In an interview, Vladimir Arnold talks about his teaching experience in France and condemns the formal thinking of the students. In the end he concludes that although their reasoning is logically correct, they understand nothing. If their deduction is correct (as he says), what is wrong with the given solution? What do the students not understand (or where does exactly their mathematical understanding fail?)? How can oneself make sure to "understand properly"? He says:
For example, at a written examination in dynamical systems for fourth-year students at Paris-Dauphine, one problem was to find the limit of the solution of a system of Hamiltonian equations on the phase plane starting with some given initial point when time goes to infinity. The idea was to choose the initial point on a separatrix of a saddle, with the limit being the saddle point.
Preparing the examination problem, I made an arithmetical error, and the phase curve (the energy-level curve containing the initial point) was a closed oval instead of the separatrix. The students discovered this and concluded that there exists a finite time $T$ at which the solution returns to the initial point. Using the unicity theorem, they were able to deduce that for any integer $n$ the value of the solution at time $nT$ is still the initial point. Then came the conclusion: since the limit at infinite time coincides with the limit for any subsequence of times going to infinity, the limit is equal to the initial point! This solution was invented independently by several good students sitting at different places in the examination hall. In all this reasoning, there are no logical mistakes. It is a correct deduction which one may also generate by a computer. It is apparent that the authors understood nothing.