There's a saying in math: Analysts understand what they're talking about but find it hard to proving things - while algebraists don't know what they're talking about but find it easy to prove things.

I find this to be true, at least at the introductory level. Once you get to topology you forget what you're talking about, but that's the structural ("algebraic") view of math resurfacing. Maybe you're right though - perhaps there should be a progression in algebra from concrete -> abstract the same way there's a progression from concrete (real analysis) -> abstract (topology)