What should a math graduate know? [closed]

There was a discussion this morning on BBC Radio 4 by Jim Al-Khalili in a programme called The life scientific, and the education of scientists was discussed with the next government chief scientific advisor, Sir Mark Walport, who stressed the importance of writing, presenting, and communicating, as well as getting to grip with scientific issues. Tim Porter and I wrote an article ``What should be the context of an adequate specialist undergraduate education in mathematics?'', The De Morgan Journal 2 no. 1, (2012) 411--67, which discusses many issues such as the aims of an undergraduate course. A notable feature of a course we ran at Bangor on "Mathematics in context" was the enthusiasm of the students: a free discussion with them on the ideas of the course was like opening the floodgates!

Part of the reason for the course was to provide students with some background and language so as to be able to have a view and talk about the value of the subject in general. This is important for the subject so that the students can be able to speak for the subject in their future careers.

Another take: whatever the concrete (well, or abstract!) content is, the most important point is that you should know what you know very well. Depth is more important than breadth! you should know not only the theorems and definitions, but their use (well, some of it ...), understand proofs, being able to do proofs, understand counteraxamples, understand why a concept is defined in some way and not in some other way (why does the other, appealing way brake down?). As Halmos said, you should fight it: not only do exercises, make your own exercises, proofs should be worked until you understand every little detail better than the author ... and so on.

One criterion that could be used is to ask if this or that subject (and the manner in which it is taught) help the student to learn to learn mathematics. We cannot determine all the needs for the future mathematics or statistics graduate but if they are trained to learn mathematics (and not by just sitting in a lecture watching `us' write on the board or click a button to go to the next slide), then they are set up for future learning.

... and having learnt to learn, as Ronnie said in his reply, they should learn to write, and communicate the mathematics (and any conclusions that it gives) in an intelligible way, (and a Maths in Context course is an excellent way to do that).

So do not just look at the content, look at the context!