Different variations of exercises are currently offered. Each variation has a distinct configuration and combination of three components - group exercise, central exercise, and homework. Here we describe two variations, whose elements can also be combined:
A. Supervised classroom problems: During each group exercise session, a series of classroom problems is handed out, each of which can be solved in a very short time. These problems are solved together by several small groups of 3 to 5 students. The students discuss the problems and solution approaches and devise the solutions together. The instructors act as moderators and consultants. Some of the classroom problems can also help the students prepare for completing their independent homework, and they can address typical errors that students made in their homework. The contents of the accompanying lecture can be addressed without losing any time. The goal of this type of group exercise emphasizes active and cooperative learning.
The purpose of group exercises is not to present solutions for the homework. This is done in the central exercises, where the methodological aspects are also presented. Central exercises should not just show sample solutions, but also recapitulate the subject matter, answer subject-related questions and stimulate discussion. Sample solutions are made available in typically or electronic form. The students receive annotated feedback on all or some of the solutions they wrote for the homework problems. If the students are able to independently review their solution methods on the basis of the annotations or the model solution, then a central exercise for conveying this information is no longer necessary.
B. Homework Discussion: This type of exercise emphasizes dealing with the homework. In addition, subject matter from the lecture can be recapitulated, and the students can ask questions on the lecture’s subject matter. In particular, dealing with the homework entails developing and discussing the methodology of the solutions. Sometimes sample solutions are shown to present complete, formally exact solutions. The instructors encourage the participating students to present their own solutions and solution approaches and to open them for discussion. The instructor also corrects the written solutions and comments on them. If many students participate in the group, only some of the tasks are selected for review. In this type of exercise, a central exercise is not necessary. The goal of this type of group exercise emphasizes teaching the students how to methodically develop solutions of even very complex tasks whose scope would go above and beyond the range of classroom tasks.
Exercises can also entail hybrid forms made from variations A and B that combine the elements "discussions of subject matter," "review of selected homework problems" and "working out classroom problems" or handle them in alternating one- or two-hour tutorial sessions. One of the reasons why these hybrid forms are a fitting alternative is that this more diverse exercise type is more attractive to students.