Maria Gaetana Agnesi: new way to teach maths is in the past?

Maria Gaetana Agnesi was born in Milan, on May 16, 1718. She was highly talented and excelled in the art of philosophical disputation. In 1738, Agnesi concluded her studies with the publication of her thesis Propositiones philosophicae (Philosophical Propositions), mimicking the academic path of male students in contemporary colleges. Profoundly interested in mathematics but still unclear about the nature of her possible contribution, Agnesi began by planning a commentary on Guillame de l’Hospital’s treatise on curves, to make it more accessible to students. Gradually, however, she worked on a much more ambitious project: an introduction to calculus that would guide the beginner from the rudiments of algebra to the new differential and integral techniques. This would be a great work of synthesis, aiming at a clear presentation of materials that were written for specialists, in Latin, French, or German and published in hard-to-find journals. During the making of her book, Agnesi interacted with leading Italian experts, such as Jacopo Riccati. At a time when the practice of calculus on the continent was moving away from its immediate geometrical meaning, Agnesi aimed to rediscover those techniques of Cartesian geometry designed to bridge the gap between the geometrical and analytical fields. Her teaching method can be still interesting. Maria Gaetana and her contemporaries do not care about logical rigour; they prefer to present the concept in an intuitive way. The question of giving a more formal and logical approach to the definition of limit was addressed by Augustine-Louis Cauchy and, later, by Karl Weierstrass. According to any scholars, this abstract approach can be didactically inefficient for the beginner student, observing that, among other things, a mathematics course uses different notions of limit for which it would be appropriate to introduce and define the limit so that applications come from a single concept. The question of the fundamentals of the analysis cannot be considered completely closed; indeed someone thinks that the system of axioms to be considered is even variable with the type of problems that are faced.