I need to give a lot of quite basic background to this question because I think (at least from conversing with fellow graduate students) that most mathematicians have not thought about fractions deeply for a long time. I think that there is an interesting germ of an idea in here somewhere, but I cannot exactly pinpoint it. Essentially there seems to be two canonical ways to solve division problems and there does not seem to be a "natural isomorphism" relating the two ways. I am interested in framing this duality formally: is there a "categorification" of the rational numbers where this duality can be precisely framed?

I TA a class for future elementary school teachers. The idea is to go back and really understand elementary school mathematics at a deep level. Hopefully this understanding gets passed on to the next generation.

We were discussing division of fractions. Rather than say "Yours is not to wonder why, just invert and multiply", we try to make sense of this question physically, and then use reasoning to solve the problem. Take (3/4) / (2/3). When doing this there seems to be two reasonable interpretations:

1. 3/4 of a cup of milk fills 2/3 of a container. How much milk (in cups) does it take to fill the entire container?

This is a "How many in each group" division problem, analogous to converting 6/3 into the question "If I have six objects divided into three equal groups, how many objects will be in each group?"

The solution that stares you in the face if you draw a picture of this situation is the following: 3/4 of a cup fills 2 thirds of a container. That means there must be 3/8 cups of milk in each third of the container. One container must have 9/8 cups of milk then, because this is 3 of these thirds. Note that the solution involved first dividing and then multiplying.

2. I have 3/4 cups of milk, and I have bottles which each hold 2/3 of a cup. How many bottles can I fill?

This is a "How many groups" division problem, analogous to converting 6/3 into the question "If I have six objects divided into groups of two, how many groups do I have?"

The solution suggested by this situation is the following: 3/4 of a cup of milk is actually 9 twelfths of a cup . Each twelfth is an eighth of a bottle. So I have 9/8 of a bottle. This solution involved first multiplying and then dividing.

Now I come to my question. This pattern persists! Every real world example of a "how many in each group" division problem suggests a solution by first dividing and then multiplying, whereas each "How many groups" division problem involves first multiplying and then dividing. It seems that solving the problem in the other order does not admit a conceptual realization in terms of the original problem. This is interesting to me! It suggests that the two solution methods are fundamentally different somehow. In the standard approach to rational numbers (natural numbers get grothendieck grouped into integers, which get ring of fractioned into rational numbers) ignores this kind of distinction. Is there a categorification of the rational numbers which preserves the duality between these two types of question?

I think that Qiaochu's answer -- which is enlightening and on point -- shows that your question is worth asking. This is an actual math education question ("actual" because one of my minor pet peeves is misuse of the math-education tag) with some underlying mathematical sophistication that makes it appropriate for an audience of research-oriented mathematicians. I say go for it.

It is posted.

@Qiaochu Could you post your answer on MO as well? I would like to discuss it but meta doesn't seem to be the right place.