> I feel there needs to necessarily be a separation of the doing of mathematics and the teaching of mathematics in these kinds of matters. In the teaching of mathematics, especially in the more 'abstract' areas, there is not nearly enough driving of intuition, and the 'problems' students are given often are unrelated to the 'problems' the theory they are learning about was created to solve. Pushing things in the concrete direction is probably the right direction for pedagogy.

Indeed. By far the best teachers I've had for math courses spent a good deal of time discussing the history of the topic and motivating its creation.