The subject matter of Mathematics is "Thinking clearly about problems" (not counting most problems related to understanding and relating to human behaviour and culture). Teachers can't teach maths well without having this focus. It isn't (mostly, and for most people) a cultural activity like music. Math tends to invent a terse language to help express itself, but teaching the language without clearly relating it to problem solving is what makes math seem weird and pointless to many students.If we could base mathematics education on this definition then we would see many immediate benefits:

- Teachers and students would know why they were learning mathematics;
- A problem based approach would help everyone see the difference between the important and the merely conventional aspects of the language and methods of mathematics;
- It would be clear why mathematics should be compulsory, and why efficacy should be a key requirement for higher education courses (outside the Humanities);
- It would integrate mathematics with computer education to the benefit of both.

### Update:

In John Baez's blog I appended this to a comment I made:My New Year’s resolution is to have another go to sell the idea that “The subject matter of Mathematics is how to think clearly about problems (mostly excluding human interaction issues like culture)”. Teachers and students are hopelessly confused by an education system that treats mathematics as a collection of facts (about Platonic entities) which is sometimes useful in the real world. My definition will give Mathematics its rightful place in the core of a modern education. I’m not going to make any progress until I can find a real Mathematician to endorse the idea.And I got an endorsement from John Baez himself. Initially his comment was (as wordpress emailed it to me): "I hereby endorse your idea. Please make progress.". But now the reply reads:

I hereby endorse your idea.

When I go back to UC Riverside in the fall of 2012 and start teaching math again, I’m going to teach it in a new way, informed by everything we’ve been discussing on this blog. I think the kids will enjoy it. I never taught math as a collection of ‘facts’, and that’s probably why the students liked my classes, but now I’m more keen on real-world examples that illustrate the big problems facing our civilization, rather than examples of the sort that pure mathematicians (like my former self) most enjoy.

Sometime before that, I plan to write a paper with the mild-mannered title “How Mathematicians Can Save the Planet”. I’ll put drafts here, and I’d appreciate your comments.I'll continue this subject area in a new post soon.