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High school material

Documents for teachers and for students in pre-university introductory analysis courses

These documents are used in some high schools in Geneva (Switzerland)  In these classes students have not studied calculus before and the syllabus requires that they must be able to understand and do proofs.


Student handouts can be directly given to students. In many cases, proofs are not written out since they would be studied in class. Hints and guidelines are usually provided - these hints may also help for results not given in the teacher manual.

student handouts

The standard level maths class is 4 hours a week over the two last years of hight school. The higher level maths is 6 hours a week.

teacher's manual

The  is a presentation of the main results presented in an analysis course, fully proving some of the theorems and leaving others as exercises to help new instructors to find their way.

For examiners.

Mathematicians who would need to read a proof without necessarily being able to write it out themselves  (this is the case for exam external jurors who are not necessarily familiar with ultracalculus) may find a crash course here:

What sort of numbers are these?

The teachers new to this approach may ask where, in this extended universe, are the numbers they are used to? This questioning - particular to teachers, not to students -  sometimes gets in the way of understanding. There are two ways to see this and either way, the mathematics are the same. For a discussion of how to interpret ultralarge and ultrasmall numbers


why we consider this approach to be of interest at this level.