It's interesting to me that we assume, somehow, that a person who simply sits in enough mathematics classes will become the kind of mathematical thinker that can relate ideas to small children, enjoy children's insights, and allow space for children's own pace of development.
What do we mean, "think mathematically"? or "mathematical thinker"? I'm going to be fascinated by what we MEAN when we use those phrases. One mathematician I know claims not to know what mathematical thinking is. So????
And, finally, what is the connection from mathematical thinking to teaching-to-think-mathematically??