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I have been reading a book called The Joy of X by Steven Strogatz, which I recommend for anyone that enjoys math. In the chapter about calculus he mentioned that integration was, for lack of terms, established in 250 BC and derivatives in mid 1600s. Yet we are teaching derivative first.

I know it is easier to do this, but what if we start with integration set the foundation for it, go to derivatives, then delve into the heavy integration part? How much will this change our understanding of math?

Truly for, like, 1800 years people did integration without the foundation of derivatives. Can we do the same?

Blue
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Feeras AlDheiman
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    People found areas and volumes.... – Angina Seng Aug 10 '19 at 07:06
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    In my opinion, the main reason we care about integrals is because by integrating the instantaneous rate of change we can compute the total change. ("The total change is the sum of all the little changes.") But to understand that idea, you first need to understand the idea of instantaneous rate of change. Also, without understanding derivatives (and antiderivatives) our ability to evaluate integrals is severely limited. There's no need to mimic the historical development of a subject; I think we should organize the material in whatever way makes it seem most easy and simple. – littleO Aug 10 '19 at 07:57
  • We need the excerpt/quote in which this mentioned – gen-ℤ ready to perish Aug 10 '19 at 08:53
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    Apostol's book does start with integration. There's no conceptual reason not to. – Kevin Carlson Aug 10 '19 at 09:14
  • There is, a priori, no reason to believe that teaching things in the order they were discovered is necessarily a good idea. – Arthur Aug 10 '19 at 09:26
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    The Mathematics Educators StackExchange may be a better place for this question. Also, History of Science and Mathematics might have insights on how the teaching of calculus has evolved. – Blue Aug 10 '19 at 09:31

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Because just about the second interesting thing you do in the modern study of integrals is understanding how powerful antiderivatives are finding more than an estimate of the area. I don't want to put down the work of the ancients, especially if you want to understand how to solve hard problems with limited tools, but it won't really help your understanding of calculus.

  • I was hoping that you would expand your answer rather than delete it. With only a bit more elaboration it should be rigorous enough and accessible to beginners. Apologies if my tone appeared less constructive than intended. – Bill Dubuque Sep 08 '19 at 03:03