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This is a more general pedagogical question than specifically teaching mathematics.

As a student, I have always been frustrated that some things were kept from me and not thought deeply. For example, in school, I was taught matrix multiplication as "this is how we multiply matrices", after a bit of discussion about what is a matrix. However, there is a beautiful world behind the simple algorithm we mindlessly applied all over again that attaches a lot of meaning to every operation and I have to wait years until I got to find about it. The same goes for determinants, limits and much more concepts.

One might argue that there is not enough time for all this. But there is. Between the tens and hundreds of matrix multiplication exercises we've done (by we I mean me and my colleagues), there was time for a few tens of minutes of explanation.

Some time ago I started tutoring/mentoring students on an individual basis in maths and computer science (more of the latter than the former, but the question and the problem is still the same). What I found is that my requirement to go and understand the fundamentals before doing anything seems to frustrate and demotivate them. They won't have the patience of looking at the results and trying to get the meaning behind them, and my attempts were received with frustration.

After a while, those who "survive" get to thank me for this treatment, but I would like to make this process more enjoyable for them. How to keep their passion and curiosity and guide them to go for the meaning rather than for the result?

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    $\begingroup$ You can lead a horse to water, but you can't make them drink. The truth is, most go through math courses as something they must suffer through. No matter what you do. Those who see otherwise you should encourage to go into more (real) math. $\endgroup$ – Daniel R. Collins May 18 '17 at 1:50
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    $\begingroup$ Are you working as a paid tutor? If so, then the population you're working with is selected to be people who hate and fear math and are not good at it. There are other dynamics that can happen in this situation that will also work against you. The subject has already been defined for them by someone else, so they think that is what the subject is supposed to be. They have hired you to help them pass the class, not to help them understand. They are failing because they don't understand more basic math topics, so they don't have the prerequisites to understand your motivational explanations. $\endgroup$ – Ben Crowell May 18 '17 at 4:16
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    $\begingroup$ @BenCrowell I worked one summer as a paid math tutor and found that nearly all of my students were among the best. They hired me not because they hated or feared math but because they couldn't understand something and wanted to get it right. When I showed them the deeper insight they were missing, they were thrilled. I've heard that this is common: the best often seek out help; those who need help the most usually don't seek it. Did you have a different experience tutoring? $\endgroup$ – Ben Kovitz May 18 '17 at 7:50
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    $\begingroup$ @BenKovitz: Teaching at a community college, we have a math tutoring center, and I keep statistics on how those students do. They are perennially the weakest students. If a student goes to more than one tutoring session, it's very easy to predict that they'll likely fail the course. $\endgroup$ – Daniel R. Collins May 18 '17 at 15:05
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    $\begingroup$ @DanielR.Collins Interesting. I didn't know that. Clearly my little one-on-one paid tutoring service and the school's tutoring center are sampling different populations. :) $\endgroup$ – Ben Kovitz May 18 '17 at 18:54
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The problem is that you assume that people want to gain understanding. That is not what many students want. Many students simply want it to be over as soon as possible. The idea that there is something deeper, or, that they can understand more than is asked of them is completely foreign to their thinking.

In Star Trek terms, you're violating the Prime Directive. Your assumption is that your culture should be their culture. The audacity.

Of course, I jest, I think the best thing is to be passionate. Share what you know, be positive, and it may wear off. They may find out that they can do more than just mimic math. They can actually understand it. That's the dream.

Incidentally, the people who are annoyed that you do more than just explain how to do problems are exactly the people you should probably never let guide your way of being. There will always be critics. Rarely are their motives worthwhile.

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  • $\begingroup$ While I agree with you, I think quite a lot of students lose their passion at some point due to bad teaching. I am one of those. Fortunately, I met the right person to help me regain it. Also, I'm not speaking about students who are just trying to get over it, they have some motivation to learn more. Just that they are not willing to learn in depth. $\endgroup$ – Paul92 May 25 '17 at 9:14

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