I think we should teach children to understand mathematics at an algorithmic level and let them see that programming is a tool/hammer for executing those algori

I think we should teach children to understand mathematics at an algorithmic level and let them see that programming is a tool/hammer for executing those algorithms. Computers are useful for doing math, but for most kids, and even most adults, that's one of least interesting things they can do. A program isn't just a hammer. It can be a paintbrush, a tape recorder, a story, a symphony, or a lab assistant. And a program is more than any of these things -- a real-life paintbrush can't change the size of its bristles, and you have to draw every stroke by hand. On a computer, everything's just code. You can run your "paintbrush" routine billions of times, vary its parameters using your voice, or earthquake data, or a physics simulation, or anything you can imagine. As a programmer, maybe I'm biased, but I can't imagine any job beyond the most menial where the ability to program wouldn't make me more productive.