In answering another question What is the justification to teach the (redundant) use of parentheses in multiplications? I was left wondering what we actually know about students' progression in terms of how they see algebraic expressions.
Is there any research work that has been done into the progression of students' ability to parse math expressions?
My impression is that as students progress in their mathematical careers, the way that they mentally process mathematical expressions changes from seeing them as a sequence of characters (and having to consciously think about order of operations) to automatically seeing their structure of nested subexpressions.
In other words, for early students (5 x 10) + (5 x 8) and 5 x 10 + 5 x 8 are processed differently, but for later students they are processed nearly the same.
But at what time scale does this happen? My impression is that students make substantial growth in their parsing skills during calculus.
This is analogous to going from reading a sentence in a foreign language and having to puzzle out the grammar to this process happening subconsciously during the reading stage.
I suspect if there is any research at all along these lines it would be eye-tracking research, but I couldn't find eye-tracking research in math focused on formulas specifically.