its implications for the nature of the universe are still hotly debated among physicists and philosophers. In his paper he showed that a universe with local hidden variables and statistical independence should satisfy Bell's inequality, but this is violated by quantum mechanics. Testing Bell's inequality turned out to be difficult, owing to a number of loopholes. These were closed in experiments in 2015 which violated Bell's inequality, putting local hidden variables with statistical independence to rest. In this paper we discuss two conceptually simple versions of Bell's inequality as a game (Maudlin and GHZ) and simulate them using Python. Using local means these games are unlikely to be won, but by utilizing the non-locality of quantum mechanics the game is expected to be won. In our simulations we find that the probability of winning decays exponentially as a function of the length of the game for local strategies in Maudlin's version and we prove that the memory loophole (also known as daily updating) will not help the participants to win. In simulating GHZ as a game we find that the game length follows a geometric distribution.

Hoewel het probleemoplossen in het wiskundecurriculum is opgenomen, hebben docenten vaak behoefte aan meer praktische ondersteuning bij het onderwijzen ervan. De recent in het Nederlands gepubliceerde didactiek ”Building Thinking Classrooms” van Peter Liljedahl met daarin veertien praktisch beschreven principes kan daarbij van betekenis zijn.

Het doel van dit onderzoek is om meer inzicht te verkrijgen in de effectiviteit van twee van de veertien principes ten aanzien van de beoogde leeropbrengst binnen de specifieke lespraktijk. Hierbij gaat het om het maken van aantekeningen met behulp van een ontworpen werkblad en het consolideren van een les door ontworpen lesactiviteiten waarin leerlingen zelfstandig reflecteren. De specifieke lespraktijk waarin dit onderzoek wordt uitgevoerd bestaat uit zes onafhankelijke lessen in ”Thinking Classrooms”. Deze lessen staan in het teken van het onderwijs van het oplossen van algemene wiskundige problemen. De bijbehorende leerdoelen zijn gerelateerd aan het ontwikkelen van metacognitieve vaardigheden en een geschikte wiskundige houding, wat volgens Alan Schoenfeld behoren tot twee van de vier essentiële bekwaamheden voor het wiskundig probleemoplossen. Om het doel van dit onderzoek te bereiken is de volgende hoofdvraag opgesteld.

Welke leerlingactiviteiten binnen de principes “note-taking” en het “consolideren” in een “Thinking Classroom”, ontwikkeld door Peter Liljedahl, dragen bij aan een verhoogde leeropbrengst met betrekking tot het oplossen van algemene wiskundige problemen door leerlingen een gemengde havo/vwo bovenbouwklas?

Dit onderzoek is uitgevoerd door middel van het cyclische ”Design-Based research”. Deze onderwijsbenadering richt zich richt op het ontwerpen, implementeren en evalueren van onderwijsinterventies in echte leeromgevingen. Naar aanleiding van de verschillende iteraties kunnen er patronen worden ontdekt in het leergedrag van de leerlingen. Zo blijkt er dat een individueel werkblad, na het gezamenlijk oplossen van het probleem, een passende niveauverhogende activiteit kan zijn. Dit is het geval wanneer er op het werkblad gevraagd wordt om de stappen voor het vinden van de oplossing van een probleem individueel volledig uit te werken en daarop te reflecteren. Om meer uitgebreide betekenisvolle reflecties van leerlingen te stimuleren blijkt dat het afhankelijk is van de vordering van een groep leerlingen binnen het proces van het probleemoplossen of een docentgestuurd klassengesprek of een leerlinggestuurd groepsgesprek relevanter is. Een bijzonder resultaat hierbij is dat wanneer leerlingen het centrale probleem uit een les al snel hadden opgelost, zij in sommige gevallen uit zichzelf met elkaar gaan discussiëren over hoe de gevonden oplossing toegepast of geoptimaliseerd kan worden en dit vervolgens gaan uitvoeren.

Vervolgonderzoek zou zich kunnen richten op de invloed van deze lessen op de leerprestaties van de deelnemende leerlingen binnen het curriculum. Hierbij krijgen namelijk wiskundige kennis en onderwerpspecifieke heuristieken, de overige twee essentiële bekwaamheden die Alan Schoenfeld beschrijft voor het wiskundig probleemoplossen, meer de nadruk.

The Bayesian approach is a very important approach for tackling problems in statistics. It involves choosing a distribution that reflects the prior knowledge and  thus takes all knowledge into account in contrast to the frequentist approach. It also assumes that the parameters (the regression coefficients) follow a distribution called the posterior distribution instead of fixed constants. When a specific choice of this prior is made, this needs to be justified as the prior directly influences the posterior distribution of the regression coefficients. It is also possible to consider priors that do not carry a lot of information and such priors will be compared in this project. 

In this thesis, the Bayesian approach will be used to apply a multinomial logistic regression model to data concerning students’ study habits and beliefs. The data is provided by a research group called PRIME and they focus on mathematics education at the TU Delft. Multinomial logistic regression is used to find predictions of the choices expressed in probabilities. Bayesian statistics is not only useful in a sense that it offers the possibility to specify the prior knowledge, but also because the Bayesian way of thinking can be incorporated in evaluating results. This can be done by constructing credible intervals for the predicted probabilities. Overlap between intervals can then give insight on prediction quality.

In this project, the models are coded in R and here two packages are used: the UPG and the BRMS package. The priors that are compared are the Gaussian and Cauchy distributions. Other than that there are also default priors used in the packages, which can be compared to the Gaussian and Cauchy priors. In the end, a conclusion can be drawn about the performance of each model based on the prediction accuracy. It can be concluded that the BRMS package outperforms the UPG package in terms of accuracy both using default priors and overall using default priors gives more accurate results than specifying the prior. However, the difference in the accuracy of the model using the BRMS package is not significantly higher than the accuracy obtained from the UPG model and the running time is a lot higher for the BRMS package. From the models with a specified prior, the model with the Cauchy distribution as prior performed better.

In 1908, the mathematician Felix Klein published a book Elementary Mathematics from an Advanced Standpoint: Geometry. This title aptly characterizes the focus of this thesis. This thesis introduces the axioms for Euclidean and projective plane geometry. Afterwards an arithmetic of lengths, based solely on these axioms, is constructed. By establishing a connection between the geometric axioms and the field axioms, it is demonstrated that these lengths form a field. It is shown that the original geometric plane is isomorphic to the Cartesian plane of lengths. Additionally, the thesis highlights the direct relation between two geometric propositions, Pappos’ theorem and Desargues’ theorem, and commutativity and associativity of the induced field of lengths.

This study examines which factors in the behaviour of math teachers in their contact with high school students contribute to the developing, worsening or alleviating of math anxiety during mathematics activities.
The study was conducted in the form of a survey involving $888$ high school students during their math lesson. The survey included several scenarios corresponding to the behaviour of math teachers to which the students could indicate their level of stress. Results show that students indicated to feel significantly higher levels of stress in a scenario where a teacher shows her belief in the growth mindset by forcing a student to try an exercise again after failing once, in comparison with a teacher who shows her belief in the fixed mindset by stating that some students remain to have difficulties with math. In addition, the students indicated to feel significantly lower levels of stress when the teacher writes down the correct answer in their book immediately. The effect sizes of these differences are small and moderate, respectively. Furthermore, it was found that students indicated to feel significantly lower levels of stress, with moderate effect sizes, when a teacher evaluates their work by writing down hints, instead of drawing lines through their mistakes or grading their work in a formative manner. Another important finding of this study was the significantly higher levels of stress students indicated in a scenario where the teacher shows her belief in the fixed mindset by concluding after a few mistakes that a student finds mathematics difficult in general, in comparison with a teacher showing her belief in the growth mindset by emphasizing the learning goal. In conclusion, this study suggests that students might not be experienced in handling exercises alone and feel higher levels of stress when they are forced to do so, even though the teacher shows her belief in the growth mindset.

Het doel van dit onderzoek is om leerlingen te helpen bij het beantwoorden van de vragen ‘Waar moet ik heen?’ en ‘Waar sta ik nu?’ en gaandeweg factoren ontdekken die leerlingen hierbij belemmeren. De eerstgenoemde vraag gaat over het deel van het leerproces waarin leerlingen een compleet beeldmoeten vormen van de eindtermen: leerdoelen en bijbehorende succescriteria. De andere vraag slaat op het maken van een juiste inschatting in hoeverre een bepaald leerdoel, inclusief zijn eindtermen, is behaald. Naast het helpen beantwoorden van deze vragen, wordt belemmerende factoren zijn voorleerlingen om deze eindtermen te behalen, i.e. gericht op het opschrijven van een uitwerking geheel in overeenstemming met de succescriteria. In het geval belemmerende factoren zijn ontdekt, is middels het ontwerpen en uitvoeren van een klassenactiviteit, ook wel een interventie genoemd, onderzocht of en in hoeverre leerlingen hiermee geholpen zijn bij het beantwoorden van de twee eerder genoemde vragen. Zo’n klassenactiviteit beruste meestal sterk op peer learning. Dit vanwege de invalshoek van het onderzoek die zich richt op wat peers voor elkaar kunnen betekenen in eerder genoemde fasen van het leerproces. Hiermee is de onderzoekshoofdvraag uitgekomen op het volgende: 
Hoe kan een docent, door gebruik te maken van peer learning, de mate van inzichtelijkheid en begrijpelijkheid van de leerdoelen en SC bij leerlingen verhogen, zodoende dat hun geleverde werk in hogere mate aan SC voldoet en zij beter in staat zijn dit te kunnen beoordelen op correctheid en volledigheid in vergelijking tot die leerdoelen en SC? 
Deze onderzoeksvraag is iteratief beantwoord volgens de methodologie van een hypothetisch doceer- en leertraject…

Dit onderzoek beoogt de veelal overheersende productgerichte focus van leerlingen in het wiskundeonderwijs te verschuiven naar een procesgerichte focus. Hiertoe is een Hypothetical Teaching and Learning Trajectory (HTLT) ontworpen waarin volgens een iteratief proces onderzocht is hoe docenten leerlingen kunnen helpen om uitwerkingen te noteren zodat deze voldoen aan succescriteria en leerdoelen. Allereerst is verduidelijkt aan leerlingen wat succescriteria zijn met betrekking tot het oplossen van wiskundige problemen aan de hand van voorbeelduitwerkingen. Vervolgens is van een aantal doceerstrategieën de effectiviteit onderzocht in het trainen van leerlingen in het toepassen van deze succescriteria en leerdoelen. Deze strategieën zijn het trainen van leerlingen een complete schets te maken, het toepassen van scaffolding in een vlekopgave en het toepassen van de goal free strategie. Deze interventies dragen bij aan het verduidelijken, delen en laten toepassen van succescriteria in het werk van leerlingen.

In this thesis the differences in performance scores of experts in the Classical Model for structured expert judgement are analyzed. The underlying assumption in the Classical Model is that variance in performances of experts in a panel is at least partly resultant of the expert's ability to quantify uncertainty. This assumption is tested against the so called Random Expert Hypothesis, that states that these differences are solely resultant of random fluctuations. At the five percent significance level it is concluded that the variation in the combined score of experts cannot exclusively be explained by random fluctuations. When the assumption is tested individually for three different subject fields, health, policy and science, the Random Expert Hypothesis cannot be rejected for both health and policy related studies. Lastly it is shown that the variation in performances between the best and worst expert in a panel strongly correlates with the performance of the best expert against random panels. This indicates that the aggregation of experts according to the scoring rule in the Classical Model may primarily work to diminish the influence of low performing experts.

Representation theory is a branch in mathematics that studies group homomorphisms between a group and the automorphism group of a vector space. A special representation that every group has is the regular representation. This representation permutes all elements of the group in a vector space which dimension is equal to the order of the group.
Within this vector space there are group invariant subspaces. There are several methods to finding representation invariant subspaces of this vector space.

This thesis aims to do two things: First of all, this thesis aims to give the reader an introduction to representation theory, presenting various key concepts, definitions and theorems. Moreover, ways to construct character tables are presented along with multiple worked out examples. Second, the regular representations of D4 and Q8 are decomposed into representation invariant subspaces of C8. To this end, two methods were used.

The first method (A), proposed by dr. Jeroen Spandaw, works out all the possible decompositions of the vector space a regular representation acts on. This is quite a laborious process in which change of basis matrices, expressed in several parameters, will have to be made for all generators of the considered group. The second methods (B), proposed by dr. Paul Visser, is known as the grand orthogonalization method and makes use of an extended version of the character table of the considered group.

Both methods are perfectly fine to make a desired decomposition. However, method A
takes a lot more computational effort than method B to come up with the desired result. The benefit of using method A over method B is that method A considers all possible decompositions, whereas method B only considers one of the infinitely many that are possible.

Peer assessment (PA) is a process that involves students assessing each other’s work and providing their peers with (elaborated) feedback. This thesis investigates how an elaborated form of formative PA can be implemented in the mathematics class to promote students’ achievement in mathematics. A mixed methods investigation at a third-grade mathematics class was conducted. For this investigation, a teacher’s guide and a student guideline were developed. The teacher’s guide helps the mathematics teacher in implementing PA in the class. The students’ guideline was developed to guide students doing online PA during distance learning due to the COVID-19 pandemic. Using this student guideline can require more effort for the teacher to monitor students’ participation and more effort for students to do elaborated online PA. Students were trained with the technique “two stars and a wish” for providing peer feedback. With this feedback technique, students could provide their peers with two reinforcing feedback and one suggestive feedback. The feedback has to first be written in their peer’s work and later be elaborated when the work is being returned. As a result of this investigation, implementing PA promoted students to interact and learn from one another during online distance learning. Evidence was found that by implementing PA, students were more involved in their learning process, and students’ achievement increased. PA can be used as an effective learning method for students. However, it is preferably for PA to be implemented during physical class, especially when it is implemented the first time.

A pentagon is an example of a highly symmetric polygon in two-dimensional space. The three-and four-dimensional analogue of these polygons are the regular polyhedra and the regular polytopes. There exist five regular polyhedra in three-dimensional space and these are called the Platonic solids. These five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron and the icosahedron. In four-dimensional space, the regular polytopes are the 5-cell, the 8-cell, also called the tesseract, the 16-cell, the 24-cell, the 120-cell and the 600-cell. The main focus in this thesis is to describe the symmetry group of the icosahedron, to introduce the Icosians, which are related to the rotation group of the icosahedron, and to study the action of the symmetries of the 600-cell on the twenty-five 24-cells it circumscribes. Firstly, the symmetry groups of the Platonic Solids, the regular polytopes in three dimensional space are established. Then it will be shown that there eixsts a two-to-one map from the the group H₁ of unit quaternions to the group SO(3), the group of 3 x 3 orientation-preserving matrices. This map will be used to describe the binary groups, which are double covers of the rotation groups of the Platonic solids. After that, the symmetry group of the tesseract will be studied both via an isomorphism between G := {± 1 }⁴ x S₄} and the symmetry group of the tesseract as well as geometrically via rotation planes. Then, the 24-cell and the 600-cell will be defined as the four-dimensional regular polytopes whose vertices are the quaternions from the binary tetrahedral group and the binary icosahedral group, the Icosians. It will be shown that twenty-five 24-cells inscribe a 600-cell and that there are 10 ways to decompose the vertices of a 600-cell into the vertices of 5 disjoint 24-cells. Next, it will be shown that these 10 decompositions are chiral, 5 being 'left-handed' and 5 being 'right-handed'. Finally, it is shown that the symmetry group of the 600-cell acts on these 5+5 decompositions by permutation, each permutation being described by an element from A₅ x A₅ ⋊ {±1}, where -1 acts on A₅x A₅ by interchanging the factors of A₅x A_5.

Eerst introduceren we het rekenkundig-meetkundig gemiddelde (RMG), een snel convergerend algoritme dat een verband heeft met π en nog veel meer constanten en functies. Hoffmann toont dit in een bewijs aan, echter is deze niet waterdicht en amper onderbouwd. Daarom gaan wij ons verdiepen in dit bewijs en met name een stap die twee ogenschijnlijk verschillende integralen aan elkaar gelijk stelt. Het blijkt dat hiervoor Landens transformatie gebruikt kan worden. Om hier op te komen en hem ook nog eens te bewijzen moeten we de wereld van elliptische functies induiken. We gaan vooral werken met de zogeheten thèta-functies die we grondig gaan onderzoeken. Vervolgens gaan we toewerken naar de Landen-transformatie-relaties die de basis zullen vormen voor Landens transformatie. Hiervoor moeten we veel eigenschappen en identiteiten aantonen. Dan stappen we over naar Jacobi's elliptische functies die de brug vormen naar elliptische integralen. Aan de hand van deze integralen en de Landen-transformatie-relaties kunnen we bewijzen dat het RMG inderdaad verborgen zit in een elliptische integraal van de eerste soort. Ook volgt hieruit dat er een verband is met π. Tot slot kijken we nog naar een aantal toepassingen van het RMG.

This Bachelor thesis is on Covariant Emergent Gravity (CEG): a covariant formulation by Sabine Hossenfelder of the ideas of Erik Verlinde on gravity as an emergent force. In this work, the ideas of Erik Verlinde are presented as well as the field equations of CEG. The aim of this thesis is to find experimentally verifiable results for CEG. From the field equations we derive a general gravitational lensing formalism for CEG that is applicable to general lensing systems. This thesis also includes an attempt at an expanding universe model for a vacuum or matter dominated universe. Subsequently, a numerical algorithm is presented to solve for the non-linear differential equations in CEG and MOND in Newtonian regimes for general matter distributions using Fourier transformations. A faster version of this algorithm for cylindrical symmetric matter distributions using Fourier-Bessel transformations is also presented. This algorithm is tested on both the Sun and galaxy NGC6503. Next, the rotation curves as predicted by CEG and MOND of 132 galaxies are compared to the observed velocities using the SPARC data set. The velocities are fitted using three fit parameters and a Markov Chain Monte Carlo algorithm. Both theories show good fits to the observed velocities. We conclude that no differences between the predictions of CEG and MOND can be made on the basis of rotation curves, but that covariant features (such as gravitational lensing) might be used to distinguish between the two theories. An important test for both theories can be provided by a comparison between strong lensing measurements and rotation curves for a single galaxies.