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This is such a vague topic that I hesitate to post. I constantly struggle between the time-tradeoff between motivating a topic, and delving into the rigorous details necessary to fully "grok" the topic. For example:

What I try to avoid is the "just do it" attitude, e.g., divide one polynomial by another without any indication of why one might every need to or care to do this. But it is too easy to spend all class time on motivation and leave not enough for the rigorous results needed to undergird the applications.

Q. How do you balance motivation vs. rigorous mathematical development? Have you any principles that might apply across several domains?

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  1. I think there are more like three (or even four) aspects of education here:

    A. Theoretical understanding (basis, range of applicability, exceptions, etc.)

    B. Motivation (why care, what's it good for, is it fun, will it make me money, etc.)

    C. The basic technique (what you do)

    D. Practice (we are not biological computes--we burn grooves in the brain with practice, not just with a program).

  2. Obviously there's a balance, but I think the key is to look at what you're doing and how well it works. And to consider cost/benefit. And the audience. Emphasizing theoretical niceties or rare exceptions (1A) when students don't have 1C or 1D down is probably a waste of time and even frustrating. For one thing, it's not even good on the cost/benefit side (perfect is the enemy of better). For another, it's probably going to be a waste, since it is hard to process more difficult aspects of a topic before getting arms around the basic aspects of the topic. Learning something once and imperfectly is better than nothing. And it doesn't mean you never return to it and learn it better.

    However, it's also true that too much emphasis on 1C and 1D can be a waste of time for some students or even turn them off. How to know which is which? HAVE THEY MASTERED THE BASIC TECHNIQUE. If in general, they are struggling, then don't kill them with abstractions. If on the other hand, they are crushing it, than move on and enrich. (This would seem to be brutally obvious pedagogy, but somehow we see questions and such where it eludes many commenters and questioners here, especially those who are basically good at math part of math ed but not the ed part of math ed.)

  3. Think about the audience. How smart are they? How advanced are they? How motivated are they? Pick your battles strategically. And please, please realize they are different from you. And that "what I like" may not be what is best for the general class. Or even what is best for the smartest of 30 kids may not be best for the bulk of the class. Think of it as an optimization problem. https://en.wikipedia.org/wiki/Linear_programming By which, I mean think of it that way intuitively, not as an actual calculation. But at least you have some ideas of the factors, perhaps priorities, etc.

  4. I have not really address motivation, 1B. My advice is to do a very minimum of motivation and move rapidly to 1C. In many cases, "motivation" ends up being some physical word problem. But guess what? Word problems are harder than math problems! Lots of tests have shown this. You are compounding a need to understand some basic physics or econ (even if simple) with the math and also adding the need to "translate" word relations into math relations. So I would generally eschew specific application to set up the discussion. (And I say this as a guy who is an application ADVOCATE.)

    One of the issues with why many PDE texts are hard for engineers is that they start out with engineering applications. Rather than the way ODE texts start with just the algebraic technique...and then do applications of it.

    Now, there are some motivations like advancing to the next class, supporting science courses, etc. But I don't think that you need to dwell on these. Occasional en passant references are plenty. Really "you have to do this because it will be tested" is fine (yes I mean it) as an immediate motivator.

    One useful "motivator" is just the ability to do something we haven't (in the course) been able to do before. So, we learned a specific technique of integration or the like but it doesn't work for some type of problem. Now we give you another tool that will solve those. This feels like a natural connection to earlier work and like we are getting something useful. But still, it doesn't take huge time to say this. OK to mention the connection and how the new tool will help with a harder but related problem. And then move on and show them the darned integration by parts or characteristic equation or what have you.

    Some general expressions of sympathy, humor, camaraderie, "in it together" and "been in your shoes" are useful as well. By which I mean don't be a robot and have some warmth. But also don't dwell on this stuff. Lots of students tend to be very immediate in terms of wanting to get the show to move on.

  5. I think there is also something that is a blend of 1A/B/C, which is to have an intuitive concept of the new material. Note this doesn't mean an exact understanding. It's the opposite of that actually. But having a quick mind picture or label to be familiar with a topic. So for instance, in algebra, we solve algebraic equations for y and in ODE, we solve ODEs for y. [Yes, there is more to each course than that. Yet the generalization is still a very useful first order description.] Examples of analogies are things like "darn doesn't the second order ODE with constant coefficients look a lot like a quadratic equation? Wouldn't it be nice if we could use the quadratic to help solve it! These may be rather far from 1A perfection or proof-based explication. But I assert they are damned close to "groking".

P.s. "What I try to avoid is the "just do it" attitude, e.g., divide one polynomial by another without any indication of why one might every need to or care to do this." I would encourage you to have some self questioning here. Often questions on this site, espouse a position and look for sympathetic support. And you may get some (likely will based on dynamics of the community). And you might even be right. But at least continue to consider if you are wrong. An analagous example: if you were teaching lock-picking, perhaps some technique-based description and even drill might be preferable to start, before the "why". (But it's a balance. Sometimes the "why" makes the "how" easier to remember/absorb. But not always...sometimes it makes it harder, for new students. If the why is easy, by which I mean not complex, not counter-intuitive, than it's more likely to help, initially.)

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    $\begingroup$ I think the point you call "blazingly obvious pedagogy" is accepted by most of us, regardless of the types you describe. The trouble is that mastery of a basic technique that the students do not understand at all the reason for the technique can also be off-putting. Nobody is seriously suggesting, I'm sure, that it is a good idea to overload weak students with abstraction...but abstraction is a necessary condition for mathematical thought, or one would never be able to approach an unfamiliar situation using prior experience. $\endgroup$ – Jon Bannon Jul 22 at 8:11
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    $\begingroup$ Incidentally, I do like this answer. $\endgroup$ – Jon Bannon Jul 22 at 8:18
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    $\begingroup$ Nice answer. Just a small point to add: In some cases, 1A can also be used as a motivation. Take a rather difficult problem and let the students solve it (as a homework maybe, taking a long time). Come back to the same problem a few weeks later and show that, with the theory you developed, it can be done in a few lines. $\endgroup$ – Dirk Jul 22 at 9:29

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