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On a math exam recently, the students were asked to use the definition of a limit of a sequence to prove that the sequence given by 3n/(3n+5) converges to 1. Given a positive number Ɛ, the definition requires proving the existence of some number N such that if n>N then |3n/(3n+5) - 1|<Ɛ.

As a consequence of the definition, once a sufficiently large N is found, any larger value of N will also suffice. Many students set |3n/(3n+5) - 1|=5/(3n+5)<Ɛ and solved for n to find N = (5-5Ɛ)/(3Ɛ). However, the professor decided to include an extra step: 5/(3n+5) < 5/n <Ɛ, which leads to another sufficient value N = 5/Ɛ.

Although most students gave a correct proof (consistent with the definition in their book), the lecturer took off points because they didn't find the "best" value of N. The lecturer claims that the author would have used some (unnecessary) inequalities to find the "better" N, which is probably true.

When students complain about losing points, I tell them that their answer is correct and that they should seek full credit for their work. The lecturer suggests that I am putting the students in a position in which they may "pick a side" and that ultimately the lecturer is in charge.

Who's wrong here?

Update: I was not notified about the lecturer's decision to remove points until after I gave the midterms back to the class. Once students started asking me about the missing points, the only written justification left by the lecturer was "not best N."

By "best N," the lecturer was referring to the N value found by using the additional inequality 5/(3n+5) < 5/n <Ɛ. By "best," he does not mean "smallest" (and by definition, there is no largest N).

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    There's a lot of confusion in the answers below: Please note that the lecturer's answer generates the larger N, as compared to the students. E.g.: for ε = 0.1, student N = 15, but lecturer N' = 50. – Daniel R. Collins Feb 18 '18 at 17:39
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    Is it possible that the lecturer didn't realize that his answer is not "best"? It gives a worse N than the students' N, and the proof is no more trivial than the students' proof. So what does he mean by "best"? Is there some criterion that I'm not understanding (perhaps not even imagining)? – Andreas Blass Feb 18 '18 at 21:43
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    123 says in his/her answer, "If students answer a question correctly [in mathematics] then they deserve full credit". I'm commenting here because this attitude seems to be implicit in other answers, and perhaps in the OP too. I strongly disagree with this opinion. If a correct but poorly written answer, excessively complicated, hard to read and full of irrelevancies, is given full marks purely because it is "correct", this is severely unfair to a student who has taken the trouble to find a simple argument and explain it clearly. – David Feb 19 '18 at 4:16
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    Let's see if I got this right. The professor graded the exams on his own? And you weren't involved in grading them? The two of you didn't meet to go over the official solution he was going to post? – aparente001 Feb 19 '18 at 5:12
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    Are you absolutely sure that the lecturer wasn't trying to claim that 5/(3Ɛ) was a slightly more elegant solution. I just cannot conceive of any mathematician claiming 5/Ɛ was best. – TheMathemagician Feb 19 '18 at 14:28

12 Answers 12

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Mathematics allows for objective truth. If students answer a question correctly then they deserve full credit. I do not think it is wrong for you to advocate for your students or for you to encourage them to advocate for themselves.

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    The whole point here is that there are multiple correct answers and the lecturer is insisting that students must give the "best" answer, without saying so in the question. An analogy would bethe question "Fire regulations require that there be at most 100 people in this room. Are the regulations satisfied?" The students have answered, "They're satisfied because there are fewer than 100 people in the room" and the lecturer is insisting that they say "They're satisfied because there are only 67 people in the room" for full credit. – David Richerby Feb 17 '18 at 13:33
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    realized latex does not work here* ... So, I will just say this: The question is about proving that a given sequence converges. Clearly, the students have done this by solving the first inequality. – 123 Feb 17 '18 at 15:31
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    This answer merely asserts an opinion. It offers no useful advice for what the OP can do in this situation. It offers no answer to the title question. – jpmc26 Feb 17 '18 at 23:30
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    @DikranMarsupial If the students really understand the topic, they know that if you need to prove that something holds "for all large enough n" it is almost never necessary to identify the exact meaning of "large enough". So this is not testing understanding. – David Richerby Feb 18 '18 at 2:09
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    @DikranMarsupial : "Do we really need to say 'show all working' for all questions we ask?" Yes! Until you teach them otherwise. I struggled hard for a week before mental anguish relented that "5 or -5" was a valid answer... that a math problem could have multiple answers. Maybe that is a valid assumption among higher math tiers, but college is a learning environment. Some students may not have gotten used to this yet. Furthermore, lots of students highly value grades (sometimes overly so, perhaps due to older baggage) so points off can be a very (overly?) harsh way to teach this lesson – TOOGAM Feb 18 '18 at 4:07
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The nature of the dispute makes this problem difficult.

As a mathematics (BS) and computer science (MS, PhD) student I have done numerous exercises that required proof of the existence of a natural number N such that for all n>N some inequality is true. In addition to limits in mathematics, they show up in computational complexity analysis of algorithms.

Every time I have done one of those exercises I have picked a value of N that made the proof as simple and clear as I could. Often, I was aware of a smaller value of N that would have required a longer proof. I have never been marked down for picking an unnecessarily large value of N.

Any finite value N, no matter how large, such that the inequality is provably true for all n>N is equally good. That is an important aspect of these definitions, something the students should understand and apply.

If smallness of N were going to be a grading factor, despite its irrelevance, it should have been announced in advance.

That said, it would have been better for the OP to discuss the matter privately with the professor, and perhaps with more senior professors. The OP should not encourage protests directly, but should state the professor's decision and recommend that follow-ups be forwarded directly to the professor or offer to forward them on the students' behalf.

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    Of course. The smallness of the constant can't matter if you care about what's going on in the infinity. In this case the professor on the next course will now wonder why are these students doing all the unnecessary steps to find smaller constant :D – Džuris Feb 17 '18 at 8:13
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    I fully agree with this. As somebody who works in this area, I've never come across a situation where it's important to find the exact cut-off for a "For all sufficiently large n"-type statement. And, even more generally, I've often been in a situation where I've just proven something strong enough for what I need, even though I know that something stronger is true. This lecturer seems to be teaching students to waste time coming up with unnecessarily precise results. – David Richerby Feb 17 '18 at 13:39
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    My understanding of the question is that the students picked a smaller value than the professor which works, but the justification requires extra inequalities (which maybe the students didn't explain?). – Kimball Feb 17 '18 at 14:22
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    I've +1ed you answer, but I think it would be better to not outsource the communication aspect to another answer that disagrees with yours. Even if much of the information is the same, the fact you disagree may mean some details will differ and will probably result in a somewhat different presentation of the available options. – jpmc26 Feb 17 '18 at 23:35
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    @Patricia: The professor didn't actually find a smaller N, but a larger one. There is a slightly larger one which simplifies the proof and only increases N a tiny bit, and the professor's change which makes N three times larger without any need. – gnasher729 Feb 18 '18 at 16:02
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Mathematically, you are clearly right. Any reasonable person should agree with you. The problem asked to prove that a limit holds, they proved it, period. "Find the optimal N for a given epsilon" has nothing to do with the question asked[0]. Since your professor doesn't agree with you, it makes me suspect he's not a reasonable person.

Having said that, it is still annoying for him if you "go against him" by telling the students to appeal the grade (appeal which they would win, if it is done honestly). Have you ever discussed this with him prior to you discussing it with the students? What did he say?

So why don't you propose to your professor a compromise? Ask him to change the question from "prove the limit" to "find the optimal N such that this inequality holds". Or "Once you prove the limit, give an estimate of smallest N such that the error is lower than epsilon. "

You can sort of add some context to the question to make it more sensible, for example by saying that f(n) is the percentage of criminals arrested as a function of the amount of money spent, and you want to get to a certain percentage.

In short, if he wants to ask a question about the optimality of N, make him ask that question, not an unrelated one.

[0] Personally, I would argue that it is actually harmful. Understanding that any finite intervals can be ignored and that we should focus on what happens for N arbitrarily large is a crucial point to understand convergence and limit at infinity. This obsession on the exact optimal N is harmful, because it gives the impression that it matters; it would be more beneficial to instead show how a complicated inequality, for example, can be simplified by simply considering N incredibly and unreasonably big. It doesn't matter, because we are only concerned about what happens at infinity.

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    "Not a reasonable person..." I agree with this. – 123 Feb 17 '18 at 15:32
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    While I agree with the beginning of your answer, I think you've misread the OP's description of the situation. As I understand it, what happened is actually the opposite of the scenario your suggestions address: the students calculated the optimal N, while the professor used a (somewhat arbitrary) simplification that yields a valid but non-optimal N, and then took points off the students for not using the same simplification. – Ilmari Karonen Feb 17 '18 at 22:36
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    @Ant, I wasn't notified of the scoring until students started asking me why they lost points. Since the only justification left by the grader was "find best N," I gave my honest opinion. – The Substitute Feb 17 '18 at 23:33
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    @TheSubstitute: With the information you gave, the professor's solution is actually not the smallest N by a long shot. – gnasher729 Feb 18 '18 at 16:05
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    @TheSubstitute Since you never said it, I'll say it now. In a problem of this sort, the smaller N is better than the larger one. The best proof might produce an unnecessarily large N, to keep the argument or the calculation simple, but that's not the best N. – Andreas Blass Feb 18 '18 at 21:38
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I think the only thing you may have done wrong is to send the students to the lecturer. That could be (but not necessarily) construed as undermining his authority, and TA's have to watch that carefully.

But I have always instructed my TA's to advocate for the students. I want the TA to come to me with my errors or any other problem they find. At least once per semester I begin a lecture with, "Mr. Johnson has informed me that....and so here is what we'll do... And I want you all to remember, when student evaluation time comes around, that Mr. Johnson advocated for you, at great personal risk to himself." Warm fuzzies all 'round.

Anyway, I think the way to handle such things is for you yourself to debate with the lecturer. If you lose the debate, you can tell the students that you agree with their complaint, but that you've talked to the lecturer about it and he's not changing his mind. You might inform them of the departmental avenues for grade appeal, but advise them that such a minor issue is probably not worth it.

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    I'd leave out the "at great personal risk" part. As a student, I'd feel nervous about asking a TA to advocate for me if I thought the might harm their prospects. – David Richerby Feb 17 '18 at 13:44
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    @DavidRicherby No. My students have a decent sense of humor (which seems to be missing from large swaths of academics. It's sad, really, that we've gotten to this point where we over-analyze every word in Cheka fashion.) – B. Goddard Feb 17 '18 at 13:53
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    Tone doesn't come across well in written text. If you'd mentioned that it was in a humorous tone, I wouldn't have said anything. – David Richerby Feb 17 '18 at 14:01
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    +1 as this answer focuses suggesting the TA and instructor discuss the issue since the grading of a particular problem seems to be causing confusion for several students. Perhaps the answer could be improved with adding some material @DikranMarsupial comment at top of page - (that is, there may be several things the professor is trying to accomplish at once with the grading rationale. Could be she is trying to encourage good students to go deeper and farther next time). Sure, the professor could be just being obnoxious. – Carol Feb 17 '18 at 18:22
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Personally, I think you are right; other people who have answered think you are in the wrong. Allow me to offer some additional advice about what to do now:

  • It's probably not worthwhile to escalate the situation further. Probably neither of you will change the other's mind.

  • You might meet with your graduate director, department chair, or other person with responsibility for supervising graduate teaching in your department. Ask them what you should do in the future, when the instructor makes a decision you feel is wrong and students complain to you about it.

    One possible consequence is that, in the future, you would be asked to TA under a different professor. Presumably this is a consequence which you would welcome.

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    And what can the students do about it? – ClassicEndingMusic Feb 17 '18 at 18:18
  • @ClassicEndingMusic Take the class with a different teacher if they're so inclined and the school provides that option. I had that happen to me as a TA, I worked with one professor who graded his students harder/assigned harder assignments than others and a lot of them dropped the class and took it again later with an easier teacher. – JAB Feb 17 '18 at 22:06
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    @JAB But the issue here is not about the class being harder, but the grading being unfair and arbitrary, and the instructor hiding behind their nominal authority rather than acting responsibly. It is really not a comparable situation to the one you describe. – Andrés E. Caicedo Feb 17 '18 at 22:45
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tl;dr- You're mostly right, but it'd probably be best to approach this diplomatically.

The basic question is whether it's appropriate for you to voice your disagreement with the instructor given your role as a TA. I'd argue that, in academia, it's entirely reasonable for you to express your disagreement; that academia isn't the place for subservient silence.


You're mostly right

It seems like we can fairly uncontroversially establish a bunch of stuff:

  1. Mathematically, you're right.

  2. This is mostly the course instructor's call to make.

  3. Students who disagree with the grading policy need to speak to the course instructor.

The controversial point would seem to be whether or not you're permitted to voice disagreement with the instructor's decision. Reasonable people may go either way on this issue.

In typical business contexts, employees are generally expected to avoid expressing disagreement with their higher-ups. In yet more authoritarian environments, e.g. in a military chain of command, such disagreement is actively punished.

However, one of academia's core tenants is academic freedom. It'd seem inappropriate to require an academic (like you) to not share their opinion on an academic matter (like an exam question) to students.

This can be approached diplomatically

When you share your personal opinion, you might express it as a personal perspective as an academic in the field. This would seem well within your rights.

Then, students might ask why, if you agree with them, you don't fix it. The simple answer is that you can't; that it's the instructor's decision, not yours.

Reasonably intelligent students will tend to understand that that means that they need to talk to the instructor without you explicitly directing them to do so.

Professional consequences

Be warned that your instructor or other job-selector may prefer to have unquestioning loyalty and may opt against giving you a position in the future, or write a weaker recommendation letter (if at all) if they're upset enough. Standing your ground on issues like this have inherent risks.

That said, personally, I've opted to do this in the past. When students have complained about a decision that I've disagreed with, I've bluntly told them that, yeah, the instructor's wrong, and that they'd need to take it up with the instructor since it's still their call to make.

  • How you frame the dispute will make all the difference: "You're wrong because xyz" is unlikely to succeed. "I'm having trouble explaining this to our students because I don't understand in the context of xyz (reasons why I think they should get credit)" is much more likely to succeed. From my experience in the military, I was never punished for a tactful question/suggestion, although my warnings were not always heeded and ultimately, as you say, the decision rests on someone else's shoulders. Dissent should be made to superiors in private; doing so in public is what gets you punished. – TemporalWolf Feb 19 '18 at 23:45
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You are getting two answers:

  1. The lecturer is your superior, he makes the decisions
  2. Mathematically you are correct

Since this is a course in mathematics, not in management, politics, or the military, he seems to me that clearly #2 is the correct answer, and that you are right.

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    It's a course in mathematics, but the question is about interpersonal dynamics within academia (else it would have been asked on a different stack). – Susan Feb 17 '18 at 19:41
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    -1 The mathematics (unfortunately) cannot solve an interpersonal dispute -- only people can -- and therefore this fails to answer the posted question. – Daniel R. Collins Feb 18 '18 at 0:00
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    The purpose of academia is to teach correctly, not to satisfy your own ego. I hope neither of you teaches, you are not good examples. – user Feb 18 '18 at 2:11
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    And yet, @user, you have failed to answer the question asked. The question wasn't Who is right?, the question was How do I resolve this dispute? This is an interpersonal question, not a mathematical question. – TRiG Feb 18 '18 at 18:38
  • @TRiG According to your logic, no-one answered the OP's question. That's a BS comment. – user Feb 19 '18 at 0:30
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When I first read this question, I was astonished by the requirement to find an "optimal" N to prove convergence as it shows lack of understanding what a limit is. In my class (I did TA work) a student would get full credit even for the factorial of the reference answer.

But then I noticed that I had misread the question. Actually, the professor's N is larger than the student's so it is definitely "non-optimal". But the answer 5/Ɛ is simpler to write and to use further if it was needed.

I think there is some pedagogical value in showing that you can weaken your statements to make calculations simpler. One can find such “unnecessary” (as OP calls them) steps in many real complicated proofs. How much this knowledge should cost to the students in question is up to their professor.

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I agree with many sentiments in comments/answers here, but---and I be misreading the question---my first guess from what you've said is that the students who lost points lost points for using inequalities that required justification in the professor's mind, not because they didn't use the same bound the professor did. Does this fit in with your situation? Deducting points for incomplete justification is of course reasonable for proofs, though where to draw the line is a judgement call, and one that is left up to the professor, though you may disagree.

In any case, if you're not sure why he took off points, then you should either ask him or direct the students to. You should never tell students to campaign for a different grading rubric.

  • How do you relate that interpretation to the third paragraph of the question? – Patricia Shanahan Feb 17 '18 at 14:39
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    @PatriciaShanahan Based on the 2nd paragraph (which indicates the professor used an extra inequality which won't give the smallest N), I think "best" here means easiest to prove, not smallest. – Kimball Feb 17 '18 at 15:07
  • I think the the students used $\frac{5}{3n+5} < \epsilon$ to find that $n > \frac{5}{3 \epsilon} - \frac{5}{3}$ which leads to $N = \ceil{\frac{5}{3 \epsilon} - \frac{5}{3}}$. Instead of the preferred solution of the professor: $\frac{5}{n} < \epsilon$ to get that $N = \ceil{\frac{5}{\epsilon}}$. So even though the first $N$ is correct as well, it is not the smallest. And that's why the professor seemed to have taken points off. – ClassicEndingMusic Feb 17 '18 at 18:28
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    @ClassicEndingMusic I'm a little confused what you mean by "the first N." The first one mentioned in your comment in smaller than the second one (corresponding to the professor's solution). – Kimball Feb 17 '18 at 21:02
  • @Kimball Yes, but to be honest I am not exactly sure either why the professor insisted on the second N. But anyways, I don't think that incomplete justification was the reason. – ClassicEndingMusic Feb 17 '18 at 21:45
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It's a bit difficult to answer your question because I don't find it completely clear what the point of contention is. But reading between the lines I think I can find two.

  • The lecturer says "ultimately the lecturer is in charge". He's dead right here. You are working under his supervision. You can discuss and disagree with his opinion, in fact you should do so (as long as it's feasible: perhaps not if there are 1000 students in the course and marks have to be absolutely definitely finalised by lunchtime). But ultimately it's his decision. If you're still unhappy with that decision - if you think it's mathematically and educationally wrong - then you could take the matter up with higher authority. But this is not something you should do lightly.
  • The lecturer says you are enabling students to "pick a side". He's dead wrong here. As long as you are giving the same advice to all students in this position, you are leaving all decisions with the lecturer - which is his job anyway. There are not two sides the students can choose between. It rather sounds here as if the lecturer is saying "you have to support what I say because I say so" - which is unscholarly, unprofessional and unmathematical.

You didn't actually ask what you should do, but in case you want my opinion - don't do anything about the first point, unless (as I said already) you feel strongly enough to take it higher. But I wouldn't recommend that. About the second, I would suggest you courteously point out to the lecturer that you are not suggesting to students that their marks should be altered, but are referring them to him to make the decision, as is his right. (And his duty - but it might be more tactful not to mention that.)

Also, keep a sense of perspective, and see if you can encourage students to do so too. I imagine this is probably a small part of the mark for a small part of a small assignment.

For the record, I have some sympathy with the lecturer's attitude (mathematical that is - I have no sympathy with his professional attitude). Mathematics, especially for advanced students (you didn't say what level this is) should not always marked as right or wrong and nothing else. That said, I doubt that I would have marked the assignments as he did in this particular case.

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The TA's answer is mathematically correct. However human society involves a hierarchy, based on the sole rule that the boss is always right.

There are other values of N (for example 6/epsilon) which also proofs the convergence. The only mistake in this context would be to proof it based on the fact that 1/n converges to zero. In that case, one can be accused of a circular proof.

The fact that the lecturer believes his/her approach is the only right one is an evidence of not understanding the topic (in my case, studied in the ninth grade).

My advice: bite the bullet and let the lecturer claim rightness. On long term, work for somebody you have something to learn from.

0

I thinks the outcome should depend on the exact question that was asked:

  • if the students were required only to provide proof, which they did, they should get full credit.
  • if the question mentioned that the "best" value of N had to be found, and defined what was considered best, the professor is free to take off points for answers which don't meet the criteria specified in the question.

It would be inappropriate to penalize students just because they didn't guess what the professor had in mind.

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