It is very difficult to determine the full automorphism group of compact Riemann and Klein surfaces. The automorphism groups of hyperelliptic surfaces, and those of (compact, non-orientable, unbordered) Klein surfaces whose genus is less or equal to 7 are known, but a lot of hard work is left to do. To amplify the knowledge in this field, in this paper the full automorphism groups of the surfaces of genus 8 are calculated.

Research on the relationships between the main constructs underlying inquiry-based learning is rarely reported in mathematics education research. Considering this as a complex problem which is worth to be investigated, the present study aims to provide some empirical evidences that might serve as an insight to support further investigations on the relationships between attitudes towards mathematics and inquiry-based learning approaches. Thus, this study adopts a descriptive research design where no variables are manipulated but observed and measured in order to identify changes depicted in data collection. An instructional design focusing on the nature of mathematical inquiry is carried out with the participation of 304 secondary and high school students, and a clustering approach is used to look at how participants are grouped around certain attitudinal profiles before and after such mathematical practice. The results show how the heterogeneity of attitudinal profiles present in the classroom evolves positively in terms of perceived usefulness of mathematics and mathematical self-concept as perception of competence in mathematics. This fact provides some basis that might be used for further research on the idea that certain forms of development in inquiry-based mathematics education (IBME) based on greater immersion in the nature and culture of mathematics can help students to improve their attitudes towards mathematics.

Algorithmic thinking is a type of thinking that occurs in the context of computational thinking. Given its importance in the current educational context, it seems pertinent to deepen into its conceptual and operational understanding for teaching. The exploration of research shows us that there are almost no studies at university level where algorithmic thinking is connected to mathematical thinking, and more importantly, to characterise it and be able to analyse and evaluate it better. The aim of this research is to characterise algorithmic thinking in a university context of the Bachelor's Degree in Mathematics by unplugged tasks, offering a model of analysis through categories that establish connections between mathematical and algorithmic working spaces in three dimensions, semiotic, instrumental and discursive. The results confirm the interaction between these dimensions and their predictive value for better programming performance. The study also adds novel considerations related to the role and interaction of mathematical and computational thinking categories involved in algorithmic thinking.

Computational thinking (CT) is a key skill in the 21st century. However, it is not clear which is the most effective way to acquire and improve CT. Big research efforts are made to determine which pedagogical means should be used. One research trend is based on the idea that teaching programming since Primary Education suffices to improve CT. In our previous work, we proposed and validated a methodology based on metaphors and used of Scratch (MECOPROG) to teach basic programming concepts to children. It is our hypothesis H that by applying MECOPROG, students will develop their CT. To check H, we carried out an experiment with 132 Primary Education Students (9-12 years in age). At the beginning of the experiment, all students were asked to fill in a programming concepts test and two tests to measure their CT. During the sessions, all students were taught according to MECOPROG. Finally, they took the three tests again. A significant increase in the results on all the tests has been measured, supporting the use of metaphors and Scratch to teach computer programming concepts to Primary Education students to develop their CT.

Using information and communication technology (ICT) in childhood education is becoming more relevant as research shows that it can be used to foster children's academic and non-academic skills. ICT can help build environments where children can communicate and collaborate, but creating effective learning environments is not trivial. It is thus necessary to study which configuration is the most appropriate for encouraging collaboration. In this work, we present how two different ways of interacting with a multitouch tabletop, taking turns without having to agree on the answer and working simultaneously but having to agree on the answer, affect group communication and children's satisfaction. We have carried out four different learning experiments involving 180 children between 6 and 11 years of age who had to solve math problems in groups of three and four at a multitouch tabletop. Our results suggest that turn-based interaction makes students communicate more with each other when solving activities in groups. In addition, children's satisfaction is high when they perform activities at a multitouch tabletop, but learning outcomes seems to not be impacted by the way of interacting with the device. Thus, while multitouch tabletops can be used to create collaborative learning environments, it is the way in which students interact with the device that may impact group communication.

The adoption of information and communication technologies in primary education is crucial for adapting traditional classrooms to the digital era. Over time, young children are increasingly using touch screen technologies such as tablets at home and interactive whiteboards at school, either for leisure or for academic activities. However, the literature shows that there is still a gap between what is known about the benefits of using this technology and its real use in primary education settings. Most researchers have focused on the pedagogical theory behind using touch screen devices, but there are few empirical studies about how these technologies and different approaches affect students' learning processes. This paper presents two learning experiences in a primary school in Fuenlabrada (Madrid-Spain) where primary students performed mathematics activities using a multi-touch table with two different methodologies: turns and consensus. The results show that both methodologies help students acquire meaningful learning, but there is no statistically significance between them.

Research shows that teaching computer programming to children help them develop important 21st century skills such as planning, problem solving, and flexibility of thinking. However, teaching these skills to young children is not an easy task as the existing tools in the market do not seem to be well adapted for them. The aim of this work is to present BlueThinking, an inclusive application to learn programming at early stages. A preliminary evaluation with 5 and 6 years old children was carried out to get the first impressions of the application and detect possible issues. The results show that the degree of children satisfaction with the application was high. Therefore, we believe BlueThinking could be easily used with young children to help them developing the aforementioned skills as well as to introduce them in a subtle way to the world of computer programming.

Every finite group G acts on some non-orientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of G. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain the groups with symmetric crosscap number less than or equal to 17. Also, we obtain six infinite families with symmetric crosscap number of the form 12k + 3.

Every finite group G acts on some non-orientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of G. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we study which natural numbers are the symmetric crosscap number of an Abelian group. This set will be called the Abelian crosscap spectrum. We obtain a full result for even numbers and describe properties satisfied by odd numbers in this spectrum.

To determine the full automorphism group of compact Riemann and Klein surfaces is a hard problem, although some partial results are known. For example, the automorphisms groups of hyperelliptic surfaces, and those of (compact, non-orientable, unbordered) Klein surfaces whose genus is less or equal to 6 are known. In this paper the full automorphism groups of the surfaces of genus 7 are calculated.

An important problem in the study of Riemann and Klein surfaces is determining their full automorphism groups. Up to now only very partial results are known, concerning surfaces of low genus or families of surfaces with special properties. This paper deals with non-orientable unbordered Klein surfaces. In this case the solution of the problem is known for surfaces of genus 1, 2, 3, 4 and 5, and for hyperelliptic surfaces. Here we explicitly obtain the full automorphism group of all surfaces of genus 6.

Every finite group G acts on some nonorientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of G. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain group presentations which allow one to find the actions realizing the symmetric crosscap number of groups of each group of order less than or equal to 63.

Every finite group G acts faithfully on some non-orientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of G. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain necessary conditions for n to be a gap. According to them, the smallest value of n which could be a gap is in this moment n = 699, and there remain eight possible candidates for n < 2000.