Computing science could be classed as math, but most programming does not require knowledge of computing science. I learned math without even having a computer. It is worth noting that quite a few numeric algorithms were developed well before computers existed. Doing the calculations by hand was a frightful slog, but that is what people had to do in the old days.
For example, there is the Euler method, which is an iterative method for solving differential equations, still much in use today. Wikipedia says this dates between 1768 to 1870. No computers in those days. But does anybody learning programming learn the Euler method? In order to understand it, you would have to know what a differential equation is, and to understand that, you have to understand calculus. Now that really is math.
You can write programmes without understanding the underlying math, because you have libraries implementing algorithms and data structures. This is not at all good for learning math. In fact, treating all this stuff as magic black boxes that crank out results given inputs could result in serious errors and inefficiencies, if misused.
I used to calculate Fourier series with pencil and paper and an electronic calculator. When I came across the Fast Fourier Transform algorithm, I thought it is really cool, and one the first programmes I wrote was an implementation of the FFT. It was a very bad implementation, but it did work. What this exercise did teach me was better programming.
In the case of numeric algorithms, programming is a way of doing something faster, but don't actually add anything to the math.