There comes a point in mathematics, at all levels, where understanding of "why" needs to stop and being able to "do" becomes more important. Ultimately, we learn mathematics so that we can actually solve problems, learn technologies which make calculation simpler and which given us a robust platform for moving on to more powerful techniques.
I don't think that is true. In fact, I think it is a huge disservice to students to teach like that. Every year I see incoming college freshman who think they are good at math because they have gotten straight As and everyone has said, "oh Billy is so good at math." Well, it turns out what they are really good at is following directions. They don't understand anything that they are doing, and they don't have the tools or creativity to do actual college level math. At the same time, there could be great potential mathematicians that were turned off of math at an early age because of the paint-by-numbers way it was taught to them. If you don't understand why you are doing something then you really don't understand what you are doing and shouldn't be trusted to do it.
In computing, the mean time to failure keeps getting shorter.