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Why is it so uncommon to find material that shows students how everything is connected?

A recurrent question from students is why are polynomial so important? If we shown them how these are related to real functions and to differential equations, they would be more interested in learning I think.

Otherwise, processing polynomials is just a mechanical ability. The same applies to second degree equations. Many, really many students know how to solve them, but it's quite rare to find those who know what real world phenomena they describe.

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Unfortunately, it is difficult to give such a general treatment as you describe without at least a little groundwork first.

For example, the connections between polynomial and differential equations is interesting, but it would hard (or perhaps impossible) for a student to appreciate this without first knowing what a polynomial is, and what a differential equation is. By all means, when discussing linear differential equations it's a great idea to emphasise the connection between them and polynomials - but is it reasonable to do motivate the solutions to polynomials with differential equations, before the student even knows what differentiation is?

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  • $\begingroup$ He does not have to dive deep into the applications/uses. It's like a trip: I don't expect to visit all places but the high view is still pleasing. $\endgroup$ – Mark Fantini Mar 22 '14 at 17:37
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    $\begingroup$ It all depends on how much experience the students have; they can only go so far into unfamiliar territory before getting lost. $\endgroup$ – preferred_anon Mar 22 '14 at 17:44
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It is certainly important for motivation to know where you are heading in teaching, and why. But perhaps trying to reach too far is counterproductive. Sure, if I'm learning mathematics as such, I'd like to have an overview of the whole soon (but as comments/other answers say, that requires some non-trivial background). For non-specialists, the outlook will necessarily be more narrow (but might need to include a wider view of another field altogether).

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Perhaps it is just that there are too many connections. What is intensely relevant to me (mostly in computer science) would have gotten a blank stare from my classmates in other fields. What even some run-of-the-mill themes like polynomials' relevance is to me has almost no relation to their use in e.g. chemical engineering.

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  • $\begingroup$ Perhaps these two answers should be combined. $\endgroup$ – Brian Rushton Mar 23 '14 at 11:41

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