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Question: Why was advanced calculus removed as the first proof-based course in favor of real analysis in most curriculums?

I regularly see in advanced calculus books either that:

  1. its purpose is, along with teaching certain knowledge, to introduce undergraduates to their first proof-based course after a usual computation-based calculus sequence (mostly older books); or

  2. it has been superseded by traditional real analysis courses.

What motivated this? I feel much geometric thinking was lost in the process, as most advanced calculus usually started with vector calculus.

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    $\begingroup$ Interesting question, but I'm afraid it needs a bit of localization. What school(s) have this course sequence? Is it common in a particular country? When was the other sequence common? $\endgroup$ – vonbrand Jun 1 '14 at 1:59
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    $\begingroup$ I started to write an answer, but, honestly, Advanced Calculus is not well-defined. I know about 3 different courses which you might call "Advanced Calculus". At my school, the reason for the shift is in part due to an extension of the calculus sequence to 4hr courses which get to the vector calculus. Also, the introduction of a "transitions" course with proof writing etc... removed some of the other motivation. $\endgroup$ – James S. Cook Jun 1 '14 at 1:59
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    $\begingroup$ I recognize that one of the problems of the so-called "Advanced Calculus" course is that it is ill-defined. Most of what I've seen involve vector calculus but the rest is probably chosen by taste. I'm in a rush now, but I'll see to add that information later, vonbrand. $\endgroup$ – Mark Fantini Jun 1 '14 at 2:15
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    $\begingroup$ In addition to the changing profile of calculus courses as described by @JamesS.Cook, I think the expansion of courses in discrete mathematics (e.g., combinatorics, cryptography, graph theory) without the corresponding expansion of departmental resources has had an effect on the number of "Advanced Calculus" courses. $\endgroup$ – ncr Jun 1 '14 at 4:27
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    $\begingroup$ Advanced calculus is extremely ill-defined. At several schools it is identical to "undergraduate real analysis" and the word "real analysis" is reserved for a graduate course in Measure Theory + Functional Analysis. Some places it is merely an honors calculus sequence that has some proofs. Other places it is a sort of soft real analysis designed for math education majors and quantitative scientists who are interested in math. And of course there is also the vector calculus type. $\endgroup$ – WetlabStudent Jun 1 '14 at 22:25
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This is somewhat of a hypothesis rather than a definitive answer, but one reason why vector calculus may no longer be the first proof based course at many colleges is because vector calculus involves teaching a lot of new material*. Real analysis, is mostly (at least in the first quarter/semester), material they have already seen before in calculus - minus all of the abstraction.

In math there are often two axes for the content being taught. (axis - 1) the abstractness and (axis - 2) the familiarity. The goal of introducing people to proofs is to venture down the "abstractness" axis that is rarely explored in high school and lower level college classes. To really focus on this abstraction it is best to do so with content that is familiar (i.e. 1-dimensional calculus). This way the students can really focus on the proof writing and the abstraction because they already have some intuitive (calculation based) understanding of the concepts from calculus. However, if you were to introduce rigorous proof writing in vector calculus you would have to teach a lot of new material along the way. This distracts from teaching the basics of proof writing; students can easily get overwhelmed if they have to do both at once. In addition when a student doesn't understand something, there is the added confusion of "do I not understand how to do the basic vector calculus calculations?" or "do I not understand the 'intuitive idea'?" or "do I not understand the proof" or all of the above. Often the struggling students can't even answer this question correctly, which makes it nearly impossible for them to overcome their difficulties without some major intervention (because they don't know what to work on).

I stole this idea from professor Steve Strogatz (the author of the somewhat recent math column in the NY times), from a talk he gave about what makes a "pop-math article" popular. He pretty much said, take something really familiar to the audience and dive into it a bit more abstractly, or talk about something unfamiliar but in the language of a concrete example that the audience can easily follow. If he wrote something that was both unfamiliar and abstract, he lost his audience. I think this general comment holds for students transitioning to higher math. Eventually, once they are comfortable with abstract thinking, you can then introduce to them new material abstractly, but not until they master proof writing.

My guess is that the real answer is going to vary a lot from school to school and have to do with how they sequence their other courses, but I feel like the above guess is a pretty good one. Note that there are also obvious pitfalls to this method, especially for advanced students, who really could benefit from going abstract and unfamiliar at once. However, I'd argue that such students are most likely already familiar with the abstract, through their own mathematical investigations. This is why often big universities have two mathematics major tracks, one geared towards prepping for graduate school and one for future high school math teachers and others who are interested in math but aren't ready for the really abstract stuff.

*Note I am taking liberty here and changing the question to "why isn't intro to proof writing emphasized in vector calculus as frequently as it used to be?" based on several people's comments on how "Advanced Calculus" is ill-defined. Proof based vector calculus seems to be the type of class Fantini is referring to as Advanced Calculus.

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    $\begingroup$ this need for familarity may be true for some students, but, for others the introduction of the abstract notion of metric space along with some proofs of easy proposition would make the real analysis course infinitely more interesting than it is currently structured most places. As a student, I was completely bored in the course and yet frustrated. New material which isn't already obvious helps motivate folks like myself (we exist, if only in a minority). $\endgroup$ – James S. Cook Jun 2 '14 at 1:45
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    $\begingroup$ I never said this is ideal. The user asked why it is done, is this what should be done. In many universities, this type of "advanced calculus sequence" is designed for future high school math teachers and engineers and such and is considered separate from Real analysis. $\endgroup$ – WetlabStudent Jun 2 '14 at 3:57
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    $\begingroup$ In my opinion the best way to introduce rigorous proof is through a separate course, that assumes no prerequisite material per say beyond high school math. This way the course can be taken early (by freshman), skipped (by advanced students), or taken later (by students who aren't quite mathematically mature enough), without disrupting the flow of traditional math courses. $\endgroup$ – WetlabStudent Jun 2 '14 at 4:00
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    $\begingroup$ @JamesS.Cook and I'd argue for the advanced students, abstractness is already familiar through their own self study, so my comments above don't negate the general principle, I don't think. I edited the answer though to incorporate the point that this is not ideal for advanced students. $\endgroup$ – WetlabStudent Jun 2 '14 at 4:09
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    $\begingroup$ @JamesS.Cook Also at my undergraduate university we talked about metric spaces in the first quarter of "advanced calculus." I agree with your first comment about advanced calculus being so ill defined its really hard to answer this question. $\endgroup$ – WetlabStudent Jun 2 '14 at 4:54
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I am an undergraduate student at Portland State University, having finished my major requirements I have decided to add a math minor. I did quite well in the lower division calculus, differential equations, and intro to linear algebra courses; yet I have had no experience with proofs. I am planning on taking a "Intro to Math Reasoning" course first for precisely that reason. If I had to jump into vector calculus with proofs as described above I would be ill prepared. Perhaps it is for folks like myself that there are courses like "math analysis" (which specifically includes: "Special emphasis is placed on the ability to understand and construct rigorous proofs")

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