One thing you might learn in math is to avoid making overgeneralized statements that you can’t support.
A valid substitution in your statement for “programming” is writing a compiler. And for “undergraduate mathematics taught in math departments”, basic differential calculus.
Yet we regularly teach smart high school students and first-year undergraduates calculus, and almost never try to teach them to write a compiler, contradicting your proposition.
But what do I know? I’m just a dumb programmer. I can’t read your mind, so maybe you had something a little more specific you wanted to say.