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One time I watched a student solve the equation $0 = (x-2)^2-9$ for $x$ like this. $$\begin{align*} 0 &= (x-2)^2-9 \\0 &= (x^2-4x+4)-9 \\0 &= x^2-4x-5 \\0 &= (x+1)(x-5) \\\\x &= -1 \text{ or } x = 5 \end{align*}$$ It would have been quicker, and a bit more natural, to instead just start moving things over the equals sign (add nine to both sides, take a square root, etc). I'll occasionally see students do something similar to this example: what they do isn't wrong, but there is a way to do it more efficiently or more naturally. In cases like this:

  • Should I stop the student while they are writing out the calculation to point out there is a more natural way?

  • Should I wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part?

  • Should I say nothing at all, and instead be satisfied that the student understands what they are doing? These details are often independent of the students' conceptual understanding after all.

Maybe the answers to these questions depend on how strong the student is, or on the specific problem they are working on? I expect that there is no good all-encompassing answer to this question, but I would appreciate any general guidelines or advice anyone has about this.

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    $\begingroup$ It looks like the student is capable, but has slipped onto autopilot. I would let them continue, but show them the easier solution afterwards with a gentle reminder that they are, in fact, allowed to think. $\endgroup$ – Adam Jan 9 '18 at 1:46
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    $\begingroup$ @MikePierce Don't frame it as a "correction", that will just confirm they've been doing it wrong and lower their confidence again. $\endgroup$ – immibis Jan 9 '18 at 3:05
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    $\begingroup$ Isn’t this rather subjective? I’d argue that it would be even quicker and more natural to realise that the left-hand side is a binomial identity that can be expanded to $(x - 2 + 3)(x - 2 - 3)$, which immediately gives us the two roots, no need for any further transformation. $\endgroup$ – Konrad Rudolph Jan 9 '18 at 13:53
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    $\begingroup$ I think this way is safer than taking the square root where you have to pay attention to the $\pm$... $\endgroup$ – Surb Jan 9 '18 at 18:31
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    $\begingroup$ Given the answers, it seems important to recall: we have a general method to factor quadratic polynomials and it goes exactly as the student did, but backward from bottom to top, before using the intended method for the exercise to conclude (complete the square/discriminant method). So, apart from what you tell the student, this is a good opportunity to explain this method if it is within the scope of the class. $\endgroup$ – Benoît Kloeckner Jan 10 '18 at 9:28

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I like your second option the best:

...wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part?

Instead of just mentioning the easier way, however, you could first applaud them for using good algebra to find the right answers, then remind them of the new method and work through it together, and then up the ante by having them solve $$(x-1)^3-64=0$$ and then $$(x-5)^4-16=0$$ by a method of their own choosing. You might even suggest they try it their way so they feel the pain of having just one tool for every job.

Test them with $$(2x-5)^9+4=0.$$

I think you're not just selling them a new technique that they can take or leave, but something they really should practice since doing it their way could be difficult or impossible.

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  • $\begingroup$ Why beat around the bush... just replace ${}^2$ with ${}^n$ and tell them $n \in \mathbb{N}$... $\endgroup$ – Mehrdad Jan 9 '18 at 9:35
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    $\begingroup$ All of your examples have imaginary solutions. Not sure the OP's class is that far along... $\endgroup$ – jpmc26 Jan 10 '18 at 2:35
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Foremost: It depends on what the lead-in lesson/topic/direction was. If this was the essential point being exercised, then I would interrupt ASAP and refocus them on the lesson/direction that just occurred. Otherwise I would let them complete their process (unless obviously totally infeasible) and then compare to a faster way afterward.

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  • $\begingroup$ Yeah, when this sort of thing comes up, it is seldom the essential part of the exercise. But you would never just, "let it slide" though? The last time this thing happened to me, it was with a student who severely lacked confidence. They were working at the whiteboard, and while working would regularly turn around to get some positive confirmation from me. I decided to "let it slide" in this case because I was afraid that the student, at me showing them the "faster way," would take it as "the correct way, the way they should have done it," maybe damaging their confidence further. $\endgroup$ – Mike Pierce Jan 9 '18 at 2:02
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    $\begingroup$ @MikePierce: The primary mitigating factor for me might be time management. Assuming time is available, I would almost surely discuss a faster strategy at the end, likely prompting the rest of the class ("did anyone have a shorter method?"). In particular, I think the conceptual issue of "the whole point is to get the $x$ isolated in one place, so expanding the binomial is really making things worse" would be too juicy to ignore. $\endgroup$ – Daniel R. Collins Jan 9 '18 at 4:21
  • $\begingroup$ In my own practice I practically always show an optimal workflow for any exercise we look at before moving on. $\endgroup$ – Daniel R. Collins Jan 9 '18 at 4:27
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An answer regarding your last idea:

Should I say nothing at all, and instead be satisfied that the student understands what they are doing?

No. I'll explain:

I remember that by the time I got to calculus, I made this kind of silly mistake all the time. In my case, I tended to get overwhelmed with all of the formulas and procedures I'd learned by this point in my education. I often wouldn't recognize the difference between a nail and a screw, so to speak, and would apply the wrong tool.

As you pointed out, using a suboptimal approach can still lead to the best answer as long as the student isn't actually making any mathematical mistakes. However, using a suboptimal approach during a test will often lead to the student spending too long on that problem and not having time to finish the test. This means that the student isn't able to adequately demonstrate what he's learned. This happened to me with at least one problem on nearly every test. In your example, the student's suboptimal approach probably adds less than half a minute to the solution. However, in later courses where problems are much more complicated than simple algebra, suboptimal paths can cost the student many valuable minutes.

Because tests are almost always timed and slow work leads to poor grades, I think it is very important to point out to a student when they are using an inefficient strategy. Help them recognize the optimal approach and learn to think efficiently. You don't want them to think "what is any way I can solve this problem?" You want them to think "what is the most efficient way I can solve this problem?"

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    $\begingroup$ Even in a non-test situation, the added mental load of an inefficient strategy might overwhelm the worker and cause them to fail. $\endgroup$ – Daniel R. Collins Jan 9 '18 at 4:34
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I will talk about the cases when I ask one of the students to come to the board and solve an exercise.

If the solution is fine (not accidental, not wrong) I let the student complete the solution. The solution that you posted is a good one. It is more general than what you suggest and shows technique that is applicable to more problems. Besides students probably are just learning these techniques - the pracitce won't harm. Perfectly they would pick up the alternative routes after mastering the main one.

In your case I would wait for the completion and tell "Thank you, your solution is correct. You may take your place now."

And then take the marker myself and say to class "I will remind you of a hack that you could use some specific cases including this to save some time. See that the brackets are squared and 9 is also a square? You all know that $a^2-b^2=(a-b)(a+b)$. So you could rewrite the right side to (x-2-3)(x-2+3) to obtain (x-5)(x+1) faster."

I would try to add the note in a manner that wouldn't offend the original solution but provide an additional tool that might be useful for them.

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    $\begingroup$ My instinct is that the student's solution is not more general. Assuming our context is quadratic equations, then they got lucky that the simplified polynomial was factorable in integers; most, of course, are not. If that hadn't been the case, then they would need to complete the square and use square roots, i.e., effectively do work to recreate the starting line all over again. $\endgroup$ – Daniel R. Collins Jan 9 '18 at 4:33
  • $\begingroup$ @DanielR.Collins: I would argue it was more general. When you take square-roots, you have to actively watch out for fallacies like this. You don't have that kind of problem with factoring. $\endgroup$ – Mehrdad Jan 9 '18 at 9:44
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I would just mention it as an alternate solution method and not make a huge hairy deal out of it. I don't think your way is that much better/faster but even if it were, I would still be low key about it when giving an alternate faster way (if it is so much better, that will be evident). But definitely here it's like 5.7 versus half dozen to mangle a saying. Also, they will encounter times later, like ODEs, where keeping things on one side is the better way. I would think of it more as different methods to catch a fish versus the all best fishing method.

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I suggest that you put aside a bit of time to show the student(s) how to solve the problem both ways. I'm assuming that this is part of the lesson plan, so you can just write-up both solutions manually ahead of time and present them to the class (which shouldn't take much time out of the lession since people read much faster than write). I'd even argue that the way you have presented here is something that they should learn how to do, since it's a near-necessity for polynomials of higher orders (e.g. quartics, quintics, etc.).

If this is more than just a pedantic issue, you might be able to give a hint before the student(s) start(s) to solve the problem, such as "try solving the problem directly, you shouldn't need to do anything fancy" or "work smarter, not harder" (something that a teacher of mine has used).

Interrupting someone while they're working is generally not considered polite either, so I'd advise against it.

DISCLAIMER: I am not an instructor.

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  • $\begingroup$ Both ways!?!? There are at least three ways to solve this, two of which I can trivially do in my head (for this simple example). Difference of Squares needs to be introduced at this point, as it will then lead into illustrating the proof of the General Binomial Theorem. $\endgroup$ – Pieter Geerkens Jan 10 '18 at 14:33
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This might be a good time to have them working in groups, so they can compare with their peers. Great luck if both methods are done in one group. What are the pros and cons of each method? Or just ask students who did each to come to the board and write their steps, and then ask the class the pros and cons of each. (What percentage of quadratics are factorable? Not many.)

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  • $\begingroup$ Usually you can only see "pros and cons" that range from simplistic to wrong when you are just learning the method and when most of the students manage to only think up one of them. At least for me, I could only properly judge the tools when having some proficiency and experience using those. $\endgroup$ – Džuris Jan 8 '18 at 23:23
  • $\begingroup$ Yeah, when there are other students about, this happens much less often. Fellow students are quick to point out the faster way. But what about when this happens in a one-on-one situation, say in office hours? $\endgroup$ – Mike Pierce Jan 9 '18 at 2:12
  • $\begingroup$ I would always congratulate them on what they did well, and then talk about how to connect it with the new idea. $\endgroup$ – Sue VanHattum Jan 10 '18 at 2:30
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This could honestly be due to both cleverness or due to weakness.

Maybe the answers to these questions depend on how strong the student is

Very much so. In fact, it's not clear to me that this was a suboptimal way to solve the problem:

If you believe the student did this due to lack of mastery of the "obvious" method, then by all means, follow what the other answers on this page tell you to do.

But if the student is relatively strong, this might have been due to brilliance, not ignorance. Why spend a second worrying about caveats when you don't have to?

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  • $\begingroup$ Factoring most certainly can go wrong; it doesn't work when the answer is irrational. Moreover, it's mostly a brute force method; while there are various facts, such as Eisenstein's criterion, that narrow the search space down, ultimately it's a matter of guess and check. $\endgroup$ – Acccumulation Jan 9 '18 at 16:06
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    $\begingroup$ @Acccumulation: "It doesn't work" is not "going wrong" (at least not in the sense I meant it, which I thought was obvious). "It gives a wrong solution" is "going wrong". And sure it's not the best idea for every quadratic, but nobody suggested that. I'm saying if they had the intuition to see it would work out nicely for this problem then it was a good idea. $\endgroup$ – Mehrdad Jan 9 '18 at 16:13
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My first words are that the third option is the worst one, I guess. When you think your student is doing something wrong, or even inefficiently, you should tell him or her. Even in the case you are wrong, you'll have enough time to reformulate your explanation and/or reaccept his/her answer.

Now, the better option depends on the subject and the context. If the student is writting the formula in a blackboard, what I would do is to let him/her finish -unless he/she is wasting a lot of time- and the explain that there is a more efficient way. On the other hand, if it is a writting excercise, I would note him/her the inefficiency if I was asked, else I would not to say anything -because I consider a inefficiency is not a big problem. Obviously, while correcting an inefficient answer it can be penalized as much as the problem needs -I would be harder in maths than in physics, for instance.

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Don't stop the student, let them finish. Also, do not refrain from interjecting, especially if you are an educator. Students seek to learn from you. If they learn your methodology from someone else in the future, what you've called the "natural way", they will wonder why they never learned that from you and it may invalidate your credential or standings with them.

I see a lot of complicated answers here. I'll make it simple for you.

In life, there are many different ways to perform a task. Sometimes there are more efficient and effective ways to do something.

My advice is to offer the more efficient way as a suggestion. Let them know that this is the method you employ and that it is faster for you, but also make it clear that it is not the only way. Explain the reasoning behind your methodology.

If the student understands and adopts your method, then that means they are capable of mirroring your thought process. This will allow you to make subsequent suggestions. If they continue doing it using their method, then so be it. The way people think varies from person to person and you can't expect someone to do something your way unless you know what's going on in their head.

I do this with my wife all the time ;)

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