Intuition before rigor

There is an old anxiety in mathematical teaching that pictures lie. They do, sometimes. A curve that looks smooth may be nowhere differentiable; a region that looks bounded may not be; the classic false proofs all depend on a diagram drawn slightly too helpfully.

The response to this danger has often been to withhold pictures until the formalism is secure. I think this gets the sequence backwards, and loses far more than it protects.

Rigour answers the question of whether something is true. Intuition answers the question of why anyone suspected it might be. A student given only the first has no way to generate conjectures of their own, which is most of what doing mathematics consists of.

The productive arrangement is to lead with the picture, be explicit that it is a guide rather than a proof, and then use the formalism to test exactly where the picture breaks. The places it breaks are usually the interesting part — pathological examples are far more memorable when they contradict something you were tempted to believe.

Rigour first produces students who can verify. Intuition first, with rigour close behind, produces students who can imagine and then verify. Only one of those is doing mathematics.