# What to do with "wild goose chase" or "quantum leap"-types of incorrect solutions when you ask students to prove/show something?

So in an advanced mathematics course for engineers, there are often problems of the type:

• Prove claim A.

• Given equation A, show that you can obtain equation Z.

I am frequently faced with a problem where students would write possibly large amount of derivations/calculations, essentially taking you on a wild goose chase and then at the end (no matter if what they wrote is correct, logical or even coherent), they will always write:

"We have proven claim A"

or "We obtained equation Z" (directly copy equation Z from the question itself)

The worst is that all steps are correct, but then a quantum leap happens (key steps skipped), and then the "We have solved the question." It always make you think if going from their final correct step to the conclusion is trivial, therefore it is myself who is unable to see such an obvious conclusion.

What do you do with this type of behavior?

• I give a 0 for this, usually. If there are 5 steps, and they have 3 right, but skip 2 (together), they might get some credit. But extra junk? Nope. Nov 18, 2023 at 14:29
• To avoid this problem, you can phrase the questions like (for example) "Show that the result is of the form $\ln{a}+\frac{\pi}{b}$ where $a$ and $b$ are integers to be determined." (The downside is that, if they cannot find $a$ and $b$, then they might not be able to do subsequent parts of the question.)
– Dan
Nov 18, 2023 at 22:45
• @SueVanHattum: I'll note that partial credit motivates wild goose answers, since students don't always how many of their wild goose steps are right. Jan 3 at 19:37
• As I said, I usually give a 0. If I think they get the idea of how one would prove a thing, but don't see it all, maybe a bit. Jan 3 at 21:00

It's quite likely to be a consequence of the belief that they have to answer the question. When they can't work it out, when they've gone around in circles and got lost, but still they have to give an answer, what are they supposed to do? They believe that at least giving something in the form of an answer is preferable to saying: "I don't know. I couldn't fill in this step. I went around in circles and got lost."

It means they haven't understood the purpose of doing exercises, which is not to get a high score, or a passing grade, or to 'get the answer', or to 'follow the rules'. It is to enable both you and them to identify gaps in their understanding, so you can go back and fix them. The goal of education is not to get good grades, but to learn a skill. If they understand this, then it should be obvious that the right response is to say you can't do it, identify the gap, and perhaps try to explain what it is about it that you find so confusing.

They should suffer no detriment from doing so. Only apply tests that they're going to be graded on once they have had a chance to fix their understanding as far as it can be fixed. With graded tests, the purpose then is to measure/demonstrate their ability having been educated (or not), and here they should simply skip any question they can't do, and not waste time. Here again, they need to understand that there is no point or benefit in pretending to be able to do something you can't actually do. You might get a pass, and even a job, but you'll get caught out as soon as somebody asks you to do it for real. And the consequences of getting caught lying/cheating are infinitely worse than they consequences of saying "I don't know. I can't do this."

Unfortunately, the belief that grades are the goal is an incredibly common misunderstanding, which the structure of school/university education often makes worse.

--

Very occasionally, you might get someone who picked up the lesson that because you sometimes perform leaps that they don't understand, that seem to have no rhyme or reason, but just jump to the conclusion, that this is how mathematics is done. Again, they need to be given the confidence and moral licence to say "I don't understand".

It used to drive me mad when students obviously didn't understand something, but nobody dared ask for fear of looking stupid in front of their peers, or making themselves unpopular by delaying the class escaping at the end of the lecture. There is no point in having a live person there doing the lecture, rather than watching a video, if you're not going to ask questions. Nevertheless, the attitude was nearly ubiquitous.

• +1 for this: "...you might get someone who picked up the lesson that because you sometimes perform leaps that they don't understand, that seem to have no rhyme or reason, but just jump to the conclusion, that this is how mathematics is done." I never considered that this might be the case for some students. Thank you, this is now something I can be alert about. Nov 23, 2023 at 19:24

The answer to "what to do?" is to discourage it as much as you can, arguing from the position of authority, if normal explanations of why it doesn't work fail (but certainly you should try the latter too).

The first step is to get rid of the idea of "partial credit" entirely or to reduce it to the minimum. In undergraduate courses I usually announce in the very beginning that every problem will be scored from 0 to 3, 3 being a full solution, 2 being a full solution with a minor glitch (like a stupid numerical error in an otherwise correct computation scheme or an erroneous sign flip near the end of the derivation), 1 being an essential progress in the right direction, 0 being everything else and that I'm the sole authority to identify all the adjectives here (full, minor, essential, right, etc.).

That may sound a bit harsh, but that allows me to shut down all the potential whining of the type "I wrote 5 pages of correct computations/derivations; don't I deserve some points for that?" from the beginning. Note that I never give a problem on a written test that would require more than half a page to explain the solution in full if one knows what he or she is doing and I tell students that, though occasionally long solutions are correct and will be given a full credit in such cases, going beyond 2 pages without any clear ending in sight should raise some red flags in their heads on an exam.

I fully realize that spending a huge amount of time going in a completely wrong direction is an essential part of every problem solving attempt both on an exam and in the real research. So, I believe that one needs to explain to the students that it is entirely normal and I would not introduce any punishments for that (especially if they are just trying to get that 1 point in my grading scheme). However, one needs also to make it clear to them that if somebody wants to get from New York to San-Francisco and goes 4000 miles south, that wouldn't count as progressing towards the goal regardless of the time, effort, or money spent on that part of the trip. As working mathematicians, ninety percent of the time (if not more) we do find ourselves exactly in the situation of that hapless traveler, and we often have to scrap it all and return to square one in such cases. The students should understand that they can hardly expect anything else from mathematical (and not only) problem solving beyond the most routine exercises where the answer or a solution algorithm is immediately clear from the problem formulation itself and get used to it if they want to succeed in the course or beyond it.

As to what to do in the cases of completely routine problems whose way of solving they should just know, I usually merely tell them that such things should become like a knee reflex and if they fail to recognize and follow the standard routine, they just demonstrate that they failed to learn the routine stuff, so they are getting nowhere in the subject and, unless they are able to compensate for that with ingenuity (i.e., to solve the problem completely by some alternative method), they don't really deserve a decent grade either.

Just my two cents :-).

• I'm not sure to which extent I can agree with various positions in this answer but, as often, I find your perspective quite refreshing. +1 Nov 25, 2023 at 23:29
• @JochenGlueck "to which extent I can agree with various positions in this answer" Just tell us what you do or think appropriate to do and the extent of our agreement/disagreement will become apparent to us both (and everyone else too :-) ) Nov 25, 2023 at 23:42
• I'm mainly referring to the first two paragraphs. But it's hard to be specific because I meant the words "I'm not sure" literally. I'm really not sure with which of your statements in these two paragraphs I (dis)agree to which extent. Arguments by authority make me uncomfortable and try to avoid them (but even that might not always be true; maybe I happen to just hide them). But then again I see your point why you use them. I think we look at things from quite different angles. That's why I'm not sure with which of your points I agree and that's also why I find your perspective refreshing. Nov 26, 2023 at 0:06
• I tend to prefer finer scales for "partial credit" and am less reluctant to grant such partial credit, but I try to follow the same overall idea that you describe: partial credit is only due if the "partial solution" is headed in the right direction. Nov 26, 2023 at 0:12
• @JochenGlueck Arguments from authority have to be used when another explanation will be just not understood (usually you'll see that quite fast when attempting it). I hate to say it, but being consistent usually convinces people more than being right (though the easiest way to achieve consistency is to say something that makes sense). A short consistent authoritative message has more chance to get through than an elaborate explanation adjusted to each case. As to finer partial credit, it is OK as long as you can clearly distinguish between n and n+1 and explain the difference to your students. Nov 26, 2023 at 0:22

I think the easiest way to deal with behavior like this is to tweak the grading method to disincentivize it (and of course have a conversation with the class where you explain the behavior, why it's bad, and why/how the grading is going to be tweaked).

For the specific situation that you describe, I would probably make a list of the key components that I want to see in the proof, and then grade using some kind of plus-minus balance for relevant vs irrelevant information in their proof.

For instance, if there are 3 key components, and they addressed 2 of those components, but also threw in 1 unnecessary component, then the amount of credit would be $$\dfrac{2 - 1}{3} = \dfrac{1}{3} = 33\%.$$

This generalizes naturally to weighting: if there are 2 minor components that they addressed and 1 major component that they did not, and they threw in 1 large unnecessary component, then the amount of credit would be $$\dfrac{0.5 + 0.5 - 1}{0.5 + 0.5 + 1} = \dfrac{0}{2} = 0\%.$$

And you can of course tweak bits of the calculation as desired to get more nuanced behavior:

• Maybe you have a ceiling of $$50\%$$ on the partial credit that can be awarded, so that if there are actually 5 equally-important components to the proof and the the student got 4 right, then they get $$50\%$$ instead of $$80\%.$$

• If the student also inserted 1 component of junk, then the "ceiling" approach could generalize to $$\min\left( \dfrac{4}{5}, 50\% \right) - \dfrac{1}{5} = 50\% - 20\% = 30\%.$$