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I am taking an abstract algebra course and I am really interested about the topic. So much so that I spend most of my time reading supplementary materials. I consequently know a lot more theorems than the ones covered in class.

In a recent quiz, I used a theorem in one of my proofs that was not mentioned in the lecture since. My professor gave me partial marks for my answer for the reason that we didn't cover this theorem in class even though the proof was completely valid! I can't really see where he is coming from. How would someone be able to use a theorem correctly if he doesn't know its proof? I was quite baffled by his comment.

Do you agree with this?

EDIT: The professor just Emailed me and said that after thinking for a while, he decided to award me the full mark for the question. He also mentioned that he didn't want discourage me from studying the subject (since he saw most of my previous quiz grades were full marks) and I was passionate about it, but kindly requested that I include any proof for a theorem that I cite from now on in his exams. He also sent a broadcast message to the whole class indicating this.

  • Comments are not for extended discussion; this conversation has been moved to chat. – eykanal Dec 5 '16 at 18:52
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    Nit-picky comment: "How would someone be able to use a theorem correctly if he doesn't know its proof?" This is extremely common in mathematics. 'Assume the Riemann hypothesis is true...' – rhombidodecahedron Dec 6 '16 at 17:36
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This depends on what is announced in the syllabus and on the kind of test.

If the syllabus explicitly says that tests should be solved using the course material only, then, yes, any answer using anything else than course material is not totally correct and should get deductions. If the syllabus does not say so, the answer is not so clear anymore.

If the test is kind of a final exam on the whole course where the students are tested on the whole subject, than I (and this is a personal opinion) would not deduct anything if the proof is technically correct, whatever tools have been used (unless the problem says "solve this with this method"). However, in an intermediate test it may well be that the instructor wants the students to show that a certain technique can be applied or a certain concept can be used. Going beyond the course material spoils this idea and (again, personal opinion) one may deduct points, although it would be much better if these policy has been stated beforehand.

The above should answer the question "Why could points be deducted?" but not the "Should points be deducted?". I think the answer to the latter question is really opinion based (and may be a question for matheducators.stackexchange.com).

As a side remark on "How would someone be able to use a theorem correctly if he doesn't know its proof?" I (and I guess most other working mathematicians) do that all the time. I would not get anywhere without using theorems of which I haven't even dare to read to proof. Some examples from outside my field: People use Fermat's last theorem, the "Kepler conjecture" (also a theorem now), the four-color theorem, the "10,000 pages theorem" on the classification of finite simple groups, the Poincare conjecture (proven as special case of the geometrization theorem)…

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    Some people even use results that nobody knows how to prove. – T. Verron Dec 2 '16 at 15:51
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    Thanks for the answer. I didn't even know matheducators.stackexchange existed so thanks for the info. Also, I totally agree, if it was mentioned clearly before the exam, I would without a doubt have included the proof to avoid any problems. – Coconut Dec 2 '16 at 15:52
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    There's also the possibility that OP did something like (for example) using the fundamental theorem of arithmetic to prove Euclid's lemma (which is not only pointless but circular, because a typical proof of FTA uses Euclid's lemma). – Kevin Dec 3 '16 at 0:31
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    Sure, that's what I meant with "technically correct". – Dirk Dec 3 '16 at 1:38
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    "If the syllabus explicitly says that tests should be solved using the course material only" - this is a position that will kill any education system (it already has?) It's the "no child gets ahead" solution to the "no child left behind"... use it long enough and the next generation of teachers won't know anything else. – Adrian Colomitchi Dec 4 '16 at 22:38
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I think that this is expected behavior, and that most professors would score it the same way.

A first point is this: As Landric says in a comment, the point of an exam is to assess mastery of basic knowledge covered in the course. If one uses a more powerful outside theorem, then the steps that they've skipped likely contained important concepts or techniques, that there is now no proof that the student has mastered. So the professor needs to ping the student to demonstrate those basic techniques before progressing.

A second point would be: Consider this to be modeling working within a particular tradition or development. Many mathematical textbooks and papers may be using competing (even contradictory) definitions, axioms, assumptions, etc. It's important to use only those results which are developed directly from that chain of reasoning. So in a sense this grading protocol tests the "focus" of the student, if they are aware of exactly what results are supported/in effect in the "field" represented by the course.

A third point (related to the preceding) is this: This tests your ability as an explainer/writer/teacher of future students. At any time, it's important that we detect and "dial in" in our explanations to the level of abstraction/knowledge possessed by the other human(s) in the transaction. If we cite some result or theorem completely outside and beyond the awareness of the other person, then it fails to serve as an illuminating explanation. We have an obligation to be aware of what the other person has developed, and keep our explanations in that context, so that they have a chance to truly understand. Write assignments as though the intended audience was another student in the same course.

Here's a programming anecdote that serves as an analogy, from a friend a few weeks ago: Professor gives an assignment to code a hashing function, or a basic search algorithm. Students routinely turn in work that's a single line long, calling the equivalent pre-existing library function that does that job. Obviously that's not what's expected, fails to show the low-level understanding desired, and is pretty comical if you think about it.

Maybe your case doesn't seem so egregious, because you saved just a few steps. But if the professor doesn't score in this way, that would always be the end result; one-line proofs citing some outside source, and failing to show mastery of the basic techniques.

  • R.Collins Referring to your programming anecdote,if that student wrote that library before, or showed in some way that he is capable of writing it, why wouldn't you allow him to use it? Maybe I get this wrong but the point of a course for me is to learn new things. If I have learn the intended thing, and even read more stuff, why should you stop a student from using it. The student wouldn't have understood that theorem if he didn't understand the basics would he? I don't think any student would go on searching for additional theorems until he understands the basic ones don't you think? – Coconut Dec 2 '16 at 16:15
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    The exam or quiz is precisely the venue where the student proves that they are capable of using the basic techniques. Students look to outside sources to avoid doing basic work all the time. – Daniel R. Collins Dec 2 '16 at 16:30
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    @Coconut Courses are supposed to teach understanding, and mere citation does not demonstrate this, regardless of where the citation comes from. You say you've learned the intended thing. Great. How does your professor know this? Maybe you're in a study group, some other student found it, and now you all use it to shortcut half the proof. Graders wouldn't see this. The new requirement to include the proof is a very smooth move to elicit a demonstration of understanding. As far as the annoyance of writing five steps... it's a worthwhile trade-off for maintaining academic rigor. – Jeutnarg Dec 2 '16 at 16:56
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    @Coconut: if you prove the big theorem in your answer then you can use it. Unless the question explicitly demands a particular method that you don't use, of course, so watch for the difference between "hence prove..." and "hence, or otherwise, prove...". If it was just about what you've learned, examiners would have to accept "This part of the question is a lemma proved by Cauchy in 1843, QED". The point is not to learn what statements are true, it's to learn how to prove them. So pulling in a big theorem often is begging the question. – Steve Jessop Dec 3 '16 at 15:48
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    For that matter, suppose the question in complex analysis is "state and prove Cauchy's theorem for a triangle". Surely it's obvious that citing Cauchy's theorem for a closed curve is not a legitimate answer? Even without looking into the details of the course curriculum and the examination statutes of your institution, we can intuit that "know Cauchy's theorem" is not sufficient to meet the examination criteria. – Steve Jessop Dec 3 '16 at 15:51
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It depends. Consider these examples.

Suppose during an elementary calculus exam you a question asking you to prove that a given real polynomial of degree three has a root. A standard solution would be to use the intermediate value property. You could also use the fact that any real polynomial of odd degree has a root, and technically that would be a proof, but that would clearly not be a good solution -- the proof of that general fact is just a more abstract instance of the proof for a given polynomial of degree three. In fact, you could even use the general fact that every polynomial of degree three has a root. I hope you can see how that would be very far from being a solution.

For another example, suppose you had a class in elementary number theory, and you were asked to show that some diophantine equation has no solutions, but somehow you could reduce the equation to an instance of the Fermat's Last Theorem. Technically, you could just do the reduction and apply the theorem, but would that really be a good solution?

You could even think of a more extreme example: suppose you were asked to prove a given theorem during an exam (regardless of whether or not it was taught during the class in question). Would simply invoking the theorem and saying that the proof is completed be a good solution?

For a more subtle example, suppose you were to find the limit of sine of x over x at zero. You could try applying L'Hospital rule. This is not such an advanced theorem, but using it would still be wrong (as you need to know the derivative of sine to apply it, making the reasoning entirely circular).

The point is, at least during the introductory courses, you are mainly supposed to show how well you understand and apply basic concepts. Frequently, there may be an advanced theorem which would allow you to largely or completely avoid really using these elementary concepts, and you don't show that you understand them well. Moreover, this can allow you to do (possibly veiled) circular arguments.

On the other hand, if you use some more advanced concepts to circumvent a technical problem (or just to make a more beautiful proof), while still showing that you understand the underlying elementary ideas very well, you should not (in my opinion) be penalized. Similarly, if you use advanced ideas, but prove every step "from the ground up", I would say that penalizing you would be very wrong.

During more advanced courses as well as final exams, you are expected to have more broad knowledge and are much less likely to be penalized for using even somewhat advanced concepts. However, the same basic rules apply -- using the Fermat's Last Theorem would not be OK in my opinion, not even during an intermediate graduate number theory exam (unless explicitly allowed). It's mostly a matter of common sense.

  • Presumably using FLT could be implicitly allowed, if they ask a question that, you learned from the course, was an open question waiting for FLT? It's probably an error in the question: they should have said "show that FLT proves this". But hey ho, it's clear what they mean, so if you're fairly sure all known proofs use FLT then you have to go ahead. – Steve Jessop Dec 3 '16 at 15:58
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    The sin(x)/x example is very apt - I did this one myself, only trying to use the Taylor series expansion for sin(x)... in retrospect, not actually that convincing! – AJK Dec 4 '16 at 3:37
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    This is all hilarious and brings back memories. – Simply Beautiful Art Jan 2 '17 at 1:47
  • In addition to sin(x)/x where the circularity is subtle (and depends a bit on your definitions of sine and pi), there’s the limit of x^2/x at 0 where L’Hop is obviously and undoubtedly circular. – Noah Snyder Aug 31 '18 at 13:58
  • @NoahSnyder: Sure, but using L'Hospital for x^2/x is just silly, and the correct solution is obvious. On the other hand, as AJK wrote, trying to use it for sin(x)/x is (I think) pretty common mistake (and the correct solution much less obvious -- unless you define sine by power series, but that's already pretty circular in the context of first year calculus). – tomasz Aug 31 '18 at 15:35
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An important point to keep in mind here is that mathematical knowledge is largely contextual rather than being a collection of isolated facts that one learns in some arbitrary order. Consequently, when an exam question says "Prove assertion A", it is generally implicit that what this really means is "Prove assertion A in the context of the material discussed so far in the course" rather than "Prove assertion A, and you may use any result that ever appeared in the mathematical literature".

Note that in the former interpretation the question makes sense from the point of view of testing whether the student has learned not just why assertion A is true in some formal sense, but how this is relevant to the topic of the course and how it's related to the context in which assertion A is being discussed.

By contrast, the latter interpretation is highly problematic, since assertion A itself in all likelihood also appears in the mathematical literature, and it is obviously nonsensical to allow a proof that appeals to the result one is trying to prove. And if one does not allow that, should one allow a mild generalization of it? Should one allow a super-powerful theorem that implies it but that encapsulates thousands of pages of mathematical reasoning the student couldn't possibly have studied? And so on. So, in my mind, a student who assumes this interpretation will have demonstrated that they fail to grasp this important issue of context and/or are trying to get away with their ignorance of the specific ideas and techniques discussed in the course by appealing to some external knowledge they happen to possess. The fact that they possess such knowledge may be impressive and worth rewarding, or it may not be - that depends on the specifics of the case, so depending on those specifics I can see it being reasonable to deduct points in some cases, or being more sensible to award the full points for the question in other cases.

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A point that is frequently missed by students is that, in a course, in addition to the syllabus, there are frequently a number of implicit assumptions. Whether these implicit assumptions are significant or not, depends on the professor.

These assumptions are related, for example, to the methods that are to be employed to solve certain exercises, to which theorems the student is allowed to use, to what should be assumed in case of missing data, to which models should be employed for certain devices, etc.

Usually, students unconsciously learn these assumptions from the lectures and the exercise sessions.

Failing to comply with these implicit assumptions might result in a lower grade.

For instance, when I was a student of electronic engineering, one of the main courses during the first year was that of circuit theory. When I took the exam, one of the exercises was about the transient response of a first-order circuit. There is a standard method to solve these kind of circuits, but I decided to use the Laplace transform.

When the exam committee hand us back the papers, I found that they awarded me half marks on that problem, and there was the following comment: "The student doesn't know how to solve first-order circuits".

Of course I knew it! I simply decided to use another, correct, method, because I judged that in that particular case, for me, using Laplace transform was quicker. Thus, I complained with the committee: "This statement is false!". They looked at the paper and said: "You shoot a bird with a cannon, you deserve half marks". That's it, the implicit assumptions was: "you shouldn't use a method other than the standard one".

Given your last edit, it's nice that your professor decided to award you full marks, but for the next time, beware the implicit assumptions.

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    "... in a course, in addition to the syllabus, there are frequently a number of implicit assumptions...:" If a student should be penalized for using information that isn't explicitly taught within that course's syllabus, then (by your own reasoning) these "implicit assumptions" need to be spelled out in the course. You can't have it both ways. – Randall Stewart Dec 3 '16 at 3:29
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    I agree with this answer. Interesting to reflect on the fact that the professor's never had to deal with this issue prior in his career (i.e., the implicit assumptions were understood by all other students), and it sounds likely that this requirement will get added verbiage on his syllabus in the future (as preferred by Dirk above). – Daniel R. Collins Dec 3 '16 at 4:09
  • I went for double-kills on my AP physics I exam, writing both the standard method and more unorthodox methods whenever I could. By the time I told my teacher, he was almost crying, but it turns out I got a 5 on the exam. I have to wonder what the grader thought. – Simply Beautiful Art Jan 2 '17 at 1:57
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There is mention on this page of "implicit assumptions" that only material taught in a course should be used in the examinations for that course. But there are enough counter-opinions on this page to show that standard is not universally agreed upon. There are equal reasons to assume that any mathematically valid methods are fair game. Perhaps the student had experiences in the past where he or she was rewarded for thinking outside of the box or for studying more than they had to. Is that really an outlandish possibility? Is that really such a bad thing? How would they know what this particular professor finds acceptable?

Think about the classic story of Gauss adding the numbers 1 through 100. Has that story EVER been recounted in a way that suggested he should have been penalized for not using the laborious method expected of him?

If a professor wants a student to use a particular method for a proof, then that needs to be made explicit. Students can't be mind readers. (And even if they were, they couldn't use that information anyway unless mind reading was explicitly taught in the course.)

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    I feel like you conflate "a method of proof not taught in the course" with "invoking (without proof) a theorem not covered in the course". The student appealed to a certain theorem that was not covered in the course and did not mention the proof. I think we all agree that if the student had given the proof of the theorem, he would have been golden. In fact that's what his instructor told him to do for the future (while being generous in response to an isolated incident). Others have explained well why "You can use any theorem from the literature in an exam" is certainly not tenable. – Pete L. Clark Dec 3 '16 at 7:31
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    If Gauss had written down (in German) "the nth triangular number is equal to n(n+1)/2, so the solution is 5050" then possibly he should have been penalised under exam conditions. Certainly he should be if he'd written down "I happen to know the answer: it's 5050". Either the formula or the specific result for n=100 could be cited from the literature, but the question is not "cite the result from literature", so even if he'd cited correctly he still fails. – Steve Jessop Dec 3 '16 at 16:06
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    However, if he applies a particular method in the special case of n=100 (which I think is how the story's usually told: he offered that 1 + 2 + ... + 99 + 100 and 100 + 99 + ... + 2 + 1 can be summed in a clever way), then we assign credit according to whether or not he justifies his method. What he can't do, under exam conditions, is use citation in place of proof. – Steve Jessop Dec 3 '16 at 16:11
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    @RobertColumbia: Apparently then it is a social skill, since you can fail it if you lack sufficient observation and analysis to understand what an examination question is asking you to do. If it helps to understand why this is, one can distinguish "doing mathematics" from "convincing the examiners that you meet the criteria to pass". IMO this question is about the latter. – Steve Jessop Dec 4 '16 at 13:34
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    There's actually quite a rich field of thought that mathematics is, in fact, a social explaining-convincing skill. Consider this quote from Cantor Medal winner Yuri Manin: "A proof only becomes a proof after the social act of 'accepting it as a proof'." Quoted here: rjlipton.wordpress.com/2010/08/09/… Or this blog post by Dick Lipton: rjlipton.wordpress.com/2013/07/14/surely-you-are-joking – Daniel R. Collins Dec 4 '16 at 19:07
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Penalized? No. But it's reasonable that you be required to prove or solve problems using the material you've studied rather than a "5 kg hammer" that you found lying around in some book.

Remember the professor wants to help you familiarize yourself with the subject of that course; the exam is just a means to that end - it's not supposed to be a test of how good/intelligent/knowledgeable you are in general.

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No. You shouldn't be penalized. The professor should applaud your understanding of the subject. That said, in school you should make a habit of answering questions the way a teacher wants them to be answered. Otherwise you'll have to deal with issues like this.

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