The title of this poster is a name of a Horizon 2020 project with the acronym MaTeK. The project belongs to the Twinning action, which stands for research infrastructure building and institutional networking. The general goal of MaTeK project is to strengthen the research performance and develop the excellence of the Department of Mathematics Education (DME) at the Faculty of Mathematics, Physics and Informatics of Comenius University in Bratislava, Slovakia (UK BA), in the field of mathematics teacher knowledge enhancement. Teacher knowledge is the prerequisite of the education enterprise, and student knowledge development its objective. For researchers, it is essential to understand what kinds of knowledge mathematics teachers develop, and how they use their expertise in teaching in order to help students to develop deep knowledge in mathematics. One of the ways how to address this issue is to work with pre-service mathematics teachers on lesson and learning progression design capacity (Pepin et al., 2017). The specific goal of MaTeK is to provide the DME with opportunities to work with and learn from partners and exchange best practices in the field of mathematics lesson and learning progression design, with focus on the topic of reasoning and proof (R&P). R&P is an important strand of mathematical proficiency (eg, Kilpatrick et al., 2001), and research has shown that pre-service teachers are often not adequately prepared to cultivate opportunities for students to engage with reasoning and proof (eg, Stylianides & Stylianides, 2009).