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I've seen in a lot of questions about "which textbook to use for intro analysis", and inevitably Rudin's Principles of Mathematical Analysis comes up, with the (almost cliche) rejoinder that "today's incoming US freshmen cannot handle Rudin." [emphasis mine].

As a product of the crumbling edifice of pre-college US mathematics education, I can attest that I was not well versed in methods of proof, and I found geometry, which was proof based, quite awful when I took it (I'm referring to the experience, not the subject matter). My experience certainly did not endear me to proofs.

My question is...was there a generation of high school students in the US who could jump right into Rudin without "a bloodbath" (to quote an MAA review of Rudin's PMA)? If so, was it the "new math" era in the 1960's that put the first cracks in our math system?

OR

Is it the case that the more widespread use of math in our society led to a need to "democratize" it, pulling it out of its rigorous, ivory tower to a level that is more focused on applications, of which there were becoming many. Was this "industrialization" of math perhaps the real issue?

Conjectures on my part, but I keep hearing about how "un-prepared" my generation is, so I'd like to know the genesis of this statement (besides the usual "..in my days we walked to school with newspaper on our feet.." sentiments).

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    $\begingroup$ A bloodbath is not an unusual consequence of being born $\endgroup$ – David Steinberg Oct 6 '15 at 18:41
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    $\begingroup$ As an inconsequential side-note: There are a shocking number of then/than typos in the MAA write-up. The word "than" appears only twice in the entire review; meanwhile, the then/than error appears twice in the latter half of a single sentence! more then [sic] half the “proofs” in the book amount to little more then [sic] hints $\endgroup$ – Benjamin Dickman Oct 7 '15 at 4:24
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    $\begingroup$ Having done PMA for two semesters as a senior in my undergrad program, I would laugh raucously at anyone saying that freshmen should be able to get through it effectively. Maybe some particularly gifted and well-prepared freshmen would love it, but not in general. At my university, the math class that introduced the pure mathematics foundations necessary to even understand chapter one of Rudin wasn't really offered before the end of sophomore or beginning of junior year. The university has since stopped using Rudin for the same course, which seems good and bad in different ways. $\endgroup$ – Todd Wilcox Oct 7 '15 at 13:03
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    $\begingroup$ What seems to have been forgotten over the years is that THREE undergraduate real analysis texts were published in the U.S.A. in 1953 --- Rudin's book, Real Functions by Casper Goffman, and Theory of Functions of Real Variables by Henry Peter Thielman. (continued) $\endgroup$ – Dave L Renfro Oct 7 '15 at 19:33
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    $\begingroup$ Goffman's book is far superior for anyone interested in pursuing work in classical real analysis topics (i.e. Goffman's book is an excellent prerequisite for Bruckner's Differentiation of Real Functions), but for this reason it was probably deemed too focused on topics on their way out of standard courses to be very widely used, and Thielman's book likely suffered from being less demanding and more limited in scope and slightly more old-fashioned than Rudin's book. See also Bull. Amer. Math. Soc.'s review of all three books. $\endgroup$ – Dave L Renfro Oct 7 '15 at 19:33
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Opinion.

There never was a generation of high school students in the US who could jump right into Rudin.

There would be (and still is) a small portion of the top high school graduates who could. And maybe a larger portion of the graduates from a few elite high schools. But that's it.

Baby Rudin would be used (if at all) for advanced undergraduates or even beginning graduate students.

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    $\begingroup$ Thanks Gerald. I suspected as much, but many mathematicians give the impression that Rudin is not appropriate for "kids these days"...I would say "any days"... $\endgroup$ – user5661 Oct 6 '15 at 18:19
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    $\begingroup$ Oh, the good old days were always so much better in everything... $\endgroup$ – vonbrand Oct 6 '15 at 20:05
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    $\begingroup$ @vonbrand Why, I remember when my whole office building was a computer, kids these days walk around with their computers in their pockets! I miss the good old days. $\endgroup$ – Sidney Oct 6 '15 at 21:43
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    $\begingroup$ Generally I believe the sentiment that 'things used to be better' are usually wrong, for a variety of understandable reasons which are an outgrowth of the fundamental workings of the brain. In this case, however, I would recommend deeper analysis. There is a well-evidenced social trend which easily could result in this situation: anti-intellectualism. Since World War I, industrialized nations have definitely experienced growing anti-intellectualism, a resistance to intellectual pursuits for their own sake. This could easily result in mathematics education replaced by arithmetic education. $\endgroup$ – otakucode Oct 7 '15 at 4:23
  • $\begingroup$ Gerald is right on the money. To handle Baby Rudin straight out of high school (which I did, barely, spending 40 hours/week on it) absolutely requires substantial prior exposure to proof and mathematical logic, which I believe is not part of the U.S. high school curriculum even at many good high schools. (Thanks Mr. Michna for your Math Logic course! Sorry for being such a pain, pointing out every single mistake...) $\endgroup$ – Bob Pego Oct 7 '16 at 0:23
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My recollection of the Math program at Harvey Mudd College (I was in CS, but lots of math major friends), is as follows:

  • The core Calc I included a taste of analysis-like material, in building up the epsilon-delta definition of a limit and proving it for various functions
  • Linear algebra was part of the common core taken by all students, and the first portion of it was taken in the first semester (first or second half, depending on placing out of Calc I). It was very heavy on proofs.
  • Discrete math had LinAl as a pre-req, and was even heavier on proofs. It was required for CS and Math majors.
  • Intro analysis, using Rudin, had Discrete as a pre-req. Math majors were required to take it, and it was typically taken relatively early.

So, sophomore-level math majors who've already dipped their feet in the water.

Not quite the 'advanced' undergraduates of Gerald's answer, but certainly not just-enrolled freshman, either.

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    $\begingroup$ Sounds similar to the program I went through for a BS in pure mathematics. We had a course in addition to those called "Foundations of Mathematics" which covered set theory, finite and infinite cardinality, advanced logic and proof (including induction), properties of functions (e.g. injective, surjective, bijective) and several other core concepts, although not really touching on topology. Foundations was a pre-requisite for almost everything after. $\endgroup$ – Todd Wilcox Oct 7 '15 at 13:08
  • $\begingroup$ Much of that (set theory, log, proof, induction, and function properties) was covered in our Discrete course. Other topics included combinatorics, graph theory, and number theory. There was also a small segment on partially ordered sets and lattice theory at the end, but professors varied what they covered in that time at whim. $\endgroup$ – Phil Miller Oct 7 '15 at 22:19
  • $\begingroup$ For us, Discrete Mathematics was an elective for Math, Engineering, and CS (perhaps required for CS, I don't recall) majors and like Todd, Foundations was a required math course that most students took the same semester as they took multi-variable calculus with the engineering students. The next semester, the Engineering and CS students went on their merry way and the math students took the first courses in linear algebra, abstract algebra, analysis. We used Wade for the analysis text. $\endgroup$ – Andrew Oct 13 '15 at 13:22
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Baby Rudin was (and continues?) to be used for the first semester of Harvard's Math 25ab sequence (the second hardest freshman math class). We got through the first 7 chapters in Math 25a, before moving on to Halmos Finite Dimensional Vector Spaces for Linear Algebra and Spivak Calculus on Manifolds for the second semester, Math 25b. (The (in)famous Math 55ab sequence, reputed to be the hardest undergraduate course in the country, sometimes used Rudin as well, or Loomis and Sternberg as an alternative, as well as some other supplemental texts for abstract algebra and topology).

That said, the students who made it through these courses constitute the top ~50 or so students of an incoming freshman class of ~1600. Also, the attrition rate was severe and most (~3/4) of the students in Math 25 and all the students in Math 55 were exposed to proof writing through high school math competitions (often USAMO/IMO winners in the case of Math 55) and/or university courses. Personally, I had taken Calc III, linear algebra, ODE's and PDE's in high school and was a USAMO participant (although I had already decided against being a math concentrator by the time I entered college). Nevertheless, I found Math 25a and 25b to be two of the most work-intensive and challenging math classes I had even taken. I would be shocked if any cohort of students in the history of American education was well-prepared (or talented?) enough to use Rudin as a standard freshman text.

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As a group "high school students" (at least in the United States) are not ready to jump into "Baby Rudin" (basically junior level college and up.

As a high schooler, I was admitted into a summer math program, where the two "introductory" courses were Number Theory and Abstract Algebra. The better students in that group were ready for "Baby Rudin."

But that is an exceptional group of people. One of my roommates went to Harvard, and went on to help Bill Gates found Microsoft.

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  • $\begingroup$ Why "at least in the US"....what are the Germans and Chinese doing that we are not? $\endgroup$ – user5661 Oct 8 '15 at 19:26
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    $\begingroup$ @Bay: The Germans and Chinese have "13th" grade. And they learn as much in those 13 grades as American sophomores "14th" grades. Our "kids" are less well educated than those of other countries, and one reason is that they finish "high school" a year or two earlier. After "normalizing" for ages, we're a bit behind, but not has much as it seems. $\endgroup$ – Tom Au Oct 8 '15 at 19:28
  • $\begingroup$ given that brain development, especially for males, is still continuing well into mid 20's, it seems that those two years can make a big difference. But...why do they manage to get an extra year's worth of education as opposed to a one-to-one correspondence? $\endgroup$ – user5661 Oct 8 '15 at 19:35
  • $\begingroup$ Foreign children do more homework than Americans, but they are not as "creative." $\endgroup$ – Tom Au Oct 8 '15 at 19:36
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    $\begingroup$ @Bey There is an entire literature on evaluating creativity -- including, e.g., comparisons of American and Chinese students (cf. Van Harpen, X. Y., & Sriraman, B. (2013). Creativity and mathematical problem posing: an analysis of high school students' mathematical problem posing in China and the USA. Educational Studies in Mathematics, 82(2), 201-221). You can find more about creativity in mathematics / mathematics education in MESE 7661 and its links. $\endgroup$ – Benjamin Dickman Oct 8 '15 at 20:33
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I don't see why a freshman should jump right into Rudin. Why not learn calculus in terms of manipulation with the basic explanations that are usually done in courses. And go all the way through Calc 1, 2, 3, and ODEs. Then junior year is fine to do a recursion into theoretical calculus. The vast amount of engineers will never ever need real analysis. For the few who do need it, most of them will learn it better by recursion than by trying to do everything at once.

But somehow people here think it is better to learn it as a freshman. Would you teach Dedekind cuts for arithmetic students also? Even many math majors who go into anything applied (stats, actuarial, OR, finance, etc.) say there is little value from the theoretical calc course. Conversely being able to handle some algebraically complex (no pun intended) problems is highly useful to physicists and engineers in derivations in their courses and in their homework problems. I don't even really think the more real analysis you pack into a calc course the better it is. The more tricks and techniques, the better it is.

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