I recall working with a reasonably reputable mathematics educator once, teaching a calculus class. At one point it became evident that the guy wasn't comfortable summing a geometric series. One thing I expect from mathematics education is that mathematics educators know some mathematics. Saying this any less bluntly would lose too much of the sentiment I'm trying to convey, so there it is.

As a mathematician, I'd personally like to see work in mathematics education that helps me teach what it is that I actually do as a mathematician. I was introduced by Ken Appel to some of the ideas of [Hy Bass][1] on mathematics education, e.g. the "granularity" concept which asserts that at different levels of sophistication mathematicians allow different jump sizes in their arguments. Awareness of granularity made explicit like this really has changed the way I organize my undergraduate course material and for me was revolutionary.

Other ideas of Bass that I'd like to see followed up include the idea of a [common structure problem set][2]. Such an idea might help to get a large chunk of a difficult aspect of the mathematical aesthetic into the curriculum. 

In general, I'd like to see mathematics education address how the practices of the best mathematicians can be brought to the graduate and undergraduate population in universities, and how we can bring more of *mathematics itself as experienced by mathematicians* to our students. I'm more interested in this than this than studies of how to improve calculus course assessment, for example. I've always been frustrated that nobody seems to study the learning approaches of successful mathematicians rather than average students. I'd personally like to see more of our best practices being studied and propagated. This last paragraph is a bit ignorant, I know, but it's my honest impression.
 


  [1]: http://en.wikipedia.org/wiki/Hyman_Bass
  [2]: http://pzacad.pitzer.edu/~dbachman/RUME_XVI_Linked_Schedule/rume16_submission_3.pdf