Calculating Complete Lists of Belyi Pairs

Belyi pairs constitute an important element of the program developed by Alexander Grothendieck in 1972–1984. This program related seemingly distant domains of mathematics; in the case of Belyi pairs, such domains are two-dimensional combinatorial topology and one-dimensional arithmetic geometry. The paper contains an account of some computer-assisted calculations of Belyi pairs with fixed discrete invariants. We present three complete lists of polynomial-like Belyi pairs: (1) of genus 2 and (minimal possible) degree 5; (2) clean ones of genus 1 and degree 8; and (3) clean ones of genus 2 and degree 8. The explanation of some phenomena we encounter in these calculations will hopefully stimulate further development of the dessins d’enfants theory.

A Joint Assessment of Reasoning about General Statements in Mathematics and Biology

This contribution belongs to a larger empirical study that focuses on issues related to the implementation of inquiry-based learning and formative assessment in science and mathematics education, while it also refers to the issue of STEM education. Here, we discuss the two topics from the perspective of professional preparation of primary school teachers. We employ an educational tool called Concept Cartoons and perceive it as a common diagnostic tool for investigating modes of reasoning about general statements in arithmetic, geometry and biology. The presented qualitative exploratory empirical study maps and codes various kinds of reasoning that can be identified with the tool and investigates possibilities of a joint coding procedure. As a result, it provides a conversion table between various modes of reasoning in the three subject domains. The arisen code categories cover the field of generic examples, including the initial stages so that they can be used for scaffolding the process of learning the foundations of deductive reasoning. The joint approach to reasoning in mathematics and biology shows how argumentation and formative assessment can be understood equally and developed simultaneously in both school subjects. It helps us to see how the two school subjects can be integrated didactically.

Shadows in the Wild - Folded Galleries and Their Applications

This survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.

TEORIA DOS CAMPOS CONCEITUAIS: INTEGRANDO ARITMÉTICA, GEOMETRIA E ÁLGEBRA NO ENSINO DAS FRAÇÕES / CONCEPTUAL FIELDS THEORY: INTEGRATING ARITHMETIC, GEOMETRY AND ALGEBRA IN THE FRACTIONS TEACHING

A presente pesquisa visa analisar as potencialidades de uma aprendizagem pautada na Teoria dos Campos Conceituais, elaborada por Gérard Vergnaud, e desenvolvida através de uma sequência didática, envolvendo o conteúdo de Frações, a uma turma de 6º ano do Ensino Fundamental do Ensino Regular. Trata-se de uma pesquisa qualitativa ancorada na própria prática, por meio do desenvolvimento de um estudo de caso, e na bibliografia de renomados educadores matemáticos e pesquisadores da área. Nesse contexto, o pesquisador atuava como professor de matemática na turma em questão, focando-se na constante análise acerca da construção do conhecimento discente a partir de um ensino previamente planejado com vistas às diferentes representações e estruturas de frações. Os resultados evidenciaram uma aprendizagem significativa, na qual a relação dos conceitos construídos encaminhou a assimilação de novos saberes, oportunizando o desenvolvimento de diferentes habilidades.

Rank-2 attractors and Deligne’s conjecture

In this paper, we will study the arithmetic geometry of rank-2 attractors, which are Calabi-Yau threefolds whose Hodge structures admit interesting splits. We will develop methods to analyze the algebraic de Rham cohomologies of rank-2 attractors, and we will illustrate how our methods work by focusing on an example in a recent paper by Candelas, de la Ossa, Elmi and van Straten. We will look at the interesting connections between rank-2 attractors in string theory and Deligne’s conjecture on the special values of L-functions. We will also formulate several open questions concerning the potential connections between attractors in string theory and number theory.

Blick auf Schulcurricula Mathematik: Empirische Fundierung? : Tagungsband des AK Grundschule in der GDM 2021

Mathematische Bildung in der Grundschule ist eine herausfordernde und langfristige Aufgabe und unterliegt curricularen Wandlungen. Mit Fokus auf Erkenntnisse aus empirischen Forschungen werden im Rahmen von drei Hauptvorträgen aus verschiedenen Inhaltsperspektiven curriculare Wandlungen im Mathematikunterricht der Grundschule betrachtet und diskutiert. Zusätzlich setzten sich sieben Arbeitsgruppen mit den Themenfeldern ‚Arithmetik‘, ‚Geometrie‘, ‚Daten, Zufall und Wahrscheinlichkeit‘, ‚Kommunikation & Kooperation‘, ‚Lernen, Lehren und Forschen mit digitalen Medien‘, ‚Frühe mathematische Bildung‘ sowie ‚Lehrer:innenbildung‘ intensiv mit aktuellen Forschungs- und Praxisfragen auseinander. Die zentralen Inhalte der Arbeitsgruppen sind in diesem Band ebenfalls dokumentiert. The Proceedings of the 2021 Conference of the Research Group on Primary Mathematics Education (Arbeitskreis Grundschule in der GDM) focus on empirical foundation of curricula in mathematics education in primary school. Three invited talks addressed the main theme in plenum. Additionally, workings groups on the research areas arithmetic, geometry, data & probability, as well as groups on early mathematics education, communication & co-operation, ICT education, and last not least on teacher education offered discussions on current research issues.