In order to develop and evaluate characteristics of ideal-typical patterns of private additional lessons in mathematics, I refer to investigations on mathematics school lessons of Hans Freudenthal, Hans Werner Heymann, Alan H. Schoenfeld, and Heinrich Winter. Their ideal-typical patterns are equally represented in the six fundamental dimensions ”independent work”, ”subject orientation”, ”process and problem oriented interactions”, ”structural thinking”, “discovering education processes”, and ”application orientation”. For private additional lessons, the content of the six dimensions is determined by case studies, consisting of the execution and analysis of remedial teaching, and evaluated by synopsis to the characteristics of school teaching. As an outlook I present a concept of school-organized remedial teaching.