**3. The project MaCo**

The project "MaCo" ("Mathematik aufholen nach Corona"—Catching up in Mathematics after Covid; https://maco.dzlm.de/projektinfos) focuses on students who are exposed to learning delays. Within this nationwide German project, basic competencies for particularly affected children and adolescents at primary and lower secondary level are reworked. However, in order to meet these learning difficulties as quickly as possible after Covid, professional support is required. Therefore, MaCo not only targets learners and their parents but also focuses on the support and further training of teachers and facilitators, or any kind of support staff. By addressing all the different levels (classroom level, teacher professional development level, and facilitator professional development level), the project follows a multi-level perspective on teaching and learning, which at the same time offers different possibilities for research strategies [16]. In general, the project's research activities were guided by a design-based research approach (for example, see [17]).

In total, materials were developed for 14 modules by different teams of researchers (one introductory module, six modules on primary level and seven modules on secondary level), concentrating on basic competencies in mathematics like understanding numbers and operations as well as place value, percentages, variables, and functions. In each module, professional development, instructional, and support materials were developed. For all modules, five principles for high-quality mathematics teaching were taken into account: conceptual focus, cognitive demand, students focus and adaptivity, longitudinal coherence, and enhanced communication [18]. With these principles concrete jobs for teachers emerge: Identifying conceptual basics, diagnosing conceptual basics, and supporting conceptual basics. For example, having identified the principle of bundling as central for understanding place value, ways of diagnosing and supporting this specific competence are essential. In general, the following questions are guiding: What encompasses a sustainable understanding of place value (natural numbers)? How to diagnose if learners have a sustainable understanding of place value at their disposal, and what are typical difficulties? How to develop a sustainable understanding of place value?

The following chapter gives insight into the different developed materials for the module "understanding of place value" and shows exemplarily how the relevant basic competencies are supported.
