How should mathematics be taught to non-mathematicians?

Gowers Weblog:

Michael Gove, the UK’s Secretary of State for Education, has expressed a wish to see almost all school pupils studying mathematics in one form or another up to the age of 18. An obvious question follows. At the moment, there are large numbers of people who give up mathematics after GCSE (the exam that is usually taken at the age of 16) with great relief and go through the rest of their lives saying, without any obvious regret, how bad they were at it. What should such people study if mathematics becomes virtually compulsory for two more years?
A couple of years ago there was an attempt to create a new mathematics A-level called Use of Mathematics. I criticized it heavily in a blog post, and stand by those criticisms, though interestingly it isn’t so much the syllabus that bothers me as the awful exam questions. One might think that a course called Use of Mathematics would teach you how to come up with mathematical models for real-life situations, but these questions did the opposite, and still do. They describe a real-life situation, then tell you that it “may be modelled” by some formula, and proceed to ask you questions that are purely mathematical, and extremely easy compared with A-level maths.
One comment on that post particularly interested me, from someone called Joseph Malkevitch, who drew my attention to an article he had written in which he recommended a different kind of question both from the usual sort of symbolic manipulation that most people would think of as mathematics, and from the sterile questions on the Use of Mathematics papers that pretend to show that mathematics is relevant to real life but in fact do nothing of the kind. The main idea I took away from his article was that there is (or could be) a place for questions that start with the real world rather than starting with mathematics. In other words, when coming up with such a question, you would not ask yourself, “I wonder what real world problem I could ask that would require people to use this piece of mathematics,” but rather, “Here’s a situation that cries out to be analysed mathematically — but how?”
Inspired by Malkevitch’s article, I decided to write a second post, in which I was more positive about the idea of teaching people how to use mathematics. I gave an example, and encouraged others to come up with further examples. I had a few very nice ones in the comments on that post.