We propose an expanded definition of Threshold Concepts (TCs) that requires the successful acquisition and internalisation not only of knowledge, but also its practical elaboration in the domains of applied strategies and mental models. This richer definition allows us to clarify the relationship between TCs and Fundamental Ideas, and to account…

The universities and faculties which educate teachers of mathematics for teaching pupils/students of any age group from pre-school age to higher education carefully monitor and compare valuable results of this research, detect the areas in which the mathematical achievements of pupils should be improved at the national level and propose the ways…