Does a programmer design computer programs?

Yes. A computer programmer will at least use simple arithmetic. Most computer science degrees require a year of calculus, though I must admit that the most useful math classes I've taken were discrete mathematics. Math that deals with sequences, series, and sets tends to be the most common and useful (in my experience).

Of course it all depends on what specific field of programming you will be in.

If you will be doing a lot of network programming, then graph theory will probably be part of your everyday programming. Graphics programmers will need to know a good bit of geometry and trigonometry.