A deep understanding allows one to deal with problems that they have not only been explicitly taught, but those they have also never encountered. I don't think

A deep understanding allows one to deal with problems that they have not only been explicitly taught, but those they have also never encountered. I don't think your example regarding libraries really applies well here. I think a more appropriate example would be knowing how to do a few things with a library without a real ( or any ) understanding of the language its based upon. I'm no mathematician, nor am I a (professional) programmer for that matter, yet I've seen real benefits going back and really learning some of those topics that were kinda passed over in high school ( so what exactly is sine doing to my numbers? Taylor series? Fourier series? Optimization functions?). Besides, math teaches you to think, memorization doesn't.