Interesting thread!

Yes, it is! I've been glancing at it all weekend, but my son was visiting, and I didn't want to take time away from him. Now he's safely on his plane back to CA, though.

But you're only talking about a very small percentage of students even at a school like Stanford. Only 2% of Stanford's undergraduates major in math (16th most popular)

My daughter is a rising junior majoring in math at Stanford, and during her freshman year she took that set of honors math classes that mathwonk is talking about in this thread.

It should be noted that this math sequence we're discussing (Math 51h, 52h, 53h) is not even required for the math major at Stanford, only for the honors track.

Over 100 kids started in Math 51h in September; after the first midterm almost 2/3 of the class dropped out. By the third quarter Math 53h class, only about 30 kids remained (so about 2% of the freshman class, the figure quoted by Crimson Wife). My daughter was one of four women to make it that far in her year (and the only non-Asian, non-Russian woman).

Some bright high school students to whom I have told my concerns have said that their AP courses really were an improvement in many cases over the ones that were in place before.

Here in suburban Richmond, I can't imagine any better alternatives for the public school kids, even in my county's math-sci magnet school. While the AP courses certainly have their faults, my county has done away with most 11th and 12th grade honors courses. so the kids are left with AP English (or whatever subject) or regular mainstream English. I can't even imagine what they'd learn in that regular class!

I myself once taught a vector calculus class called "beyond calculus" to students in a private high school who already had AP math, from a book by Marsden and Tromba that was used at Berkeley, and some of my students protested that they did not expect or need such a tough class.

I love Marsden and Tromba and used it with both my kids for multivariable calculus at home. My PhD advisor worked with Jerry Marsden, so I knew it would be full of good stuff.

It's one of the few books that L insisted on taking to college with her!

Unfortunately, as homeschoolers, that class had no 'documentation' in the eyes of college admissions officers - just a mommy grade and course description. We homeschoolers have to work twice as hard as schooled kids to accumulate a hard record of accomplishments if our kids want to go to competitive colleges. So in addition, my kids did do lots of APs and SAT 2 tests and math contests, but they were a tool, not an end in themselves. Like Jane in NC, I dislike the Calc AP test in particular, due to its heavy reliance on calculators and lack of proof. That doesn't mean that I have to spend more than a day or two teaching the basics of how to use the TI calculator functions, or that I can't add a heavy dose of proofs to my version of AP Calc.

Whether homeschool, public school, elite private school, gifted program, or community college, a student's learning experience is much more likely to be tied to the skill of his or her teacher than to any other factor. An AP course can be outstanding or a complete waste of time.

But if a student goes to Stanford and enrolls in the second year honors course from Apostol, or Harvard's math 55, with preparation equivalent to only a 5 on the AP test, after learning from a book like Stewart, he/she better be ready to be shocked by what will be expected. One should always ask the person teaching the course what that is. E.g. when I called the Stanford professor teaching the Apostol class he freely told me that the formal AP prerequisite was not the real prerequisite. Unfortunately I did not call him 2 years in advance. One recent success I know in math 55 prepared by taking essentially a complete math major including graduate courses at UGA before enrolling in college.

Roy, I know we talked about Stanford math last summer at camp, but I don't think that we ever talked about the textbooks used in the 50h series. Did Stanford use Apostol v2 when your son took the class?

They don't use Apostol any more ( I wish they had! I think it would have been more accessible). Leon Simon teaches 51h now, and he used this

book, which is just a compendium of his lecture notes on linear algebra and real analysis, according to this

outline. I used to have a link to the final exam given to L's class, but it's been taken off-line. As you said, it was almost 100% proofs (and given from 7 to 10 pm at night -- I don't know how those kids had the mental stamina to make it through).

The second quarter class,

52h, (

topics list)is now taught by Yakov Eliashberg, using online

lecture notes rather than any textbook. This was L's favorite course of the trio.

Math 53h, also taught by Prof. Eliashberg, uses

Ordinary Differential Equations by V I Arnold. Judging by one of the reviews on Amazon, I would guess that you like this text?

The 51h course page has several difficulty warnings these days, and most of the students are well aware of what they're getting into (there are links to a "Survival guide" and a "how to drop this class, even after the drop date" advice pages). But face it, most of the kids attempting 51h were high school superstars, and they come to Stanford believing that advice doesn't apply to them...I know for a fact that many are sad that they can't keep up...it really is a shocking experience for many smart kids). I personally felt that while there was a ton of good math material, that each of these 10-week classes could have been done more justice at a slower pace. Maybe it's just my age showing!

My daughter took many classes (epgy, AoPS, home-brewed) far more advanced than AP calculus, but she knew from years of attending Mathcamp and traveling to math tournaments like PUMaC and HMMT that many kids out there are better than she is when it comes to the really hard stuff. So she went in with that attitude, didn't expect to earn straight A's, and just tried her darnedest to *learn* something. During the process, and even after the first year ended, she felt a bit frustrated at the difficulty of it all, but in retrospect (as a really wise sophomore, LOL), she's glad that she did stick it out.

And as an aside, we see that while it's easy to earn an A/B grade in the humanities and social sciences at Stanford, the grades given in math and science (esp the honors level - she did the honors level computer science classes this year) aren't given out easily and really have to be earned. She finally received an A in her last comp sci class this past year, and she's proud as heck (as are her parents).

Someone asked about K-12 preparation in my state. I will try to keep this briefer, but I was once charged with evaluating the testing materials that were used to prepare the high school math teachers in my state for certification, so I know something of this topic as well.

I taught in the math department at your rival cross-state tech university in the mid 80s. Yeah, I'll agree that the preparation of the kids coming from the GA public schools left a lot to be desired.

More relevant to this thread is my 15yo son's future ... I was already planning to incorporate more proofs into his high-school coursework, because he is a future math (or perhaps computer science) major, and this thread reinforced the importance of that. He received a 5 on the BC test in 9th grade (having skipped a grade), and TA'd an AP Calc AB/BC class last year with over 40 students (and will again next year) ... none of which will help him in a high-level class at an 'elite' school (should he be fortunate enough to attend one)! He has had an introduction to writing proofs at two summer math camps, and he is planning to take the WOOT class offered by Art of Problem Solving. Without math camp and AoPS I can see that his preparation for college math-major classes would be woefully inadequate -- the courses he has taken at our local community college (ODEs, etc.) just manage to cover the basics. Oh, and the many math competitions he participates in always include a "power round," in which teams of students collaborate to write proofs (for which they are given the better part of an hour). I am left wondering if all this will be enough ...

Laura,this is all very good! Has your son considered participating in USAMTS next year? That alone taught my kids a lot about proof writing. It's free, they have at least a month to complete each round of proofs, and their proofs are read & commented on by real mathematicians at the NSA as well as by AoPS grading groups of university math majors who know their stuff. The kids can learn a lot if they compare their proofs to the solutions posted online after each round. It was motivating to my kids at least.

And I always recommend thinking deeply about a few problems and working hard to get past the roadblocks.