Should a student be penalized for using a theorem outside of the curriculum?

2018-03-13 17:08:02

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

  • 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.

    2018-03-13 17:13:37
  • 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

    2018-03-13 17:21:42
  • 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, bu

    2018-03-13 17:26:49
  • 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 a

    2018-03-13 17:30:16
  • 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

    2018-03-13 17:46:36
  • 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

    2018-03-13 17:47:46
  • 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.

    2018-03-13 18:07:03
  • 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.

    2018-03-13 18:21:25