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A student benefits from their attempt at a solution or proof being checked by the teacher. My own view is that, if the student's work is poor, it is best just to provide a model solution or proof in its place. In general, I think that bad work is best ignored and forgotten. The value of the failed attempt is that it has focused the student somewhat on the issues; he or she is then better prepared to understand and appreciate the good solution or proof.

An alternative view is that it is better to start from what the student has done: correcting and adding minimally so that the goal is eventually reached. Presumably the idea is to give encouragement by allowing the student to believe that they mostly got there and only needed a bit of help.

I don't like the alternative view because, even when the amended result is, in a strict sense, free of fallacy, the student is typically still left with a clumsy and badly written piece of work.

What I am seeking here is an argument for the opposing view and against my view, which I admit may be wrong.

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    $\begingroup$ I think this greatly depends on factors such as the goal of the course (which is more important, learn subject matter or learn how to write proofs) and the number of students and how much time you can devote to each student. If learning how to write proofs is important, then I think it would be best to do this the way one encounters in various humanities and history and social science courses where you have to write term papers -- rough drafts are done by the student, which are steadily improved by teacher comments. But for many math courses there isn't time to give this much attention. $\endgroup$ Jan 11, 2023 at 22:23
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    $\begingroup$ Have you thought about working with the student to help the student produce an acceptable solution? $\endgroup$ Jan 12, 2023 at 1:02
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    $\begingroup$ Perhaps a little of each? If you are asking about proofs: Cross off anything awful. If they have made a correct first step, hint at the next. Then provide a model proof for the whole class. Also, perhaps have students work in pairs? $\endgroup$
    – Sue VanHattum
    Jan 12, 2023 at 4:55
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    $\begingroup$ I don’t understand why one would need to choose between the two options; if one helps a student fix their own work first, wouldn’t comparing it to a model solution afterwards be an invaluable learning experience that neither option on its own provides? You could explain the differences to them and show them why a different approach is better/more elegant/more concise. $\endgroup$
    – 11684
    Jan 12, 2023 at 15:25
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    $\begingroup$ I haven't met a person yet that enjoys their time and effort being discarded and ignored. $\endgroup$
    – David S
    Jan 13, 2023 at 19:18

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As Besicovich once said, the reputation of a mathematician is determined by the number of their badly written papers (the pioneering work is often clumsy and follows the route far from the shortest one). When one really tries to solve a problem, the thinking process goes back and forth between several options hitting dead ends in many places until one gets some path from A to B, so the final solution may resemble a trip from Cleveland to Detroit through San-Francisco, Sidney, Beijing, and Paris (not necessarily in this order). However, one can often pick some useful stuff and see interesting things on that way as opposed to driving for 4 hours on I-80/75.

So, showing the students how they can tie loose ends and bridge the gaps in their arguments has its merits, provided that what they try is something leading in the right direction or, at least, going somewhere by some non-zero distance. Then they can learn to go further and further in their attempts until they are able to design a correct full proof (though not necessarily the slickest one). Never try to repair total gibberish though. If some step is obviously formally wrong or if what is written is a non-statement (or even a non-sentence), weed those things out ruthlessly to start with, even if they can be turned into something useful by a moderate amount of modifications.

In general, one usually derives much more satisfaction and confidence from seeing how his/her own approach can be completed than from seeing the shortest and the most elegant approach possible. The latter often just leaves one with the feeling that he/she is a total idiot and can do nothing (I'm not saying that this feeling is far from the truth: very few of us will have a reason to rate themselves much higher when compared to Archimedes, say, and that guy lived over 2000 years ago. I'm just saying that cultivating it too much is counterproductive to the learning process).

You'll have enough opportunities to present the good solutions when going over the problems in class. But when working with a student one on one or when letting them go to the board, I usually just allow them follow their own path, even if it is suboptimal, as long as what they are saying makes sense.

I hope what I wrote is a bit more meaningful than what GPT3 would write in response to this question, but I leave it to the other people to judge if it is really the case :-)

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  • $\begingroup$ "If some step is obviously formally wrong... weed those things out ruthlessly to start with, even if they can be turned into something useful" - I don't get this. If it's obvious (to you) that some step is formally wrong, but the student thought it was right, wouldn't it be best to point out where and why it's wrong? (From there, it's often a negligible step to turn it into something useful, whether you choose to take that step or not.) Total gibberish is one thing, but using your own judgement of "obvious" as a criteria for whether to address something is quite another. $\endgroup$
    – NotThatGuy
    Jan 13, 2023 at 11:36
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    $\begingroup$ @NotThatGuy We don't have a real disagreement here. It is just a question of language. "Weeding out" means for me exactly explaining that the step is wrong and where and why it is wrong and stopping there, as opposed to saying something like "you have a great idea here only $a$ should be $b$, plus should be minus and "implies" should be "cannot hold simultaneously with". As to "using your own judgement" of what is "obviously wrong" and what is a "subtle error that can be corrected", I wish I had the God help me when I make decisions, but, alas, He keeps silence more often than not :-). $\endgroup$
    – fedja
    Jan 13, 2023 at 11:52
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A lot of studies in math education have found that mathematical confidence is an important mediator of success - that is, students who aren't confident in their ability to do math are less likely to successfully solve problems, even when those problems are entirely within their abilities. In my own teaching, I place a high importance on building that confidence - and part of building confidence is building the understanding that we do not do math fully-formed. If you tell a student "no, this is wrong, here's the right way to do it", you're telling them "the ideas you had are irredeemable". If that's true, that's true, and both you and the student need to live with that - but it's hardly ever true.

If you instead help them repair the solution, they begin to believe that this is something they can do, because they got 25% or 50% of the way to an answer instead of zero. And they can see their own progress over time - "last time, we had to fix everything past the first two steps, but this time it was only the second half that was wrong". That helps!

Now, of course, the result is often inefficient or awkward, and there is a lot of value in being able to do math efficiently and smoothly. But this makes the most sense to me as a separate learning outcome, and it deserves a separate part of the conversation. For example, after I've helped a student fix their solution, then I'll point out things like "okay, you could have made a different choice here that would have saved you from having to do this" or "gee, didn't those variable names turn out to be so confusing? Would it have been easier if you'd used $w$ for width instead of $x$?"

Of course there are exceptions! There have been times when a student has come to me with a solution that was far enough off that repairing it would have taken ages, and we didn't have that kind of time. Even in those circumstances, though, I prefer to validate and then divert: "This general approach would probably work, but it'd be much harder than it needs to be; here's a different strategy that'd be faster."

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I've had foreign language teachers distinguish between learning a grammatical form either "for production" or "for recognition". The former means the student can produce a sentence using the form correctly and appropriately, while the latter means that they merely know it when they see it, and know what it means. As an example, my knowledge of the subjunctive in French is merely at the "recognition" level.

I think your choice runs a higher risk of students only absorbing the material at a recognition level.

As I would sometimes say when explaining a problem one-on-one "It's not enough to be able to nod along as I explain it - you're going to have to come up with this on your own on the exam.", and then I'd ask them to explain it back to me. And I think that getting a student to full understanding from where they currently are, (as opposed to picking them up and plopping them into the cleaner, better answer) helps them better make the material their own.

Of course, this approach is more time and energy intensive, and sometimes we just don't have that available, so sure, show them the optimal solution. But just be aware of the risk that they might be actually learning less.

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One problem with mathematics for many students is that is perceived as something complete that one learns, rather than an activity one does.

On one hand, you do want to present nice and elegant mathematics.

On the other hand, you really, really want the students to do mathematics and think for themselves. It is very easy to nod along as an expert does something, but this is unlikely to lead to understanding, unless you already are a strong student. You need some relationship to the mathematics happening, and one way of establishing that relationship is starting from and playing with what the student has given you.

On the third hand, if you not in a one-on-one situation, you should take into account all the other students present, too. They might not have much of a relationship to the attempted solution by a peer (though they might have! hard to know), and maybe seeing a muddy and unclear rescue attempt is not so useful for them.

So: as usual, you have to balance several considerations, and most probably also use of time and sometimes whether you even understand what the student is trying to do or say.

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It's a judgment call, but I think you are probably right, especially with very bad solutions.

I agree that it's important for people to try first (even if bad). Makes them learn more than seeing the result without trying first. Seen this in many settings outside of math also (history, business, engineering). In the military, I remember hearing the value of war planning is not the plan, but the planning (learning the features and issues). Same thing in corporate M&A. Kudos for making this point.

I really don't think that it's critical to make the connection from the poor solution to the strong one. Especially for weak students. And after all, they are neophytes, also...accumulating a range of insights, like the model solution. But the key is the mental/moral investment of making a trial attempt first.

The strong students will eventually make the connection themselves. Like Freeman Dyson showing equivalence in competing QED approaches. But to start with, maybe more important that they get some grist for the mill (from learning the model solution, AFTER being invested with a trial attempt.)

Also, of course...it's important to have something to "make the solution your own". Even with the investment. Repeating it a few days later. Showing applications or small variations of it, etc. This will help retention and later usage.

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There is usually more than one solution to every problem - so, if student took a less efficient way of doing it, they would benefit from both: knowing how to get through using this method and how to do it more efficiently.

In my life as a scientist I have extracted a lot of mileage from an advice given by one of my teachers: know the general method (that is a dull but sure way of solving a problem), and resorting to it whenever you are stuck, out of ideas, or cannot wrap your head around it.

Teaching students to be able to use a general method goes beyond a particular problem - it teaches them a skill that they will be able to use on other problems, including those that they will face after graduating. Teaching ingenious/beautiful ways of dealing with with specific problems fosters their ingenuity. Ultimately, it is about balance between educating an engineer and a top-class scientist - some people will become the latter, but many of them need just practical skills.

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