My experience was the opposite. The more practical Mathematics was too easy to hand wave through so no one ever bothers with proof. But then when something goes

My experience was the opposite. The more practical Mathematics was too easy to hand wave through so no one ever bothers with proof. But then when something goes wrong or you're not sure whether a particularly complicated calculation makes sense, you're SOL without a real formalism to go on. I didn't really believe in calculus except on a case by case basis until after my first analysis course. I think the only reason the less formal mathematics courses seem easier is because the thing being studied is also dead simple and the calculations are easy. But that approach doesn't scale.