hangman
almost-scientist
How does a mathematical basis contribute to thinking for oneself? Math is the most concrete field of knowledge in existence. It has absolutely no room at all for opinion...a mathematical fact is indeed a fact, and that's pretty much that.
Because it's not always that simple. We often have multiterm, partial differential equations of multiple orders that are utterly impossible to solve theoretically. It makes it necessary to "guess" at which terms are more important to construct a theoretical framework we can actually work with, and that takes some critical thinking.
There's tons of programming involved also, it's like 80% of what we do on a given day. There's often no one, "right" way to program, so organizing the problem in a way that can be solved the fastest is important. Experience with that kind of algorithmic thinking makes it easy to see how problems in other areas are constructed.
There's also statistics, which is really as much philosophy as it is math.