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I'm teaching a course at a community college called "Business Pre-Calculus", which is basically some varied rudimentary algebraic concepts integrated with some topics from basic economics and finance.

I give in-class exams for which calculators are provided and the use of a phone is prohibited. On the last exam, a student who had conceptual difficulty adding fractions, on the same test was able to solve a 3x3 linear system, a complicated rational equation, and an even more complicated rational inequality. His work in the space provided is mimnimal and makes absolutely no sense (I've already considered the "savant" possibility), and I cannot fathom how he could have come up with the correct solutions to each with any of these ridiculous "calculations". They resemble what you might see on a chalkboard in a cartoon: nonsense made to look like math to an ignorant observer. In all honesty, I feel like he gets the answers somehow, and then scribbles some "work" to avoid suspicion. But his answers are still correct, and seemingly out of nowhere.

I've watched him take the test, but didn't see anything suspect. What am I missing?

Thanks.

Update: I've removed the picture.

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    $\begingroup$ I will say, from over a decade of experience at a community college, that it's fairly common for some students to be skilled enough to pass elementary algebra, college algebra, precalculus, and starting calculus, and still not be able to perform operations on fractions. $\endgroup$ – Daniel R. Collins Mar 19 '18 at 10:38
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    $\begingroup$ Personally, unless I catch a student unambiguously cheating, I don't like to toss about that accusation. In this case, it is possible (likely, even) that the student was cheating, but it doesn't really help to levy that accusation, since the student so clearly deserves little or no credit for their work---they have written utter gibberish. $\endgroup$ – Xander Henderson Mar 19 '18 at 16:16
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    $\begingroup$ On my tests I often included something like "appropriate work must be shown to receive full credit". Then, when a solution like what you have shows up, I'd circle the expression (4x+8)/(2x-2) (has an arrow pointing to it) and write "not equal to #" (# is the original difference), I'd comment about the (+)(-) and (+)(-) ("These are not different sign possibilities"), I'd circle "at x = -3" ("Where does -3 come from? There is no 3 in anything you've written before"), and other things. I'd probably give this a 2 out of 10 points (I'm generous), and make a photocopy of it. $\endgroup$ – Dave L Renfro Mar 19 '18 at 17:42
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    $\begingroup$ You said that calculators were provided. What kind of calculator was this? $\endgroup$ – Nick C Mar 20 '18 at 2:08
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    $\begingroup$ I think @NickC's question is an important one. If it was a graphing calculator, then there are lots of possibilities here. $\endgroup$ – mweiss Mar 20 '18 at 5:03
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First off, I challenge the framing of the question. You seem to be seeking answers for the question

How did the student get the correct answer from this work?

Unfortunately, I don't think that there is any possible way to answer that question without being the student. Maybe they cheated. Maybe they got lucky. Maybe they did some work in their heads but were unable to communicate that work in writing. Maybe they saw that exact problem in a homework assignment or study guide somewhere and somehow managed to memorize a correct answer. Without being in the head of the student, it is entirely impossible to know how they got that answer.

Instead, I think that there is a related question which can be answered:

If a student writes gibberish followed by a correct answer, what should I do?

In the classes that I teach, I strongly emphasize that most of mathematics is not about obtaining the correct answer, but about communicating to others how that answer was obtained. If you cannot communicate how you arrived at an answer, then you haven't done the basic work necessary for conducting mathematics.

In an exam setting, I make it very clear (both on the syllabus and in the instructions for the exam) that correct answers without supporting work will receive no credit. In this case, I don't see any work that I would be willing to give credit for (though I might be a harsh grader; Dave L Renfro sees some merit in a bit of what is there, and he isn't wrong; I do see an attempt to combine the terms with a common denominator—that is a reasonable start; that being said, I still probably wouldn't give it any credit). I don't need to know how the student got the correct answer in order to know that they haven't explained that answer.

That being said, if the student came to my office to challenge the marking, I would be willing to hear them out. If they could explain to me how to solve a similar problem that present to them on the fly, I would be willing to give them a handful of points on the exam (say 2-4 out of 10), with the admonition that, in the future, I won't be so generous (it is quite difficult to offer the option to revise a final, since grades are due pretty quickly after the final is graded). I might even encourage this student to come and talk to me; not even under the "guise" of helping them—they clearly need help, whether it be help with their mathematical skills, or help with their cheating technique. ;)

I will also point out that I am very carefully avoiding the issue of cheating entirely. Yes, it is very possible that the student cheated. But if you don't have any evidence of that (other than some suspicious answers on an exam), accusing the student of cheating does absolutely nothing to resolve the situation. It is probably better to assume good faith and let the student prove in office hours that they have no idea what they are doing. Whether they are cheating or not, they aren't going to get credit for something that they don't understand.

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Do you give full credit for answers without clear work?? If you don't then I suggest you give partial credit and meet with the student in the guise of helping him write his work more clearly so that he can get full credit next time and possibly even a little more credit. Put the responsibility on him to explain how he got his answer and help him write what he did clearly. If he can't explain what he did, then explain that this is why he can't have any additional credit. IF he is interested, go over the method with him. Find out if he has trouble writing down his thinking in general.

In my years as a teacher, I have seen students who can do all sorts of things in their heads and have no clue how to write it down. These students didn't cheat, just learned and worked differently. All their thinking was done in their heads and trying to write things down interfered with their thinking. If you required that they wrote something down it rarely matched up with what they did in their head.

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  • $\begingroup$ I give partial credit when a student correctly does a portion of the problem correctly but not another portion. This is not my issue with his solution. My issue is that without at least mentioning the potential endpoints of his solution intervals, he was able to derive them exactly. At no point does he correctly express (let alone factor!) the numerator once zero is isolated, but he somehow knew what points turn the top into zero. And knew which intervals made the inequality true. Further, he never even subtracts 1 from both sides, however attempts to test if the left... $\endgroup$ – bloomers Mar 20 '18 at 15:00
  • $\begingroup$ ... is positive or negative, which makes no sense. All in all, if he had these calculations and arrived at a nonsense solution, I would have awarded him some partial credit. But the fact that he was able to give the correct solution, combined with the fact that this is not the kind of problem to which one can guess the answer, lead me to believe that something else is going on here. $\endgroup$ – bloomers Mar 20 '18 at 15:04
  • $\begingroup$ So does meeting to find out his thinking as I suggested a workable solution? $\endgroup$ – Amy B Mar 21 '18 at 14:41
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    $\begingroup$ I've decided that the best course of action is to give no credit or very limited partial credit to this problem and the five others on the exam that he answered in a similar fashion. If he has an issue with this, I'll meet with him and tell him that I cannot follow his work or how he arrived at the answer, and if he is able to explain his process to me and/or execute a similar problem, then I'll be more than happy to change his grade. But you and the other contributors were correct in saying that I can't come at him in an accusatory way because I don't have any definitive proof that he cheated $\endgroup$ – bloomers Mar 21 '18 at 16:08
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From the picture you gave, it seems like he wasn't cheating. The whole solution, including how he got there, is there, even though in a bad style and wrong in some places:

He started by getting the left hand side on one fraction by multiplying the first one by $$\frac{x+2}{x+2}$$ and the second one by $$\frac{x-1}{x-1}.$$ This is the first line of the calculations block on the right. In the next line, we get this, namely $$\frac{4x+8}{(x-1)(x+2)} - \frac{2x-2}{(x-1)(x+2)}.$$ Now we have an arrow pointing to the left in his solution, towards $$\frac{4x+8}{2x+2}.$$ I don't see how he got that, maybe he did a mistake substracting the fractions, maybe it has something to do with the unreadable word he put there.
Now, he tries to test where this expression is positive or negative to get to the solution. I also don't see how he got $-1$ and $3$ here, but he surely had a reason...

All in all, I think he knew what he was doing. Of course there are some errors and the solution is almost unreadable, but still, I would not think that this student was cheating. I would think it is either that he thinks himself to clever to write such trivialities down, or he doesn't know how to properly document his work. Thus, I would suggest to talk to him, find out what is the case and then explain to him that he needs to properly write down things; both for you to understand it, and for him to avoid doing mistakes.
If possible you should teach him resp. your whole class that writing things down properly, even in seemingly easy cases, helps a lot with avoiding and tracking down calculation errors. And yes, such errors happen, to everyone.
Best example I can think of right now is the Gaussian algorithm to solve systems of linear equations. It might seem boring and students always want to be clever and do a really intelligent trick to solve the system faster, but in the end, just following the algorithm strongly reduces the number of errors due to slip of the pen.

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  • $\begingroup$ The "unreadable word" is "reduces to", and explains that the next step after (4x+8)/(2x+2) is (2x+4)/(x²-1), which is another error, to be followed by yet another error. $\endgroup$ – Jasper Mar 20 '18 at 10:22
  • $\begingroup$ I do follow his logic up through that part. The problem I have with his work is that the way that I taught them to solve this type of problem is to first isolate zero, then find all values that render either the numerator or denominator to be zero (in this case, -3, ,-2, 1, and 4), and finally test points from each interval (in this case, five points). If he found out another way to do it, that would be great. But I don't see that here. And if these calculations led to an incorrect answer, that would be fine as well. But I don't see how he was able to pinpoint exactly on which intervals... $\endgroup$ – bloomers Mar 20 '18 at 14:49
  • $\begingroup$ ... this relationship holds when at no point does he even figure out what values to test around. It looks to me like he's testing what happens AT -3 and 1, when both of those values render this expression undefined. Additionally, he never mentions -2 or 4 but somehow has them as endpoints in his final answer. $\endgroup$ – bloomers Mar 20 '18 at 14:54
  • $\begingroup$ ... AT -2 and 1, when both...* $\endgroup$ – bloomers Mar 21 '18 at 19:39

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