Beyond just the importance of understanding how computers work, I deeply believe that learning to program can be a huge aid to understanding (really understanding) mathematical concepts as well as other fields of science, language, etc.

I'm currently reading Seymour Papert's amazing book _Mindstorms_ (after reading a comment on HN, appropriately enough). I am in complete agreement with his assertion that schools currently present topics as a series of disconnected rote pieces with no "point" or basis for deeper understanding. Computers can provide the platform for understanding, intuiting, and even self-teaching concepts in a way that just isn't possible with pencil and paper.

The whole book makes me unconsciously flex my hands into fists and suppress a scream, "yes! This!"

I was one of those kids who thought they just weren't made for math and I hated it and felt bad about it. Later in life, I finally figured out that all of the math I'd been taught in school was about as complex as a "hello world" program. It's laughable, really. And I don't feel bad about it anymore. I wish somebody had been able to tell me that then.