What enlightening misunderstandings of test questions have you encountered? Of course, there are jokes about this, such as literal (graphic) “expansion” of the power of a binomial, but I am asking about real situations where the student misunderstood a test question in a way that is plausible, but which was unanticipated by the teacher.
From time to time I encounter one of the following type: The student interprets “graph $|2x – 1| \leq 3$ on the number line” to mean “graph $y = |2x – 1|$ for $y \leq 3$”.
This is very enlightening, showing me that the question is ambiguous, and needs to be more tightly phrased. (But since I am not the one who creates the test, this will take a while to happen.) You might object that the phrase “on the number line” is a pretty big hint, but I don’t agree. Such a phrase does not faze a student under time pressure. Moreover, the textbook that we are using also uses the expression “on the number plane” for the 2-dimensional case, which, to a student under time pressure, can easily be the mistaken interpretation for the other phrase. Part of the situation, of course, is that in spite of the hand full of one-dimensional graphs that are done, “graphing” is very strongly, even exclusively, associated in the minds of many students with the 2-dimensional case, the idea: “If you don’t see a wiggly line in the plane, you don’t have a graph.” This lets me know that at least once I ought to mention the caution to the students: “Look at what these examples are telling us: You don’t have to have a wiggly line in the plane to have a graph!”
So, I’m sure there are a lot of other such examples out there. What are they?
This may not be exactly what you mean, but its an example of where I got unexpected answers. The problem gave two (relatively simple) functions and asked students to verify that they were inverses. I was hoping the students would show that composing the two functions (in both orders) resulted in $x$. However, about half of the students in the class used the procedure for finding the inverse of a function (which was also covered on this test) on each function, showing in each situation that they arrived at the other function. As there is nothing wrong with this method, I accepted their answers. I'm not sure how I could have reworded the question without giving too big of a hint.
I'd be interested in reading others' accounts of unexpected answers.