It still remains an important open question of the interpretation of the foundations of quantum mechanics to thoroughly elucidate the essence and significance of the correspondence principle. We focus from a new mathematical aspecton the review of the correspondence principle to gain the correct understanding of the principle. As a result, we show that there does not exist the algebraic isomorphism between the algebra of the observables and that of the quantum operators, and therefore the previous interpretation of the correspondence principle aiming to provide allthe operators corresponding to physical quantities is inconsistent from the mathematical view point. Furthermore, it is demonstrated that the correspondence between physical quantities and quantum operators is possible within canonically conjugate observables constituting the action, while classical and quantum quantities satisfy one and the same dynamical relation. Moreover, it is shown that the classical limit of quantum mechanics can be explained not by the correspondence principle but by the de Broglie relation and the operator equations.