Álvaro Nolla de Celis

This paper presents an initial output of the project “Augmented Intelligence in Mathematics Education through Modeling, Automatic Reasoning and Artificial Intelligence (IAxEM-CM/PHS-2024/PH-HUM-383)”. The starting hypothesis of this project is that the use of technological tools, such as mathematical modeling, visualization, automatic reasoning and artificial intelligence, significantly improves the teaching and learning of mathematics, in addition to fostering positive attitudes in students. With this hypothesis in mind, in this article, we describe an investigation that has been developed in initial training courses for mathematics teachers in several universities in Madrid, where students used GeoGebra Discovery automated reasoning tools to explore geometric properties in real objects through mathematical paths. Through these activities, future teachers modeled, conjectured and validated geometric relationships directly on photographs of their environment, with the essential concourse of the automated discovery and verification of geometric properties provided by GeoGebra Discovery. The feedback provided by the students’ answers to a questionnaire concerning this novel approach shows a positive evaluation of the experience, especially in terms of content learning and the practical use of technology. Although technological, pedagogical and disciplinary knowledge is well represented, the full integration of these components (according to the TPACK model) is still incipient. Finally, the formative potential of the approach behind this experience is highlighted in a context where Artificial Intelligence tools have an increasing presence in education, as well as the need to deepen these three kinds of knowledge in similar experiences that articulate them in a more integrated way.

As a contribution to the discussion of how to improve the initial teacher training in mathematics of elementary teachers, this work presents a descriptive and comparative study of the courses devoted to mathematics and mathematics education in the education degrees of Greek, Romanian, and Spanish universities. Considering the constrains of each national law and within the Mathematics Teacher’s Specialized Knowledge model framework, we analyze the official descriptions of such courses to organize them by their content, to describe their distribution along each degree, and to study the optional and specialization courses in mathematics offered by each university. Data are compared among countries, and across programs within each country, with the aim of including similar studies from other states. Moreover, they are confronted with the National Council on Teacher Quality standards for the mathematics preparation of elementary teachers, showing that, in general lines, GRS data seem not to be in tune with such standards.

We construct a consistent dimer model having the same symmetry as its characteristic polygon. This produces examples of non-commutative crepant resolutions of non-toric non-quotient Gorenstein singularities in dimension 3.

In this paper, we describe an experience to test the predominant presence of Delaunay triangulations in the artwork of Okuda, a quite famous, young, contemporary Spanish artist. We addressed this task involving, as a citizen science activity in a STEAM (Science, Technology, Engineering, Art, Mathematics) education context, several hundreds of students (of different kinds: secondary education, university undergraduates, in particular, following teacher training degrees). Each student was asked to select an Okuda archive and, with the concourse of a dynamic geometry program provided with some computational geometry commands, to measure the ratio of coincident triangles in Delaunay's and artist's triangulations, over an ample region of the chosen artwork. The results show a very large percentage of coincidence ratios. We conclude with some reflections about how to interpret this fact, and about the potential role of future, enhanced, dynamic geometry systems to automatically address similar issues, concerning mathematical properties of figures from the real world.

This study presents an experience that combines problem-posing and Math Trails in the context of future teachers’ instruction. Pre-service teachers in the third year of their studies were faced with the design of tasks to be included in Math Trails for primary school students. The study analyzes, from a quantitative approach, 117 tasks contained in 11 Math Trails. The analysis was performed on the basis of classification variables (grade, mathematical content and object or real element involved in every task) and research variables which provide information about the nature of the tasks (procedural vs. problem-solving, level of cognitive demand, degree of contextualization, openness and creativity). Additionally, relationships between the different categories of analysis have been studied. The results reveal certain biases in the tasks in relation to the contents addressed (an abundance of tasks with a geometric component and a scarcity of tasks involving algebra or probability concepts). Most of the tasks are presented in a real context. However, a higher percentage of procedural tasks than problem-solving tasks is observed, with a predominance of low openness, creativity and cognitive demand. These results provide useful lines of work to address difficulties faced by future teachers in the STEAM field.

Este trabajo recoge las experiencias con el uso de la impresión 3D en un club de matemáticas de Madrid que se llevaron a cabo entre los años 2017 y 2019. Tras el análisis de las producciones de los alumnos se distinguen cuatro tipos de proyectos que se pueden trabajar en un aula de matemáticas con el apoyo de la impresión 3D que fomentaron el trabajo colaborativo.