This new approach started to be written in French and is still incomplete there (about 40 pages nearly ready, 20 or 30 more pages in draft, and 50 to 100 more pages in plans). The translation into English of what is ready, just started and will be progressively completed in the next few weeks.
The ambition of this approach is to combine the following advantages by a new structure of progression:

• It is perfectly rigorous, all what is provable is proven from the beginning. (Bourbaki had this goal, but here further goals are added, which might seem a priori conflicting with this one; better than a compromise, I offer here a new intergrated solution to gather several more advantages)
• Conduct original approaches to most subjects: most concepts were already largely known here, there, but many of them had not, as far i know, been described and articulated in this way. Except about a half the contents of text 2, which has much in common with traditional courses, other parts also have some common aspects with what can be found elsewhere, but it comes with new structures and explanations, simplified proofs, and the like.
• Emphasis is placed on intuition, the "philosophical" explanation, the deeper meaning of things, the mathematical world, without being slowed down by the development of tedious formal rules. Here, rigour or formalisation will neither be missing nor dilute or obscure meaning. On the contrary, the right formalism will express the highest concentration of meaning.
• The tools developed are very powerful and general
• Very few proofs take more than half a page, and most proofs are just couple of lines long; all choices are made with great care for the shortest and elegant path as possible, better than any other existing approach.

2. Elementary constructions (translation in progress - ultimately 12 pdf pages ?)