[isanybodythere]

I'm not sure we can really guess what the student meant, but I do know for sure that she was wrong.

If you add up a sixth of a six-pack and a sixth of another six-pack you get a sixth of two six-packs -- two twelveths.

The student's misunderstanding comes from being taught fraction addition in terms of items in a collection -- which only holds if you keep to the same set (what you called scale).

This is a common choice -- "students already know how to add integers, so let's start from there", but as it did in this case, it doesn't always work as intended.

This is a great example of taking an analogy so far that the student didn't learn anything new at all. Everyone feels happy -- teacher's teaching, student's learning -- until you test what your knowledge on outside the domain of the analogy.

Fraction and integer addition are one and the same, yes -- but from the point of view of fractions, wherefrom integer addition is a special case. It remains challenging to teach and understand from the point of view of the integers, which is where the student stands.