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Teaching and learning calculus are notoriously difficult and the didactic solutions may involve resorting to intuitive but vague definitions or informal gestures offered as proofs. The teaching literature is rife with examples of metaphors, adverb manipulations and descriptions of what happens “just before” the limit. It is then difficult to leave the domain of the mental image, thus losing the training in rigour. The author (with Karel Hrbacek and Olivier Lessmann) has endeavoured a radically different approach with the objective of training students to prove theorems while preserving both intuition and mathematical rigour. Hence we change the mathematical setting rather than the didactic setting. The result (which is a by-product of nonstandard analysis) has been used in several high schools in Geneva – Switzerland – for over ten years.

Calculus using proximities: a mathematical approach in which students can actually prove theorems