# Thoughts on ad hoc answers to homework questions?

There's quite a few questions around (such as Show that 13 divides $2^{70}+3^{70}$) which ask a homework question which could correctly be answered in numerous ways. For the sake of discussion, I'd like to put them into two categories (of course, in reality, answers are not going to fit exactly in either category):

• ad hoc: where the techniques used in answering the question are likely to be unhelpful in resolving similar questions. These answers tend to be particularly slick and creative.
• pedagogical: utilising the techniques that one suspects the author of the homework question intended, written at a level chosen based on the original question. These answers tend to be unremarkable, but the techniques involved are likely to be useful to the student for subsequent questions.

To illustrate let's contrast two answers to the above referenced questions. Zarrax's answer would fit in the ad hoc category above:

...if $n$ is odd the polynomial $x + y$ divides $x^n + y^n$. So letting $x = 2^2, y = 3^2,$ and $n = 35$ you get that $13 = 2^2 + 3^2$ divides $2^{70} + 3^{70}$.

The issue I would like to raise here is that if the teacher of this subject changes the question slightly on e.g. an exam, then understanding this answer is likely not to be helpful. E.g. imagine if the teacher asked: "Prove that $2^{70}+3^{71}$ is not divisible by 13."

[To be fair to Zarrax, that answer is more of a comment-in-an-answer, and seems to be intended for the benefit of the whole community, rather than just the OP. (and it is a nice proof)]

Arturo Magidin's answer fits into the pedagogical category:

Compute $2^{70}$ and $3^{70}$ modulo $13$ separately (e.g., using Fermat's Little Theorem). If $2^{70}\equiv a\pmod{13}$ and $3^{70}\equiv b\pmod{13}$, then what is $2^{70}+3^{70}$ congruent to modulo 13?

Clearly this answer has the student as the primary focus. If the teacher then asks on an exam "Prove that $2^{70}+3^{71}$ is not divisible by 13" the student, after understanding Arturo Magidin's answer, should be able to do so.

I'm concerned that the focus of the community is too heavily on the side of the slick ad hoc answers.

• It might not be clear to the student which answers are intended for the overall community and which are intended primarily for the OP's benefit.
• There is a possibility that slick answers could inadvertently overwhelm a struggling student -- "I would never have thought of that answer; perhaps I'm not cut out for mathematics".

So with that I pass the discussion baton to the community:

Question: Is giving ad hoc answers to homework questions an issue? If so, does it need addressing?

I'd especially like to see feedback from students who utilise the forum to assist with their studies.

• Personally, I do not need what's there to discuss. Specific questions get specific answers. It benefits the community, as long there is no generalized question, as someone will find it useful sometime.
– Asaf Karagila Mod
May 8 '11 at 15:32
• "Arturo Magidin's answer fits into the pedagogical category" - I think his answers almost always are pedagogical... May 8 '11 at 16:45
• It is good to have both types of answers. The OP can then decide which answer to accept... May 8 '11 at 17:00
• Keep in mind that "ad-hoc" answers are not always "slick and creative". Indeed, far too often they are brute-force and nonconceptual (which can also hold true for what some call "pedagogical" answers). May 8 '11 at 17:51
• @Bill: I completely agree that the beauty lies in the generalized and abstract answer. For specific problems, though, that were stated in a very specific way - sometimes a specific and well aimed solution can be clean, elegant and as pedagogical as the general form.
– Asaf Karagila Mod
May 8 '11 at 18:19
• Or, shortening Asaf's comment: "use the right tool for the job" and "exploit the problem's inherent structure". May 8 '11 at 18:34
• So sometimes we see this. A question from basic calculus, followed by an answer using jet bundles. May 12 '11 at 13:50
• @GEdgar: Maybe you didn't intend your comment to be funny, but I had a good laugh reading it. :) I suppose people just like nuking mosquitoes sometimes... May 12 '11 at 14:35
• I disagree that the so-called "ad-hoc" answer is unpedagogical here. Depending on the context of students and background, it can be very pedagogical to liberate the students from the "have to calculate term 1 and then term 2 and then add them"-mindset to a "in a congruence I do some partial calculations, reduce mod m and look if it looks better"-mindset. Both approaches have a pedagogical value although the latter approach works much better in an interactive setting. Otherwise, you train students who solve $x^2-4$ with the quadratic formula (which is better than nothing, but still). May 12 '11 at 15:20
• So, I disagree with your categories and I strongly feel that for the particular example given it is valuable to have both (or more) answers. May 12 '11 at 15:22
• I realize I'm a little late here (I'm not a regular user of meta) but actually when I read the question I immediately thought the intention of the person who made the question was to use that $x + y$ was a factor of $x^n + y^n$ when $n$ is odd, and I actually was surprised everyone else started using Fermat's little theorem, etc. In other words I thought the asker was trying to create a question to be solved in this way. So what might seem ad hoc to one reader might not to another, and even to the question asker. Sep 6 '11 at 2:20
• I always distinguished them as tools and tricks. Aug 7 '15 at 2:09

People are free to post whatever (useful) answers they want. If you think an answer is suboptimal in some way, you demonstrate that by commenting and/or voting on it. There's no need for a policy decision here, if that's what you're asking.

• Yeah, I'm not asking for policy decisions here (I'm not even sure if this is worth addressing period). At the very most I'd suggest raising attention that pedagogy is a factor which might be taken into account when upvoting. May 9 '11 at 0:52

If you see an answer to a question which fits into your ah hoc category in a way that you think is suboptimal for the student, you can always post another answer, perhaps with a polite introduction explaining/contrasting/justifying the approach you are taking vs. the approaches taken in the exising answer(s).

• I like "ah hoc" as a fortuitious typo: a combination of "ad hoc" and "aha!" A good term for a slick trick that only works once. May 9 '11 at 20:25
• How do you know it only works once? May 12 '11 at 15:25

Both answers so far are basically saying the same thing: If you don't like it, do something about it. But Douglas isn't omnipresent. The question wasn't what he can do about it, but what we think about it, whether we view this as a problem; it was, as I understand it, mainly an attempt to raise awareness of this as a problem in the first place. I don't see a hard-and-fast rule for dealing with this, and I don't think Douglas was aiming for one, but I do think he pointed out a real issue and I think a bit more focus on what would really benefit the OP in cases like these couldn't hurt (explicitly including myself). One easy remedy might be to point out when you give an "ah hoc" answer that this is interesting from some perspective but there may be more generalizable/pedagogical/systematic answers.

By the way, I agree with the comment that Arturo's answers are almost always pedagogical. One might add "thorough". Certainly not ad hoc :-)

• "I think a bit more focus on what would really benefit the OP in cases like these couldn't hurt." - well said! May 12 '11 at 12:13