I actually studied things like Newton's method and even Runge-Kutta ODE solution in a numerical methods A-level module (i.e. at school). There are other bits of the curriculum that are quite different too -- like decision maths, which basically was lots of algorithms (routefinding! critical path analysis!). As a result, my university-level courses were very much based on efficient numerical implementations of stable algorithms, and looked a lot at something like how to solve chaotic ODEs, or integrating rather unpleasant functions. (I was actually taught by this person and how to solve his problems! https://mathworld.wolfram.com/Hundred-DollarHundred-DigitCha...)