My degrees are in aerospace engineering. Now I teach aerospace at a local college. My notes from 35 years ago are still handy, in writing lecture-material and homework problems. They haven't changed in the least.Sort of related, while getting my engineering degree had to pass all of the high level math required. But I joke that once working as an engineer the hardest math I had to do was adding up my expense account.
And the math is crucial. If you don't understand a smattering of measure theory, linear spaces and orthogonality of functions, calculus of variations and the like, then you won't understand the theory that we teach. A little bit of topology and abstract algebra also help.
But I'm fanatically conservative. My teaching style is patterned after how things were in continental Europe 100 or 120 year years ago. The greatest compliment that I received from my older colleagues - these guys are in their 80s or 90s now! - is when they'd say, "I remember going to college in the 1950s. There were still some of the elderly stalwart professors back then. The profs who taught them... teach like you do, today".