As a programmer you know that being able to program changes the way that you think. Algorithmic thought is different from the messy stuff that most people use to work out the problems that they face. This is, of course the best reason for teaching programming to as many people as possible.
On a more modest note, programming and computers also provides an easy an accessible route into mathematics in general. You have to keep in mind that despite what you have been told over the years, math is not about adding up or multiplying. No - that is arithmetic and it is what a cheap pocket calculator can do with no effort or intelligence.
Math is about much more complicated stuff than arithmetic even if arithmetic is a prime (no pun intended) example of something that mathematicians study. I've encountered too many mathematicians who were brainwashed by their early experiences and held the position that without being good at numbers you can't be a mathematician. This is just silly and it is nice to hear from a mathematician who gets it right.
Watch Conrad Wolfram (younger brother to Stephen Wolfram of Mathematica and Cellular Automata fame) as he discusses the error in forcing children to do sums with no real purpose other than just needed to meet an externally set goal. As he puts it