What they said wasn't wrong. Their mental model was absolutely correct.
"One third of this and one third of that is two-sixths of everything" is absolutely right. Telling them "No, you're wrong" is counter-productive. "You have to look at the whole" isn't a helpful statement because, in this case, there are THREE 'wholes.'
Their written representation of the mental model was incorrect because their instruction was focused only on abstract numbers instead of concrete labels. If those fractions (or ratios or whatever) are labeled properly the equation is completely correct.
Separate from the discussion of what is or is not counterproductive / educative, the student's mental model was wrong.
The student was taking the model for ratios and applying it to fractions. If I need to add 2 + 2, and I multiply instead, I did the wrong calculation. It does not matter whether I multiplied correctly, nor does it matter that, in this case, both operations equal four.
No. The mental model was exactly correct. Their assignment of notation to the model was non-standard when they wrote "+" for the operation of merging the wholes.
The confusion of what is wrong with what is correct but non-standard is in fact quite standard and quite wrong.
The confusion of the mental model with the notation is also a standard mistake.
Math is about learning how mathematical language is applied to problems. She is incorrect about how the math is applied. Ergo she is wrong. Her intuition is on the right track.
The only "whole" in this case is the entire set of students (or seats or whatever it is that is being talked about).
"One third of this and one third of that is two-sixths of everything" is absolutely right. Telling them "No, you're wrong" is counter-productive. "You have to look at the whole" isn't a helpful statement because, in this case, there are THREE 'wholes.'
Their written representation of the mental model was incorrect because their instruction was focused only on abstract numbers instead of concrete labels. If those fractions (or ratios or whatever) are labeled properly the equation is completely correct.