In mathematics lessons and lectures at university-level, many students - even including those who excelled in maths at secondary school – are at a loss and do not know what teachers are talking about. There is obviously a considerable gap between the average level of knowledge of a high school graduate (Maturant/Abiturient) and the knowledge required at universities. It seems that even the new trend of the past years, i.e. the call for skills-based teaching in maths, has not contributed very much towards reducing this gap.One of the ambitious objectives of skills-based teaching would certainly have to be a transition from “drilling” certain structures through mere repetition towards a deeper “understanding” of the subject. In contemporary pedagogic terms, this means that a deeper understanding of maths means a more competent approach towards mathematical models, which includes better understanding, production, handling and interpretation of these models. The key factor in achieving this ability lies in the improvement of the intellectual skills necessary to understand maths by developing a sound basis for the subject. This basis rests upon two fundamental competences, namely knowledge of the terminology and the ability to draw adequate conclusions (i.e. being able to deal with mathematical models). The approach described in the book differs from the quite conventional opinion that the basic competences involved in understanding maths develop almost automatically as a side-product of dealing with, and working on diverse mathematical problems. The terminology of the subject and the techniques of drawing adequate conclusions cannot be learned unknowingly through exercise, they are explicitly explained and trained. The distinctive feature of the book lies in the fact that all explanations are presented in terms and concepts that pupils have learned at primary and lower secondary schools (between the ages of 6 and 15). The required level of knowledge comprises only the basic arithmetic operations for fractional numbers and decimal numbers, and elementary concepts of algebra and geometry. This book is to be seen as a new alternative for all who want to improve their understanding of mathematics. The objective is not to teach certain parts or items from a curriculum but to enable everybody who is keen and willing to independently deal with any kind of mathematical subject matter.
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