I think we have a false dichotomy here. What we call 'intuition' and 'rigor' are not necessarily opposed. I think the problem with math classes is not any lack or excess of rigor, but that they focus too much on details instead of the general principles from which those details are derived. Complicated formulas and procedures are presented for students to memorize without any understanding of their origin. It's mostly 'how,' a little 'what,' and never 'why.'