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I'm currently teaching "Real Analysis I". I took a midterm exam and one of the questions was to show that there is a unique positive number $r\in\Bbb{R}$ so that $r^2=17$. One of the students tried to answer this question with a similar method that we used in the class for a similar question about the existence of a unique positive number $r\in\Bbb{R}$ so that $r^2=2$. It's based on "Completeness Axiom" and a set $S$ defined as $S=\{x\in\Bbb{R}:x\ge0 \quad and\quad x^2<17\}$, then trying to show that $(\sup{S})^2=17$. Now this specific student wrote a correct reasoning about $S$ and how it is true that $(\sup{S})^2=17$. But the problem is he didn't defined $S$ in his/her answer! What's your opinion? How should I mark this answer?

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    $\begingroup$ Consider sending an email to the student inviting them to your office. Present them with the same problem (without their writeup). Ask them to show you what they did. If they can demonstrate their knowledge in front of you, factor that in to their grade. If they cannot recall, consider showing them their exam and asking them to clarify what they meant. (You might consider awarding less credit if this is required.) $\endgroup$ – Brendan W. Sullivan Dec 3 '14 at 19:22
  • $\begingroup$ @brendansullivan07, Thank you! I'll try that. What is your opinion about a memorized answer in general? How should we grade such an answer? $\endgroup$ – Woria Dec 3 '14 at 19:30
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    $\begingroup$ Did the answer have any mention of $17$ at all? If the answer looks perfect except that the definition of $S$ is omitted, then I would only mark it off very slightly. But I'm not so sure what is meant by your "realizing s/he just memorized the answer" or that the student "tried to answer this question with a similar method." $\endgroup$ – Benjamin Dickman Dec 3 '14 at 19:42
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    $\begingroup$ @BenjaminDickman, The answer to the question of the existence of $\sqrt{2}$ was such that if we write $17$ instead of $2$ it ends up with the existence of $\sqrt{17}$. About "memorized answer", well, since that student didn't define $S$ and then wrote a lot of reasoning about it, I thought maybe he/she didn't understand the argument at all, he/she jut picked a picture of the solution of a solved problem in his/her mind and then at the exam he/she pasted a corrupted version of it! $\endgroup$ – Woria Dec 3 '14 at 20:15
  • $\begingroup$ A possibility is that the student had it all right on a draft, and forgot to copy the first part on the copy. More importantly, in the future I would rather suggest asking questions that are less prone to direct regurgitation of memorized answer; that will both measure better what the exam is meant to measure, and (in the long run) give student incentives to try to understand the proofs rather than learning them by heart. $\endgroup$ – Benoît Kloeckner Dec 23 '14 at 21:29
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In my opinion, for the purpose of grading, you should not try to guess or otherwise ascertain the student's reason for writing an incorrect answer.

I think you should grade the answer as you would grade any other answer such that

  1. there is a crucial piece missing, and
  2. you can fill in the piece, but someone reading the question and answer in isolation would not be able to fill it in routinely.

In my case, I give such answers low scores.

The student can then draw his or her own conclusions about the advisability of attempting to regurgitate memorized answers.

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Are you completely sure they memorized the answer, or might they understand the original proof just forgot to include the definition of the set? Either way, there is some benefit for them memorizing the solution as I've found that some learning has happened, and they did put in enough effort to memorize it. Anecdotally, I have found proofs that have stumped me at first, but I know I should know their steps. I find that I almost always understand what occurs when I take another look later, either the same semester or even years later since I was more than likely lacking familiarity with the subject that time helps with. Sometimes it is more worthwhile to remember the proof in order to pass to other topics and re-approach them later.

What I would do is deduct the appropriate number of points you think that not including the set definition is worth. It is critical that they include it, so some deduction might be necessary.

If you think they just forgot to include it, inviting them into your office as brendansullivan07 suggested to see if they deserve the full points might be a good option, as long as you would do it for other students as well. I would use this method more of a "I think you have the right idea, but just want to check" rather than trying to catch they memorizing.

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    $\begingroup$ Students may not know which uses of symbols are so standard that they are understood without defining, and which are not. This may be one reason for "forgetting" to include the set. $\endgroup$ – Tommi Brander Dec 4 '14 at 15:00
  • $\begingroup$ @TommiBrander It's often important to keep that in mind, but perhaps not here. Even if the definition of $S$ from class were global, the student's answer would still be wrong because the '$2$' in the definition of $S$ would need to be replaced with a '$17$'. $\endgroup$ – Trevor Wilson Dec 18 '14 at 3:50

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