Tomas Recio

We introduce the current performance of the GeoGebra and GeoGebra Discovery programs with regard to various features of automatic reasoning tools that rely on computational algebraic geometry methods, mostly in the complex setting. Then we focus on theoretical and algorithmic issues pending to extend (in the same technological and educational framework) such methods to deal with statements in the real algebraic geometry context.

Background. As the Open Science context evolves broadly and Scientific Institutions implement Open Access and Open Science policies, it is of interest to observe which are the issues that come into play when these Institutions attempt to contribute to make their research outputs visible, accessible and reusable. The present work reflects on the problematic behind the establishment and adoption of policies regarding the dissemination of Research Software (RS) as Free/Open Source Software (FOSS) in a Scientific Institution. It takes advantage, as an initial motivation, of the Request for Information (RFI): Best Practices for Sharing NIH Supported Research Software. This request was issued by the USA National Institutes of Health (NIH) organization in October 2023, seeking for responses to a set of questions regarding Research Software sharing and dissemination best practices.Method. In this work we begin by establishing an initial, general framework regarding RS and FOSS fundamental concepts and terminology. Then we include the initial list of questions for the NIH-RFI, followed by our detailed answers, argumented in the context of the provided framework. Results. Besides presenting this large collection of answers, we also include diverse reflections on what we have observed and learned in the context of the analysis of the NIH-RFI, arguing it provides also with the opportunity to explore how the previously published work on RS and Open Science can be envisioned in the lens of the definition and adoption of RS sharing and disseminating policies by any Scientific Institution or research community. Conclusions. We have detailed here a foundational framework, showing how it could be considered as the key tool for providing a systematic reply to the RS and FOSS issues arising in the NIH-RFI. This framework that can be easily adapted to more general contexts, and, thus, could be of potential interest for other scientific organizations, as well as for scientific communities with an important RS production, like the Computer Algebra and the Symbolic Computation ones, aiming to clarify and upgrade their policies and practices regarding this kind of production.

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.

We present the continuation of a study on an international MOOC for the training of mathematics teachers who were introduced to the methodology of outdoor mathematics through MathCityMap. Previously, we presented the immediate learning and professional development outcomes achieved by teachers who participated in the MOOC. Here, using the theoretical framework of Meta-Didactical Transposition and performing a mixed-type analysis, we investigate the long-term impact of the MOOC on these teachers to whom we administered a questionnaire one year after the end of the MOOC. We conclude that for these teachers, practising mathematics outdoors has become a consolidated teaching practice, but they still feel quite uncomfortable for structural reasons about using some pedagogical instruments they have learned in the MOOC.

This paper explores the application of automated reasoning tools, specifically those implemented in GeoGebra Discovery, to the perimeter-area inequality in triangles. Focusing on the computational complex and real algebraic geometry methods behind these tools, this study analyzes a geometric construction involving a triangle with arbitrary side lengths and area, investigating the automated derivation of the relationship between the area and perimeter of a triangle, and showing that only equilateral triangles satisfy the exact perimeter-area equality. The main contribution of this work is to describe the challenges, and potential ways to approach their solutions, still posed by the use of such automated, symbolic computation, methods in dynamic geometry, in particular concerning the discovery of loci of points that satisfy specific geometric conditions, suggesting relevant improvements for the future development of these symbolic AI-based educational tools in geometry.

In the context of Open Science, the importance of Borgman’s conundrum challenges that have been initially formulated concerning the difficulties to share Research Data is well known: which Research Data might be shared, by whom, with whom, under what conditions, why, and to what effects. We have recently reviewed the concepts of Research Software and Research Data, concluding with new formulations for their definitions, and proposing answers to these conundrum challenges for Research Data. In the present work we extend the consideration of the Borgman’s conundrum challenges to Research Software, providing answers to these questions in this new context. Moreover, we complete the initial list of questions/answers, by asking how and where the Research Software may be shared. Our approach begins by recalling the main issues involved in the Research Software definition, and its production context in the research environment, from the Open Science perspective. Then we address the conundrum challenges for Research Software by exploring the potential similarities and differences regarding our answers for these questions in the case of Research Data. We conclude emphasizing the usefulness of the followed methodology, exploiting the parallelism between Research Software and Research Data in the Open Science environment.

Locus computation is an essential issue in mathematics education, and a traditional feature of Dynamic Geometry software (DGS). The rising of programs merging DGS and Computer Algebra software (CAS) has fostered a combined approach to locus computation, quite performing in standard examples, but demanding an extended theoretical, and the related algorithmic counterpart, able to deal with less conventional situations. Here we formulate—and reflect about, yielding some proposals—on a few pending issues related to the protocols for the computation of parametric families of loci. Then we focus on a different source of difficulties, through the example of the very elementary and classical theorem of Thales. Thus, in the framework of the current development of automated deduction in geometry (ADG) tools, we will show how the automatic discovery of Thales's converse statement might require a locus computation that gives rise to an unexpected family of octic curves. Finally, we will exhibit how the handling (finding the equation, plotting, geometric characterization, etc.) of such curves requires the concourse of DGS and CAS programs, a mixed graphic–symbolic–numeric approach, and human–machine interaction, a cooperation that could be the basis towards achieving the required improvements concerning locus computation software.