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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?

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Totally not an answer (sorry, I couldn't resist) but there's also a well known joke where the student circles $x$ because the question was to "find $x$". –  Joel Cohen Jun 29 '11 at 22:33
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I've converted the question to community wiki, since this is asking for a big list of examples and there's no one right answer. –  Zev Chonoles Jun 29 '11 at 22:50
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Joel, I knew a guy in high school who, when asked on a physics test to "name the force acting on this object", named it "Bob". –  Michael Lugo Jun 29 '11 at 23:26
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@Michael: Was it a buoyancy force? :) –  Rahul Jun 29 '11 at 23:34
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2 Answers

Fermat's teacher asked the class to show that $n^x+n^y=n^z$ has no natural number solutions with $n >2$. Unfortunately, the young Pierre misread the question, and $\:\dots$

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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.

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I don't think this is the kind of answer the thread asks for. You gave an example of a bad (but correct) solution of a problem, rather than a plausible misinterpretation of a problem statement. There are plenty examples of the former. Yours fits the probably most common format: when asked to verify that a given value is a solution to a given equation, the student solves the equation rather than just plugging the value into the equation. –  Samuel Jun 29 '11 at 23:07
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