There is no rigorous definition of rigor.

The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present day kaleidoscope of events.

[The error in the teaching of mathematics is that] mathematics is expected either to be immediately attractive to students on its own merits or to be accepted by students solely on the basis of the teacher's assurance that it will be helpful in later life. [And yet,] mathematlcs is the key to understanding and mastering our physical, social and biological worlds.

There is no rigorous definition of rigor.

Mathematics is a body of knowledge, but it contains no truths.

Actually, most mathematics courses do not teach reasoning of any kind. Students are so baffled by the material that they are obliged to memorize in order to pass examinations.

Psychologically, the teaching of abstractions first is wrong. Indeed, a thorough understanding of the concrete must precede the abstract.