Sternberg's Advanced Calculus and math education

[zpconn]

There are several reactions that one can have upon seeing the text Advanced Calculus by Sternberg and Loomis. (Those unfamiliar with this book can find it online here: http://www.math.harvard.edu/~shlomo/)

My reaction was simply to be surprised at the extreme discrepancy between the material in Sternberg and the material taught at my university. For a book intended to be a two- to three-semester honors sequence in analysis for entering college students, Advanced Calculus seems like it would be beyond most senior math majors at my school and perhaps even a few first- or second-year graduate students. Indeed, I would say that if one worked through Advanced Calculus, then one would find the Ph.D. qualifying exam in analysis here trivial.

Now, I don't go to a particularly prestigious school. It's my state's flagship campus. It's a good school. It's probably very representative of the majority of universities around the country.

But it makes one wonder how different things could be. Personally I believe that most schools could offer sequences at the level of Sternberg to their entering freshmen if math education were reformed properly. The reform would consist almost exclusively of simply raising the grading standards in high school math classes and introducing critical thinking far earlier. There's no reason that math classes can't start emphasizing logical reasoning, proofs, and creative arguments as early as eighth grade.

I believe that such reform would work because I've observed that there's essentially a quantum leap in ability between those math students who do the bare minimum and those who dedicate themselves to their studies. Obviously a large gap is to be expected, but the gap in ability isn't just large--it's extremely huge. The students who really study math seriously could simply eat the other ones alive who still, even in college calculus, will write a / (b + c) = a/b + a/c and who think that simply giving a numerical example of a theorem constitutes a rigorous proof.

And this really doesn't have much to do with intelligence. It's simply that "better" students (the dedicated ones) have pretty much always been dedicated to studying decently from junior high on. The effect of this is cumulative and profound so that upon reaching college the gap between those who studied seriously and those who didn't is miles in length.

The reform would have to do with schools making sure that students who get As in algebra are actually very good at algebra and that students who get As in calculus actually understand why, for example, monotone functions are Riemann integrable. If getting straight As in math courses in high school signified that one had learned calculus with at least a decent bit of rigor and that one had exposure to proofs (including basic techniques like induction), then I don't see why it couldn't be a nationwide standard to have students interested in math starting college in sequences at the level of Sternberg's Advanced Calculus. Less theory-intensive sequences could also be offered (and should be offered for those who don't want to study analysis intensively), but even these could be at a far higher level than most calculus sequences at colleges these days.

What do you think? Is it realistic to reform math education so that first and second year math courses could actually be at the level of Sternberg's Advanced Calculus?

Part of the reason I'm writing this is that I personally feel a bit short-changed by the whole situation. Students who are rather serious about studying math and who are rather dedicated to the pursuit are left with a fairly poor curriculum that progresses too slowly and covers material for the most part with insufficient rigor. Most math majors at my school don't take analysis or algebra until their junior year. The result is that people like me, who need and, I think, deserve a more rigorous undergraduate curriculum but who aren't ready to skip on to graduate level courses, are stuck in the middle where nothing really fits.

[Note: there are of course other books similar to Advanced Calculus. I just picked one to use an example.]

 

[snipez90]

But it makes one wonder how different things could be. Personally I believe that most schools could offer sequences at the level of Sternberg to their entering freshmen if math education were reformed properly. The reform would consist almost exclusively of simply raising the grading standards in high school math classes and introducing critical thinking far earlier. There's no reason that math classes can't start emphasizing logical reasoning, proofs, and creative arguments as early as eighth grade.

Sorry to be blunt, but it hardly matters what you or I personally believe when the simple truth is that the reform you hinted at will never happen. Although your suggestions are rather vague, they are certainly not that simple. For one thing, America still manages to produce many successful scholars, so while the problem with our education system is fundamental, no one really sees it as a grave issue. The bureaucracy of education in America is also a huge factor that halters any reform. However, I think the first reason is more relevant to the discussion. Those who are motivated will always be ahead of the standard curriculum, regardless of what level they are at. If you have access to the internet and a library, you can always learn more math, or any other subject for that matter, outside of school. Specifically, there are many resources on the internet that are available, including this forum, wikipedia, and gigapedia, to name a few.

Also, you have to be aware that the teachers at certain institutions will simply be more outstanding lecturers. Some professors, such as the ones who wrote the textbook you mentioned, are very likely to present a course on a higher level than many others would. But again, this should not matter too much if one is willing to do the work himself.

 

[HallsofIvy]

Actually, the textbook you mention seems to be a pretty standard "Advanced Calculus" text, intended for, say, physicists and others that will work with applications of mathematics(as opposed to an "Analysis" course that would be more theoretical). The last chapter is perhaps a little advanced for undergrad work, but it is standard in texts that the last one or two chapters be more advanced and optional

 

[zpconn]

snipez, the purpose of my post wasn't to actually seriously propose a strategy for reform to be implemented. I intended this to be more of a hypothetical discussion. Of course I know that actually implementing reform would be very difficult. Since there are many incredibly talented people here, I was just interested in what people thought the general ability of non-math concentrators could be given such hypothetical reform.

Actually, the textbook you mention seems to be a pretty standard "Advanced Calculus" text, intended for, say, physicists and others that will work with applications of mathematics(as opposed to an "Analysis" course that would be more theoretical). The last chapter is perhaps a little advanced for undergrad work, but it is standard in texts that the last one or two chapters be more advanced and optional

Hmm, interesting. Sternberg has written several books for physicists and applied mathematicians, but I'd always interpreted this one as intended for a general mathematical audience (although it does have a section on classical mechanics at the very back).

Either way, I've certainly gained very much from the book. I've found its unifying presentation of various subjects extremely helpful.

 

[Civilized]

Part of the reason I'm writing this is that I personally feel a bit short-changed by the whole situation. Students who are rather serious about studying math and who are rather dedicated to the pursuit are left with a fairly poor curriculum that progresses too slowly and covers material for the most part with insufficient rigor. Most math majors at my school don't take analysis or algebra until their junior year. The result is that people like me, who need and, I think, deserve a more rigorous undergraduate curriculum but who aren't ready to skip on to graduate level courses, are stuck in the middle where nothing really fits.

My suggestion is to go to the library and learn math on your own, all that matters is research anyway. What I mean is that the advanced calculus course you linked to, compared to the courses you are taking, won't really matter at all once you start to grapple with research level problems. At that level all courses become inadequate anyway.

 

[zpconn]

My suggestion is to go to the library and learn math on your own, all that matters is research anyway. What I mean is that the advanced calculus course you linked to, compared to the courses you are taking, won't really matter at all once you start to grapple with research level problems. At that level all courses become inadequate anyway.

Oh, I do study whatever I like on my own, of course. Still, I obviously want to get A's in the classes I take, so during the semesters I have limited time to study alternate material.

 

[ice109]

i don't understand how you figure this is meant for incoming freshman? this is meant for an advanced calc class to be taken at whatever age you happen to reach advanced calc. if you look at the prereqs you see apostol's calculus; that's a freshman calc book used for example at mit.

 

[zpconn]

Well, I interpreted the preface as meaning this was intended for people who had taken introductory calculus (probably the equivalent of calculus I and II at most schools--i.e., including some treatment of integration techniques and series) and linear algebra. If you examine the content of the book carefully, you'll see it actually covers the material typically taught in multivariable calculus courses (although it does presume familiarity at least with the definition of partial derivatives) and even in differential equations courses.

Some of Harvard's freshmen courses have much higher prerequisites than that. I'm not sure about other schools.

But I was more fascinated by the breadth and depth of the coverage. There really is nothing that would correspond to this book at my school, for example. That is to say, there is simply no such unified coverage of linear algebra, analysis, differential equations, multivariable calculus, and, in the end, calculus on manifolds (which I admit may be a bit much for many undergraduates, at least before the junior/senior years).

Usually teachers have to tell students to sort of "take their word" that (abstract) linear algebra is useful in multivariable analysis. I'm not talking about geometric vectors, vector fields, and Jacobian determinants, but the theory of linear maps, the idea of the derivative (or differential) as the best linear approximation to a nonlinear map, and so forth. This book makes it totally obvious and is a testimony to the fact that, as the authors state, "the adequacy of its [multivariable calculus's] treatment depends directly on the extent to which vector space theory really is used."

That said, I don't go to MIT, so I'm unfamiliar with the teaching styles there. You should interpret my statements in the context of the school I am familiar with, which provides just a fairly standard treatment, nothing fast-paced or unusually sophisticated.