A few months ago, a redditor introduced me to the notion of "the trivium". It was mentioned in the context of reforming the U.S. education system. I think it was a neat idea, and it was the lens through which I interpreted this comment.

The trivium consists of grammar, dialectic, and rhetoric [1]. Grammar signifies knowing the language/jargon. E.g. knowing the parts of a car. Dialectic signifies logic and critical thinking. E.g. realizing that adding more oil won't fix the alternator. Rhetoric signifies application and expression. E.g. designing a safe, high-performance, cost-effective, and aesthetically-appealing vehicle.

Regarding your anecdote about not understanding abstraction of shapes. It sounds to me like your friend is comfortable with grammar, but not dialectic.

According to this redditor, a major flaw in the education system is that it teaches "subjects" (grammar) while critical thinking (dialectic) is shafted (mostly limited to math class) [2]. But I repeatedly read stories about people who can program without anything more than a basic understanding. I find this surprising because computer science is mostly math, which falls under dialectic. Why is this? Disclaimer: I'm new to programming.

[1] http://en.wikipedia.org/wiki/Trivium

[2] http://www.scribd.com/doc/36325362/The-Lost-Tools-of-Learnin...