>No, the issue here is expanding the expression by use of the >distributive property and there is no intuition involved.
In other words: it IS a notation issue. But the very fact that you need rules to expand the notation, and therefore cannot rely on your intuition, is a major roadblock in the students' paths to understanding. Furthermore, yesterday we may have needed to put a lot of emphasis on how to parse notation, but today we have computers for that purpose, our minds are free to go do something more productive.
And ah, if I have to choose between using intuition and using rule-based inference, I'll choose intuition umpteen out of umpteen plus one cases.