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Let's assume that a group of students need to learn to solve a certain type of mathematical problem for which there is two general methods of solving it, $X$ and $Y$. We also assume that $Y$ is more conceptually difficult to learn than $X$, but once understood, $Y$ makes the problem far easier to solve.

So in teaching a group of students who are capable of learning $Y$, is there any educational benefits of teaching them how to solve problems using $X$ first?

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  • $\begingroup$ Torture them! Buahahahaha. $\endgroup$ – vonbrand Mar 3 '16 at 19:16
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Yes.

The primary reason in my mind is that you want to have multiple procedures with which you can double-check work later. Knowing only a single procedure makes for very fragile understanding; on the day that you make an error, you have no way of recognizing or fixing that error. So I would claim that you want at least two or three methods of sanity-checking an answer. (E.g., a rough estimation, an algebraic solution, and a graphing solution.)

Secondly, in many places you would use the initial, tedious procedure as motivation and appreciation for how much more powerful and elegant the later procedure is. Likewise if students have difficulty understanding the sophisticated procedure, then $X$ can be used to confirm that $Y$ does give correct results.

Thirdly, if you have a disagreement with someone over a result, you should have a more basic shared understanding that you can fall back on to similarly check that the answer is correct another way.

That said, in my community-college classes I find that students frequently do resist this, and either zone out on the secondary procedure, or occasionally become actively hostile. Actually, just this week I updated my statistics final (which I've been using for a few years) to require graphing checks of all the problems. Previously that was only recommended, not required -- the weaker students skipped it, basically just hoping they were right, and had no way of knowing how far off they were in most cases. I'm hoping if they know that the secondary check will be required, they will then be forced to develop that skill, have better intuition, and produce better results.

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    $\begingroup$ This is exactly the answer I was looking for, thank you. I'm asking this because I'm planning on teaching a class with somewhat advanced material to community college students, and this was a very relevant answer. $\endgroup$ – Brandon Thomas Van Over Feb 27 '16 at 4:57
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    $\begingroup$ My pleasure! Of course, time constraints are always a balancing issue -- but personally I commit to requiring at least two methods any time I can. $\endgroup$ – Daniel R. Collins Feb 27 '16 at 5:31
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    $\begingroup$ I would add: fourthly, an inefficient method in one context may be more useful in another. Or more easily remembered. $\endgroup$ – Jessica B Feb 27 '16 at 8:35
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No

It might be useful to go through the hoops of the "hard" method (perhaps it helps in understanding the problem better, maybe it is a stepping stone in the path to the more sophisticated one, or it is in fact easier in some cases), but as a general rule try to teach the "best" method. Be careful to teach what is actually used, even if the other method is elegant/instructive. Your task is to teach useful skills first.

Your student's time (in class, self study, homework) is limited, you have to make the best of it. If by skipping the longwinded method you gain time to have them learn the shortcut better (or even at all), I'd consider it an overall win. You might give the traditional method as a homework, or publish a few examples for contrast to cater to interested students.

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This is a problem of design. Specifically, the fact that there is any difference between the method that is "more conceptually difficult" and the method which is "easier" is a problem of design. By claiming that there is a necessary distinction between what is "more conceptually difficult" and "easy" you have already committed yourself to a bad design constraint.

Think of it this way: before the iPhone came out there were plenty of ways to do things like call people, surf the internet, and listen to music. People accepted that these were things that happened on different devices, but Apple wasn't convinced and showed that you can do all of these things in a way that is actually simpler than any of the individual ways we used to do them. Now, my summary is a bit loose on the use of the term "simpler" just as your question is loose with the term "easier" or "conceptually difficult".

Outside the design of the mathematics itself, there is also ample opportunity to design a killer presentation of the methods and topics which eliminates the entire problem you're worried about. Question your assumptions and see if you can't find a way to actually reach for the ideal.

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