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A generalization of Ascoli–Arzelá theorem in Banach spaces is established. Schauder's fixed point theorem is used to prove the existence of a solution for a boundary value problem of higher order. The authors’ results are obtained under, rather, general assumptions.

First, a generalization of Ascoli–Arzelá theorem in Banach spaces in Cⁿ is established. Second, this new generalization with Schauder's fixed point theorem to prove the existence of a solution for a boundary value problem of higher order is used. Finally, an illustrated example is given.

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In this work, a new generalization of Ascoli–Arzelá theorem in Banach spaces in Cⁿ is established. To the best of the authors’ knowledge, Ascoli–Arzelá theorem is given only in Banach spaces of continuous functions. In the second part, this new generalization with Schauder's fixed point theorem is used to prove the existence of a solution for a boundary value problem of higher order, where the derivatives appear in the non-linear terms.

In March 2020, with only two working days’notice the French national education system went online due to the coronavirus pandemic. This study explores the relationship between the move to distance learning, the teaching practices employed and the socio-economic context of the learners in French schools during this period. We ask how far the changes in teaching practices during the coronavirus crisis were influenced by the social context of teaching. And to what extent this context influenced the focus of the pedagogical continuity those teachers set up.

A review of the literature situates the study within the field of mathematics teaching practices. The study was carried out through a multidimensional analysis using multiple correspondences of the responses of 368 French secondary school mathematics teachers to an online questionnaire.

We found that the unprepared move to distance learning impeded the employment of dialogic practices. The socio-economic situation of the teaching was identified as a determining factor in the teachers' different interpretations of the term pedagogical continuity. Whilst those working in more deprived areas tended towards practices which focused on maintaining pupils' links with school, consolidation of knowledge and providing social/affective support, those teaching a more privileged public favoured tools and practices which allowed them to focus on the disciplinary content of their teaching.

The challenge of maintaining dialogic activities – teacher education to combat inequalities.

A quantitative study of mathematics teachers providing pedagogical continuity through distance learning for the duration of the crisis.

The purpose of this study is to explore how critical aspects can be defined in a learning study on welding without conducting any pre-tests.

In this study, the authors focus on empirical examples from a learning study on welding conducted in six iterative cycles, with conversation analysis and variation theory approach (CAVTA) as a theoretical basis. The welding lessons have been video-recorded, and in the study, the authors analyze examples where the teachers try to identify critical aspects of a vocational practical object of learning in interaction. CAVTA permeates the complete process, where the analysis has been part of the iterative cycles and further developed when the six cycles were completed.

The results show how critical aspects can be made visible in the interaction between teacher(s) and student(s) in the enacted learning situation. In the process, the authors work with the three concepts expected critical aspects, displayed critical aspects and targeted critical features in relation to a vocational practical object of learning where conducting a pre-test to define critical aspects is not educationally possible.

Teaching vocational practical objects of learning could be seen as something different from teaching other kinds of objects of learning and the use of the traditional pre-tests in learning studies may be problematic. From that follows, that other ways of finding the critical aspects for the students regarding a vocational practical object of learning might be needed. In this study, such a way is presented.

The purpose of this study is to explore Swedish police education teachers’ informal workplace learning and its perceived value for their professional development. Two categories of teachers, police teachers and university teachers, with different professional knowledge and experience, work together at the police education unit.

The method used was in-depth interviews with teachers working at a Swedish police education unit.

Informal workplace learning was perceived by both teacher groups to be of great value for gaining knowledge about the local practice and for their professional development. Their learning emerged in discussions, observations and practically oriented activities in their daily work. Four conclusions: firstly, the teachers’ informal workplace learning was socially and practice-oriented and learning emerged in a collaborative, reciprocal and active process. Secondly, the embodied nature of the learning is evident in the teachers’ joint activities in the teaching practice. Thirdly, it takes time and active involvement in the local practice to become a professional teacher in this kind of education. Fourthly, an educational structure where academic knowledge and experience can be integrated with police knowledge and experience constitutes an important basis for teachers’ professional development in police education and training.

The study’s focus on police education and the professional development of teachers in this specific practice contributes to increased knowledge of the social, practice-oriented and embodied nature of informal workplace learning.