TITLE: What does a mathematician expect from mathematics education? 
QUESTION [7 upvotes]: Consider that my question is not a personal and/or subjective question. Perhaps, you have hired a mathematics educator in your department and you are interested in finding a way to communicate with him/her and vice versa. Or not, for some reasons you are looking at one of the writings of those peoples, and more often than not you get disappointed since you can hardly find what you are looking for. What do you expect from mathematics education as a discipline?
Addendum: After being revised by others it occured to me that my description had the potential of being interpreted as rude or offensive. Thus I decided to accept the revision done. Furthermore, to be less argumentative , it would be great to include in your answers/comments some good examples of the ideas from mathematics education that you have found useful, and perhaps an explanation of how they were useful for you (there are one or two below). I hope this would be a posstive way to go.

REPLY [20 votes]: 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 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. 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.