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The purpose of this study is to explore how MathCityMap training for teachers can be designed and implemented in a hybrid setting. Design research has been carried out involving teachers from 34 provinces in Indonesia. A four-day model training program for indoor and outdoor activities in a hybrid setting has been designed. The core concept of designing modeling tasks and math trails, as well as the introduction to MathCityMap, were delivered in a hybrid mode between Indonesia and Germany. The activity continued with the development of MathCityMap Trails in city parks. Post-training activities were conducted in their respective places of origin, in schools, and in city parks. Participants have successfully designed trails in both their schools and the city parks where they live supported using Learning Management System. The findings of the MathCityMap team's review indicate that most of the trails fulfilled the requirements. The trainee teachers then disseminate their knowledge and experience to their colleagues. They also used MathCityMap Trails for classroom learning as well as other activities such as MathCityMap competitions and public educational activities. The hybrid environment helps to accelerate the attainment of training objectives while also broadening the program's reach and network.

Mathematical arguments are central components of mathematics and play a role in certain types of modelling of potential mathematical giftedness. However, particular characteristics of arguments are interpreted differently in the context of mathematical giftedness. Some models of giftedness see no connection, whereas other models consider the formulation of complete and plausible arguments as a partial aspect of giftedness. Furthermore, longitudinal changes in argumentation characteristics remain open. This leads to the research focus of this article, which is to identify and describe the changes of argumentation products in potentially mathematically gifted children over a longer period. For this purpose, the argumentation products of children from third to sixth grade are collected throughout a longitudinal study and examined with respect to the use of examples and generalizations. The analysis of all products results in six different types of changes in the characteristics of the argumentation products identified over the survey period and case studies are used to illustrate student use of examples and generalizations of these types. This not only reveals the general importance of the use of examples in arguments. For one type, an increase in generalized arguments can be observed over the survey period. The article will conclude with a discussion of the role of argument characteristics in describing potential mathematical giftedness.

In 2020, Germany and Spain experienced lockdowns of their school systems. This resulted in a new challenge for learners and teachers: lessons moved from the classroom to the children's homes. Therefore, teachers had to set rules, implement procedures and make didactical-methodical decisions regarding how to handle this new situation. In this paper, we focus on the roles of mathematics teachers in Germany and Spain. The article first describes how mathematics lessons were conducted using distance learning. Second, problems encountered throughout this process were examined. Third, teachers drew conclusions from their mathematics teaching experiences during distance learning. To address these research interests, a questionnaire was answered by N = 248 teachers (N[subscript 1] = 171 German teachers; N[subscript 2] = 77 Spanish teachers). Resulting from a mixed methods approach, differences between the countries can be observed, e.g., German teachers conducted more lessons asynchronously. In contrast, Spanish teachers used synchronous teaching more frequently, but still regard the lack of personal contact as a main challenge. Finally, for both countries, the digitization of mathematics lessons seems to have been normalized by the pandemic.

The aim of this study is to investigate how an augmented reality mobile math trails program can provide opportunities for students to engage in meaningful mathematical modelling activities. An explorative research design was conducted involving two mathematics teachers and 30 eight grades in Semarang, Indonesia. An Augmented Reality Mobile Math Trails App was created, and several math trail tasks were designed, then students run the activity. Data were gathered by means of participatory observation, interviews, questionnaires, tests, and worksheets. Data analysis began with the organisation, annotation, description of the data and statistic tests. The findings indicate that an educational program was successfully designed, which offered students a meaningful mathematical experience. A mobile app was also developed to support this program. The mobile app with augmented reality features is helpful for students as a tool that bridges the gap between realworld situations and mathematical concepts in problem-solving following the mathematical modelling cycle. The program thus contributes to a higher ability in mathematical modelling. The study identified a link between instrumented techniques in programs and mathematical modelling, as built during the instrumentation process. Further studies are essential for project development and implementation in other cities with different situations and aspects of study.