History, an educational tool in mathematics

In textbooks or syllabi, which are used as pedagogical supports in mathematics courses are very accurate, detailing what should be taught and how. However, and even though, different mathematical subjects are given as precisely as they can, depending on the level of the students, the latter oftentimes have difficulties to get an overview of what is actually taught and why. Consequently, in the course of time, students increasingly start to question the necessity and usefulness of what they’re seeing in class. One of the most difficult tasks of mathematics teachers is to give an answer to these questions but, first and foremost, to show that mathematics makes, or can make, sense. A way to overcome these difficulties is to use history of mathematics as it provides with an inexhaustible source of real problems. Therefore, firstly, this thesis treats the purpose and necessary methodology of implementing the history of mathematics as a part of mathematics lessons. However, there is another important question that needs to be addressed: what happens if one leaves out history of mathematics? Nowadays, mathematics is simplified or changed in order to make it more accessible for students of different ages. Consequently, the core definition and how the latter was developed fall into neglect, leading to a further neglect of the richness, subtlety, and the power of a definition. The latter being, most importantly, the fruit of centuries of work and mathematical thinking, which could be neglected, as for instance the Riemann’s definition of the integral. Therefore, secondly, this thesis is also going to focus and return to the core of the Riemann’s definition of the integral by using original sources.