I want to share a theory on this mistake:

The student had an association between negative exponents and reciprocals and “half-powers” and square roots. As the student was parsing the problem he “fulfilled his obligation” to use that association on the number. I guess what I’m positing is that the mind works by making a connection, and then remaining in tension until that connection is used in a problem. (I’ve often had the experience of feeling as if there’s an insight that I haven’t used yet in solving a problem, and it’s like having a small weight on my back.) The mind comes to relief at the moment that the insight is used.

The student’s mind made the connection between negative powers and reciprocals and was in tension. He then used this insight at the first opportunity he saw, to relieve himself from the burden of his insight.

Some of you might disagree. For instance, you might think that the student had just memorized some rule poorly, had no conceptual understanding of powers, and gave the answer that he did.

But I think that the answer felt right because he used the fact that he knew. I’d predict that this student would be able to answer $x^{1/2}$ correctly.

If you think that the student just memorized a rule, is there any reason to think that a student would get a question such as $x^{1/2}$ correct?

• I think the student does apply a rule correctly but doesn’t see the differences between doing so with a coefficient vs. a variable. I agree with your assessment about this student probably correctly answering x^1/2 – even more interesting to test my theory would be to see what happens with (4x^2*y^4)^1/2.

• How would you expect a student to reply to $x^{1 / 2}$ ? You can’t do anything with it. Are you anticipating $\sqrt {x}$ ? Would you take $x^{0.5}$ ? It’s not a fraction ( yes, I understand I have been working with teenagers too long…)
I think this student knows how to find the powers of numbers (possibly by plugging them in to a calculator) but has no idea what to do with a letter. So it just gets left sad and alone. I agree with Evan – what does the student do with higher powers of variables?
I’m also intrigued by the notation in purple, presumably from the grader.

• I think the student would answer $\sqrt{x}$. The student knows that there’s an association between fractional exponents and roots, and he has no other outlet for that connection other than writing the correct answer.