Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice - George Pólya

The worst may happen if the student embarks upon computations or constructions without having understood the problem.

The volume of the cone was discovered by Democritus... He did not prove it, he guessed it... not a blind guess, rather it was reasoned conjecture. As Archimedes has remarked, great credit is due to Democritus for his conjecture since this made proof much easier. Eudoxes... a pupil of Plato, subsequently gave a rigorous proof. Surely the labor or writing limited his manuscript to a few copies; none has survived. In those days editions did not run to thousands or hundreds of thousands of copies as modern books—especially, bad books—do. However, the substance of what he wrote is nevertheless available to us. ...Euclid's great achievement was the systematization of the works of his predecessors. The Elements preserve several of Eudoxes' proofs.

The general or amateur student should also get a taste of demonstrative reasoning... he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life.