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The standard calculus course does not acquaint the student with the reasons why calculus has been and continues to be important in the intellectual development of humankind. Rather, it attempts to prepare students for other courses in which they might learn that. (To revamp the standard calculus course to do that would be at the cost of scrapping maybe 90% of what's now in the standard course. Deal with it!)

A graduate-level algebra course like Nathan Jacobson's Basic Algebra I and II likewise only prepares students for other courses (maybe algebraic geometry, number theory, and some other things?) in which they learn the motivations.

Of what proportion of math courses in the conventional curriculum can it be said that the motivations are deferred until the student takes some other course?

Just answer that and ignore the next question, unless you don't ignore the next question: What are the pros and cons of changing that?

Michael Hardy
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  • I'm inclined to give Michael the benefit of the doubt ... but my first impression is that this question is more of a “I would like to participate in a discussion about ______” question, hence not really on topic for MathStackExchange. – Greg Martin Nov 26 '13 at 18:35
  • @Greg: It’s also, in my view, primarily opinion-based, and I’ve voted to close it on that account. – Brian M. Scott Nov 26 '13 at 20:10
  • @GregMartin : It may have some elements of tendentiousness, but I actually am seeking information. – Michael Hardy Nov 26 '13 at 20:28
  • I wish my undergraduate math curriculum had had a semester or even year course based on something like Mathematics: Form and Function (Mac Lean) that would have given me a roadmap. (That book hadn't been written then, but I found it much later, to my great joy.) – davidbak Dec 02 '17 at 06:30

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This is what we have instructors for... I always tried (I don't know how successfully) to give my students some idea of the motivations behind the course. In the case of calculus, for example, I talked about the dynamical theory of planetary motions and how this contributed to the mechanistic world-view.

Robert Israel
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  • Now, 40 years later, I wish my instructors had thought that. I sure as hell didn't know what Analysis I was for when we were slogging through multiple new definitions of "calculus" for wierdly discontinuous "functions" that seemed to have no use whatsoever. (What the heck isn't sufficient about Riemann sums?) – davidbak Dec 02 '17 at 06:22
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This is not an answer, just some thoughts:

Philosophical speaking, every math course is just another brick in the wall. Why do you take linear algebra? To be able to study functional analysis or algebra or go to work and apply your knowledge.

Let me speak for the branch of pure mathematics. Gaining insights from a definition/theorem/etc. is the purpose of this branch. You do mathematics for the purpose of doing mathematics. That is somewhat recurrencing but is actually the definition of pure mathematics for me. Of course one could argue that at some point interesting applications of some theorems fall like fruits from the "tree of mathematics" but in fact, one is always eager to go further, do more research on topics. To what end? Possible to infinity.

So the ultimate question for me is? What is your aim? Understanding algebraic geometry - then go for linear algebra, for commutative algebra and maybe elementary geometry. But every course in that chain is a starting point for really a lot of different other courses.

If I would change that? It's in my opinion unchangeable and actually the source why mathematics is so fruitful to everyone. So far from me.

Tom

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