kitchen table math, the sequel: Allison on "the naturals"

Friday, May 2, 2008

Allison on "the naturals"

The "naturals" will be fine, I think. Kids with natural math talent can come from behind. I could be too sanguine about this, of course...
You are. The kids with natural math talent who are not utter prodigies DO NOT come from behind at a school like Harvard, MIT, Caltech in the math or sciences. They are completely outclassed by the Russians, Czechs, Estonians, Koreans, Japanese, Singaporeans, etc. In physics at MIT, the Russian kids were an order of magnitude ahead of the brightest American math kid in physics. In math, it was the same.

What American kids have going for them is an escape hatch: the "naturals" in math can more easily go into Investment Banking and other places that math skills are wanted than the foreigners, such as the Russians can, it seems (because of a lack of H1 B visas, maybe?)

But the only thing keeping more of the best math and science foreign kids out of MIT and the Ivies are restrictions on percentages of foreign students.

In grad school, it's almost a lost cause. There are virtually no Americans in the top programs, and white American men are almost unheard of. They have to have been the real prodigy (skipped high school, or college at 15, etc. and never fell off the train of perfection) to get there.

35 comments:

Anonymous said...

When I read this I think of the handful of people that have told me how inappropropriate it is to teach pure math (as opposed to math methods in the sciences) because "Ya know, not everyone is going to be a mathematician."

Indeed.

Catherine Johnson said...

I'm horrified that I fell into the trap of assuming that talented kids "will be fine."

That is the ENTIRE district philosophy around here.

"Your kids will be fine."

"They'll get into college."

The absolute, unshakable, indestructible belief system is: These are rich white kids, they've been given everything, they'll be fine.

(Sure, most of them will be "fine." But we don't need $22,000 per pupil funding to ensure that upper middle class white children will be "fine.")

I made the same mistake thinking about "mathematically advantaged" kids. Yes, they'll be fine in the sense that they can go into Investment Banking.

But they're not going to be getting Ph.Ds in physics at Berkeley.

Catherine Johnson said...

Meanwhile I'm desperately trying to get C. up to the level where he could go into investment banking if he wanted to.

Or do research in a social science.

Or - hey - just be able to read a social science research paper.

I want the same thing for me.

Anonymous said...

"Your kids will be fine."

"They'll get into college."


You need to read Paul Fussell's slightly outdated, yet hysterical, book called Class:

http://www.amazon.com/Class-Through-American-Status-System/dp/0671792253/ref=sr_1_1?ie=UTF8&s=books&qid=1209826603&sr=8-1

Allison said...

The problem is even greater than I suggested though, because it's not just that the "naturals" can't compete at the top of the college food chain in the sciences or math. They are not able to compete in engineering classrooms anymore either. That means engineering at UC Berkeley, U of Illinois, U Michigan, Purdue, Duke, as well as the CMUs, MITs, Stanfords, etc. is becoming out of reach as well. And as the social sciences get more mathy, they will be less able to compete there, too.

What's the main problem? That Do Not Know that they need to catch up until they reach college. But of course, it's too late--how do you remediate yourself after 12 years of inadequate math education when you are competing against students who don't need remediation?

And where would they begin? Do they really know how to study? How to *do* the hard work of learning the math? How many of their teachers know how to help them remediate, or think that it's appropriate? (At Cal, Professor Wu can't even help them remediate, he ends up flunking half his calculus class--that's why he got into fixing K-12 math education in the first place.) It's not as if we've got Direct Instruction for the remediating math classes, and the "naturals" could manage through all that, but that's not the kind of stuff they need to start pushing in these subjects. They need to have had good instruction for the prior 6 to 12 years.

Anonymous said...

My take on this is that kids come in normal distributions, i.e. they come to your classroom with a range of capabilities. In low SES districts the spread, the standard deviation, is quite large. I've had classrooms with kids ranging from 1 year above grade to 5 years below grade.

If you are in a failing district, like mine, you are blanketed with consultants, coaches (of which I am one), tight curriculum maps, walk throughs, and on and on. The sum total of this is that you're asking teachers to teach to a really narrow portion of a theoretical distribution while, at the same time, you are 'delivered' children with a 5-6 year span in abilities.

This means that if you follow the rules, and it's perilous not to, you are by definition throwing 80% of your class under the bus. Teachers adjust the curriculum to try to push as many of the distribution as possible through the eye of the needle. They teach to their median. So by definition 40% of your class is bored and 40% don't get it.

My district retention policy is "We don't have one!" Even if they had one there is no remediation program so the miniscule portion of students who are retained are put back into the very same classroom that failed them with the expectation that the second time through will be magic.

"Just stand right there in the middle of center field, Stan, and I'll hit you a few balls", says Ollie, as he proceeds to spray 60 balls at poor Stan; all at once and all over the outfield!

Independent George said...

I am a sample size of one, but this matches my experiences. I am one of those 'naturals' who got to college, and realized I had three choices:

1. Study twice as hard to become a median student in chemistry
2. Study twice as hard to become a top student in economics.
3. Keep my same work habits to stay a median student in economics.

Shamefully, I chose option number 3 (and am now paying for it by working four times as hard to become a top student in an MS program I probably wouldn't have entered if I'd wised up earlier).

The other point is all those pesky foreigners are now entering the easy subjects like economics and biology.

concerned said...

I think that the problems begin very early in a child's math education... when a child begins receiving poor scores in middle school math because their final answer is incorrect due to ONE careless error in a FOUR-step problem!

It is very discouraging to the child, and they may end up seeking avenues other than mathematics for academic affirmation.

Correct solutions are important, of course, but considering the ENTIRE attempt at the solution is much more important to the learning process! This may be why the NMP advocates formative assessments. (I hope that they are evaluated appropriately...)

I have encountered grading of the final answers only, rather than the entire solution, with my children more times than I can count on both hands and both feet!

Counterbalancing these negative influences is exhausting! (but unfortunately necessary!)

le radical galoisien said...

"
The absolute, unshakable, indestructible belief system is: These are rich white kids, they've been given everything, they'll be fine."

The irony is that (at least in my area) the system fails horribly at paying attention to the lower-income students as well.

One of my greater grievances is the way both of my countries handle remedial situations. This includes both the education administration and the parents of both nations.

Firstly, if a child is placed into a remedial class, parents might have a right to be mad and have a fit about how the school has failed the child, but now that you're at this point, if the child is having trouble, parents actually evaluate if the remedial class will actually help him/her, rather than be set on speaking to the principal about placing him/her back in the normal class. The sooner it's fixed, the better. It's better to be in a remedial class in 5th grade than to be in a remedial class in 10th grade. The prestige and the ability to boast about your child in fifth grade is overrated. Save the boasting when he gets into the better colleges.

Secondly, when administrations place children "down" into a remedial class, THEY SHOULD STOP ACTING LIKE THESE CHILDREN ARE INHERENTLY BAD AT MATH AND WILL BE STUCK THERE FOR LIFE. In both places it seems that teachers and administrators forget what "remedial" means. It is related to the word, "to remedy." It doesn't mean, "the place where you can put incompetent teachers because you think the kids of those classes are dumb at math anyway."

To me, remedial classes should *not* be the same thing as streaming. When you put a fifth grader into a remedial class you shouldn't consign him to vocational school for life. And yet that's what a lot of administrators treat remedial kids like, as though they're inherently disadvantaged at math, putting them in a "track" rather than with the aim of *reintegrating* the remedial child back into the normal class eventually. (Eventually == as soon as possible.)

"
The other point is all those pesky foreigners are now entering the easy subjects like economics and biology."

I suppose they are easy subjects if you want them to be easy.

Well, I suppose at this point I'm only a high school senior, but often the juniors ask me whether AP Bio is easier than AP Chem or AP Physics C. When I respond with some sort of, "not really," they might qualify with, "oh, I meant easier in math." And ah, they are true on that point, but I don't admit it -- it's easier in math only because the administrators *make* it so.

Anonymous said...

Allison,

A practical question for you. What does a child need to have learned by 12th grade to compete in the technical fields at a top-notch school? I don't think that the kids in $other-countries are inherently smarter than our kids -- or at least not enough to matter.

So ... what to do? This is a practical question as I am homeschooling and have a lot of control over the curriculum (and a medium amount of control over pacing). Currently, the child in question is on track for Calculus by Freshman or Sophomore year in high school. This *sounds* like all should be well...

Am I missing something?

Part of my concern here is that I went to school with a number of kids who took Calculus (BC) in high school and a few who were taking post-Calc courses at a local college by their HS Junior year. One of our local public high schools offers a senior level class in Differential Equations. You make it sound like none of this is sufficient ...

Regards,
-Mark Roulo

le radical galoisien said...

Indeed, while I haven't had any Singaporean education for 4-5 years, and having gone to one of the secondary schools in Singapore that produced "top" students, I wouldn't say that with a top-to-top comparison Singaporeans' math abilities happen to be significantly significantly ahead or anything. Distribution-wise, I think Singaporeans fare better (e.g. average math ability fares better, less inequality in faculties), but Singaporeans also study different topics. What they exactly are I can't say at the moment, because while I have a rough idea of what my friends back home are studying (I think I'm one semester behind, possibly two), I haven't exactly been asking my friends to print out their transcripts to me.

However, I am a bit tired of being a cog in the machine though. After 4 days straight of prepping for my linear algebra final, while I am fascinated with the power of the subject, my attitude is becoming, "Screw this. I want a break from the engineering track. Give me magic squares and number theory."

le radical galoisien said...

Or magic hypercubes, even.

Adrian said...

Well, what the hell -- I'll tell you all my sob story/pseudorant. I went to graduate school in math at Ohio State (not the top most echelon, to be sure, but still a pretty reputable graduate program for math, all the same). It was flooded with Chinese students. And, many if not most of the professors, for that matter, were not American. I never really had much of a problem with that, actually. To me, it is a matter of he who does the best math should get to stay. (Besides that, I thought the Russian analysts all had this "hard core" mystique about them that I liked and still like -- to this day I regret never meeting or talking to Boris Mityagin who I now hold in high regard almost entirely for his infamy as a tough advisor both then and now.)

I think I may qualify as "a natural", too, because having taken only one class in real analysis before entering graduate school, I ended up being one of only two normal students (i.e. domestic students from some American four year university) to pass the analysis qual on my first attempt. Nevertheless, both me and the other guy, both having the highly coveted position of having all our quals passed (the lack of which being one of the main reasons people were forced to leave), left after two years with our masters degrees. (I pretty much knew I was going to leave after my first year.) He left because he had a wife who got into a psychology program at the University of Michigan.

I left because academia looked like crap. For one thing, I realize now that OSU grad school is essentially run like a bad community college. It is all about teaching, actually. That is how you get summer support. (In fact, it all came down to your overall rating on student evaluations of the instructor, so it was really more like all about popularity with the students.) And, the stipend is way too small to live on without summer support. A lot of the people I knew in graduate school got along by having a spouse that was working for a living. (Both my officemates quickly ended up with people that could support them. I, unfortunately, did not have such a spouse/girlfriend/etc.) At any rate, with

* a litany of horror stories about PhDs going unemployed,

* the fact that I couldn't really support myself on the stipend,

* the fact that even if you did get a job, it seemed like it was all about "teaching" and not doing math

(* and, the fact that even if "teaching" were really teaching, it still wouldn't be teaching math)

* the fact that after years of study, I still didn't know any math -- not really, (In fact, that was the main thing I learned in graduate school -- just how much of a waste my undergraduate educaiton was.)

* and, I couldn't figure out how I would possibly be able to study the kind of thing I wanted to

I decided to leave. They all said I would be back and that I was making a big mistake. I stayed out in the working world for years as both my officemates went on to leave school, themselves, without PhDs. I was kind of vindicated by that whole outcome, perhaps, given how certain they were that I made a mistake, and yet they were the ones that supposedly "wasted" all those years just to end up with a masters degree just like me.

Now, I kind of wish I had stayed.

I realize, now, why there was so much unemployment -- not because they couldn't possibly get some kind of job but just because they were holding out for that tenure track position at a prestigious university. At the time, I figured that if there was that kind of unemployment, then there was just no way in hell I would have gotten a job at a national laboratory or something like that (which "obviously" must have been the really coveted positions). And, talking to people in the business world, they all kind of acted like getting your PhD (or even just having your masters for that matter) could make you "overqualified" for normal entry level jobs in industry. But, not having the advanced degree in the right field makes you underqualified for the other jobs (all of which was perfectly consistent with this apparent catastrophe brewing of unemployable math PhDs). (Boy, did I really not know how things worked real world back then! Most of all of this was just complete bullshit.)

At any rate, I look back on it all and realize how much of a sham a lot of it was. You had two large groups drowning out the people that really belonged there. There were ostensibly high performing foreign students that were using the math department as a means to come to America and do something else. One guy, I recall, wanted to transfer to engineering and only came to math because he couldn't get into engineering. One of the best English speaking Chinese students came after getting his Masters in china so that he could eventually make his way into being an actuary, ultimately leaving, I believe, with another masters degree from OSU to work at a local insurance company there in Columbus, OH. I think that kind of thing was quite common, actually -- older foreign students, quite often from other disciplines or interested in other disciplines, flooding in and essentially making the quals a rigged game for the domestic students. (Of course ABD students from China can come on over and pass the quals on their second day in America -- right after they flunk the driving test for their drivers license.)

So, if you think that is a sham, then you will go and lobby for the domestic students, right? Well, that turns out to be mostly a lot of whining over not being able to pass the quals by people that are all about teaching and not nearly interested enough in the actual math. (Frankly, saying you are all about the teaching in graduate school is like saying you've got lots of "people skills" in the business world.) While you have foreign students taking up a massive amount of space and money just so they can eventually go get a job as an actuary or transfer to the engineering department or something like that, you have a bunch of domestic students that are just destined to stick around long enough to realize they really didn't want to do math after all and transfer, themselves, into business or something else. A good example of that was (as I recollect) one lady who was totally into the teaching and teaching methods -- all about graphing calculators and group work back then -- she successfully petitioned to stay after not passing that analysis qual (just like most domestic students) but eventually dropped out and went into an MBA program.

Meanwhile, a couple of the most promising students ran away screaming from that circus. Well, I did, at any rate. It seems like there were some that remained and probably got there PhDs, I imagine. They were never too caught up in the social atmosphere, for one thing. Who knows what happened to them....

But I will say one thing about going on in math. It is absolutely indisputable that calculus and differential equations does NOT prepare you for the next step. All the calculus in the world will never get you to pass that (nearly ubiquitous) analysis qual. And, that is pretty much what domestic students have going into graduate school -- a whole bunch of calculus. Why is that? Because the engineering, physics, chemistry, business, statistics, etc majors out number the math majors infinity to 1. And, they are all served by calculus and more calculus (aka differential equations, vector "anlaysis", "complex variables", etc.).....

Anonymous said...

"But I will say one thing about going on in math. It is absolutely indisputable that calculus and differential equations does NOT prepare you for the next step."

Okay. So Calc and DiffEQ in high school aren't enough. What is? I'm totally unfamiliar with "analysis". Is analysis super-duper-abstract algebra?
Do I need to be targeting this for senior year in high school? Should I plan for my child taking this analysis instead of DiffEQ?

-Mark Roulo

lgm said...

Do you have any comments on the value of the Ross program for high school students?

Adrian said...

Okay. So Calc and DiffEQ in high school aren't enough. What is? I'm totally unfamiliar with "analysis". Is analysis super-duper-abstract algebra?
Do I need to be targeting this for senior year in high school? Should I plan for my child taking this analysis instead of DiffEQ?


Yeah -- it's like abstract algebra in that it is "real math". The main fields of math as it is practiced by mathematicians are abstract algebra, topology and analysis. "Classical real analysis" is essentially the "real math" version of calculus. You basically rehash calculus, including revisiting the derivative, the reimann integral and everything, only this time you are tested on the proofs of the theorems and not the applications of the theorems. For most people that have taken both, real analysis is really a completely different kind of course, though. It is amazing how much they can cover the exact same set of topics and yet almost have less in common with each other than, say, combinatorics and abstract algebra.

At any rate, I don't know that there is really a good answer for you. Real analysis is normally a strictly advanced topic in math. The main prerequisite for it is "mathematical maturity". (Clearly calculus is not pre-requisite since it is just a rigorous rehash of calculus -- if anything, calculus depends on it.) For a text, you could try Spivak's calculus. Not Apostol and maybe Courant and definitely not any other calculus text that I know of. Outside of that, you could try A Course of Pure Mathematics by G H Hardy or possibly just a regular analysis text like Analysis With An Introduction to Proof by Stephen R Lay. But, frankly, I am finding it difficult to get there with my own kids. You really need to build up to it with an extremely proof intensive program. It just seems like it is going to take too long to get there -- maybe if I had them until they were twenty. (I guess, also, our oldest is kind of a worst case scenario, temprament-wise, and I am willing to force only so much math on him.)

Also, with regard to teaching real analysis, it is something you really need to know your proofs on and something, especially for someone with a background in calculus, that is really easy to think you got the problems right on when you were nowhere close. I would definitely try to teach it to myself first if you are going to do that. In fact, I would want a solid foundation in point-set topology, too. For instance, the Baire Category Theorem and Brower's Fixed Point theorem show up as exercises in The Principles of Mathematical Analysis by Walter Rudin. One big theorem, for instance, that is routinely skipped in calculus but that is finally proven and one's knowledge of and ability to reproduce such a proof is tested in real analysis is the Intermediate Value theorem. It is based on a lot of basic point-set topology like maybe what "connectedness" of subsets of the real numbers implies, for instance. I would think you would want to be really comfortable with that kind of material before teaching real analysis, yourself.

Adrian said...

Do you have any comments on the value of the Ross program for high school students?

Oh, you might be asking me because this is apparently a program at OSU. It looks like it came there after I left, though. But, it does look like quite a nice program. It sounds like they are trying to really do some math. Emphasizing number theory is a very tactically telling choice, too (that they might really be doing what they say they are). And, those are real mathematicians doing it, too. In fact, I think Daniel Shapiro taught one of my abstract algebra classes. "What?? You should have covered the Chinese Remainder Theorem in HIGH SCHOOL!!" That was the main thing I took away from that class. I am pretty sure he was 100% serious, too -- that's what made it so amusing. Maybe he was just deadpanning, though....

le radical galoisien said...

Differential equations isn't like terribly hard. The principle is pretty simple, and in fact it doesn't require a lot of creativity to grasp -- gee: the nth derivative of your function is dependent on the value of the function at that particular point. The only thing hard about it as I see is you need to take multivariable calculus and learn multiple integration before you attempt any diffEq's higher than order one.

I suppose I'm biased because I'm taking the course now, but I'm sort of timidly approaching analysis material -- linear algebra deals pretty extensively with transformations, isomorphisms and subspaces. For a class that doesn't use calculus that often (except for fun things like where integration and differentiation become linear transformations and where polynomials can be treated like vectors), there's a bit of imagination involved. Linear combinations remind me of (though they are critically different from) NP-complete problems, and proving elements of nth dimensional subspaces require a bit of mind-expanding. Subspaces are related conceptually to fields, rings and groups (I think?)

But of course as far as I know, linear algebra is "in the machine."

Anonymous said...

Ross @ Ohio State is a summer math camp.

It's phenomenal. It's the kind of thing that makes young, excited math kids WORK HARD for the first time in their lives. And they get to meet dozens of other math nerds. And some of them are so much smarter than anyone dreamed that they see how REALLY TALENTED someone can be.

I'll write a post about math camp in a few minutes, but Ross, PROMYS (at Boston U), Hampshire College, and a few others are the gold standard. Nearly every math major at Harvard and MIT has been to a math camp like Ross, and many of the science and engineering majors, too.

Anonymous said...

Yes, Ross, PROMYS, etc. teach number theory to teens. They teach more number theory in 2 summers than I learned as a math major at MIT. For real.

Adrian, Ross has been at OSU for a long time now, as my undergrad friends at MIT were high schoolers there in the 1980s. When did you leave OSU?

Anonymous said...

Adrian, you have a funny way of thinking about analysis.

Mark, "real analysis" is a math course where you start proving things about the math you've used. It's a course that teaches WHY calculus is true by proving it.

So you learn to prove things about continuity, diferentiability. You learn how to prove sets are open and closed. You learn how to deduce and induce what's true until you've proven all of the math behind calculus.

So of course you have to have had calculus first, because for you to prove something about a function, or the derivative of a function, you need to KNOW what a function and its derivative mean. (Adrian, what kind of math grad student were you?)

I don't think anyone NEEDS to learn real analysis in high school to be ready for math or physics or engineering at an MIT, Harvard, Stanford, etc. What they need is be ready to take a sophisticated calculus course and work hard to understand all of the problems.

But I'm going to post about math camps, and other things for those who are ahead of the curve in just a bit.

le radical galoisien said...

"early every math major at Harvard and MIT has been to a math camp like Ross"

No wonder there was no hope for me. :_( I could never afford those bloody things. (I suspect a great portion of of the immigrant kids can't either...)

Anonymous said...

Some of them have good financial aid packages, I hear...but how would you ever know in the first place, right?

le radical galoisien said...

At that time I didn't know what great math camps were. (In Singapore, usually teachers might recommend something to you and the school would sponsor you, sometimes with government EduSave funds.) So I couldn't differentiate between ripoffs and good programmes. The first envelope I got in the mail (during sophomore year) was for some ridiculously-expensive leadership conference thing. After receiving several more of similar letters to different "camps," [and because my family wasn't in a position to travel anywhere anyway] I took the unfortunate position of disregarding further offers that asked thousands of dollars.

Of course, come college application time and immediately overwhelmed by how my national peers outcompeted me I did begin to wonder if I should have taken a more serious look at them.

Adrian said...

So of course you have to have had calculus first, because for you to prove something about a function, or the derivative of a function, you need to KNOW what a function and its derivative mean. (Adrian, what kind of math grad student were you?)

What kind of a graduate student were you? You act like you learn any of that in calculus and like they just assume you know it in real analysis. There really is no actual knowledge prerequisite for real analysis. If there is one it is abstract algebra and topology. What you learn in calculus definitely isn't it. In fact, name one thing from calculus you can directly use in real analysis.

You absolutely do not have to do calculus before real analysis. I know of people who didn't, and most people that do learn calculus first (which is most people) have a big problem unlearning the approach to problems they were conditioned to take in calculus. What you can't really do is just skip in maturity level too much. And, this fact is borne out by the fact that most people just flunk out when it gets rigorous.

That is probably, in fact, why we do it the way we do -- so that more people "can do it". So we state all the theorems, give lots of relatively easy applications of them and say "you now know calculus". But, those applications are nothing compared to the proofs, especially of the tough spots like the Intermediate Value Theorem, for instance. We usually draw them a picture as if that is a proof. And, it is precisely years of that mentality they must overcome to make it in real analysis when they eventually get there. Frankly, while I most certainly could have done epsilon-delta proofs coming out of high school, I probably could not have done something like Baby Rudin (in a normal amount of time). Unfortunately, all the courses are like that, though -- designed for college seniors without much of a care as to what attrition rate there might be at that point. What is needed is a much slower systematic build up to it. But, of course, if they did that, then the physics majors might have to spend two years just getting to the point where they finally know how to integrate. And that right there is why we do it the way we do.

There is nothing you were taught in calculus that you will use in real analysis that will not be explicitly covered again (this time correctly) in real analysis. Compare The Principles of Mathematical Analysis by Walter Rudin ("Baby Rudin") to Calculus by Thomas and Finney if you don't believe me.

Anonymous said...

---What kind of a graduate student were you?

I wasn't a math grad student. I wasn't that good. I was a grad student in Computer Science, doing quantum computing.


--You act like you learn any of that in calculus and like they just assume you know it in real analysis.


I'm sorry, that's not at all what I think, and if I've said or acted as such I was way off. But what I meant was that if you haven't seen functions, or continuity, or differentiation, then proving things about continuity and differentiaion doesn't have any place to land in your head. There's no there there yet.

--- There really is no actual knowledge prerequisite for real analysis. If there is one it is abstract algebra and topology.

I don't know what this means--"actual knowledge". I mean, how does one GET the mathematical maturity needed to handle analysis if one hasn't taken some math courses, even lousy ones?

--What you learn in calculus definitely isn't it.

I don't know what this means, either. I am not suggesting that 2 terms of 9th edition thomas and finney equals knowing how to prove the square root of 2 is irrational. I'm suggesting that if you don't have a basis from which you can build maturity, you won't understand analysis. And so you need to have had some calc so you've at least heard about continuity, probably some differential equations or linear algebra where you'd get a firmer grasp of functions and operators as abstractions, and maybe some complex variables wouldn't hurt. If you'd learned how to do a proof in number theory, then a proof would be easier in analysis, but it's unclear how you'd get through number theory if you didn't know the rules for a proof.

--In fact, name one thing from calculus you can directly use in real analysis.

Uh, what does this "directly" mean? The fact that you know the definition of a function already means you aren't drowning trying to understand it in your first term of real analysis. The idea that you know what an irrational number is matters, too. The fact that differentiability implies continuity means you at least are familiar with continuity.

--You absolutely do not have to do calculus before real analysis. I know of people who didn't, and most people that do learn calculus first (which is most people) have a big problem unlearning the approach to problems they were conditioned to take in calculus.

Look, you seem to be arguing that I said calc was a sufficient condition for analysis. I wasn't. My point was that grasp of continuity and differentiability is a necessary condition--and you won't get that without calculus, even if you dont' get it with it.

This is quite a straw man to what I was saying. As I've made clear, I don't think high school calc has much if any rigor, and would totally agree that kids having done that haven't learned much.

--What you can't really do is just skip in maturity level too much.

Right. And where are you going to get the maturity from if you've never taken any college math? Bad high school math?
--There is nothing you were taught in calculus that you will use in real analysis that will not be explicitly covered again (this time correctly) in real analysis.

OF COURSE IT'S EXPLICITLY TAUGHT AGAIN. DUH. Because that's how you LEARN SOMETHING. And yes, it's taught differently because service classes for engineers have different needs than math majors. What this has to do with my point that you need to come into contact with these concepts is beyond me.

Tracy said...

This is a comment thread which has made me realise how little I know about maths.

Anonymous said...

"This means that if you follow the rules, and it's perilous not to, you are by definition throwing 80% of your class under the bus."

-- hey, this "paul b." fella
is pretty good. why was i not informed? v.

Catherine Johnson said...

I think I read Class years ago---I should pick it up again.

Catherine Johnson said...

This is a comment thread which has made me realise how little I know about maths.

I haven't read the thread (yet), but in my case realizing how little I know about math would actually be a step forward in what I know about math.

Catherine Johnson said...

Paul B is damn good!

Although he may wish us to pay for his Ferrari.

Catherine Johnson said...

The rest of you will have to handle that, because I'll be busy paying my share of our school's $50,500,500 school budget for school year 2008-2009.

total enrollment: appx. 1790 and falling

Catherine Johnson said...

Currently, the child in question is on track for Calculus by Freshman or Sophomore year in high school.

shoot me

Catherine Johnson said...

(congratulations, btw -- you have to tell us sometime how you're doing this - what curricula, how much time each day -- YOU MUST SHARE!)

Anonymous said...

"Me: Currently, the child in question is on track for Calculus by Freshman or Sophomore year in high school.

Catherine: shoot me

Catherine: (congratulations, btw -- you have to tell us sometime how you're doing this - what curricula, how much time each day -- YOU MUST SHARE!)
"

No real rocket science.
1) We are homeschooling and are using Singapore Math. My understanding is that SM gets kids to Algebra in 7th grade, so I'm figuring 11th grade can be Calculus.

2) Child in question is about 1½ years ahead in math [we just legally finished 1st grade this week and are plowing through SM 3a right now]. Blame mom for starting him early. We're pretty much going at one year per year now, so I figure he's on pace for Calculus 1½ years earlier than normal Singapore 11th grade.

3) Math is about 1 hour per day, broken into ½ hour in the early morning with me and ½ hour in the late morning or early afternoon with Mom.

No real magic here:
A) We are using a curriculum that runs about 2 years faster then a typical US curriculum,

B) Mom got a bit of jump even on that, and

C) We do math pretty much every school day (and because I'm mean and because he only spends about 2 - 2½ hours per day on schooling we don't take summers off. I don't see much of a need) and it tends to be a priority subject [it is the first subject we do in the morning almost every day].

-Mark Roulo