How to resolve grading dispute between TA and Lecturer?

Question

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).

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

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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.

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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.

Answer 11

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.

Answer 12

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.