Remarks on AP courses in high school

[moose]

I myself hope to be running a program in a couple years at UGA for undergrads interested in algebraic geometry and I will be trying to recruit the brightest and most motivated undergrads I can find.

In a couple years? Where will I be able to apply :tongue2:

There are only two people in my AP Physics who know how to do a problem without being told how to do it earlier, even though they have enough knowledge to solve it. Same goes for my Calculus class, generally speaking. This angers me for several reasons, including that this causes the teachers not to asign such problems. :/

EDIT: Chroot, in some way, I agree with you because many of the students are simply not able to do what they are asked there. However, in several other countries, students do get that sort of education and they generally, by the time they graduate high school, are extremely bright. In Poland for example, which doesn't rank extremely high educational wise, random people who I met were brilliant. The type which I only see once every so often over here. This of course isn't solely because of the math, but it certainly does help. People over there are taught to think for themselves and are expected to. Here, the only reason that mathwonk's special courses in high school may not be successful to the non mathematician wannabe's, is that they were never prepared to think for themselves.

 

[mathwonk]

Warren, you seem to think that only mathematicians want to understand math. That everyone else just wants toe arn a living. You may be right but what a sad state of affairs for the life of the mind in America if that is so.

As I said, I also want to understand physics, and literature. To paraphrase the great Richard Feynman in his lectures on Physics; I am going to teach you as IF you were going to become a physicist, and so is every other teacher here going to treat you that way in his course, because that is the best way to learn.

 

[chroot]

Warren, you seem to think that only mathematicians want to understand math. That everyone else just wants toe arn a living. You may be right but what a sad state of affairs for the life of the mind in America if that is so.

It's a sad state of affairs? No, it's not. That's a ridiculous statement. The sheer volume of knowledge available to mankind has exploded in the last hundred years. The amount of knowledge required to do many modern jobs -- physician, engineer, etc. -- is immense. The education process has to change with the times.

If you want students to cope with this enormous increase in requisite professional education, continue to graduate in their twenties and go on to contribute in their chosen professional field, you have to start trimming somewhere. Or, are you of the opinion that students should begin graduating a year or two later than they do now, just so they can get the "great benefit" of a rigorous mathematical education?

There's no way we have to time to educate all our children as if they are all going to become mathematicians; it's a waste, it's pointless, and their time is better spent on other subjects that will actually help them in their careers. Save the deep math for the people who will actually benefit from it.

I will continue to make the point that you suffer from hubris in thinking that your subject is somehow much more important than any other. Do you advocate teaching "deep chemistry" and "deep history" and "deep english" with equal ardor?

- Warren

 

[Dr Transport]

My orginal comments were made because I was in graduate school in an area of the US where I actually encountered students who although were carrying very good averages taking AP courses (3.9 out of 4.0) their parents were transferred within their company to another part of the country. Some of the students I ran into were set back 1-2 years in high school because their new state of residence tested them and placed them there. Their parents lived in two areas of the country because it was detrimental to move the high school students that would be nearly 20 before they got thrier high school diploma.

Actually saw it happen to a kid, she was going to be a senior, started to move, got tested in what was to be her new high school. The school placed her in the sophomore class and said that to earn her diploma, she needed to take 3 years worth of courses. Mom stayed behind in the old house with her and allowed her to complete her final year and graduate.

Sad but true.

 

[Hurkyl]

There's no way we have to time to educate all our children as if they are all going to become mathematicians; it's a waste, it's pointless, and their time is better spent on other subjects that will actually help them in their careers.

If you're going to advocate that only a little bit of time be spent on mathematics, then shouldn't it be spent wisely? If I had to choose between:

(1) Teaching algebra deeply enough so they understand it, and might even be able to apply it in the odd circumstance.

(2) Breezing through algebra so there is time to put them in a calculus class. (Which they won't understand because it's not taught in depth either, and their poor algebra skills are an obstacle too)

then I would advocate choice (1).

 

[chroot]

If you're going to advocate that only a little bit of time be spent on mathematics, then shouldn't it be spent wisely? If I had to choose between:

(1) Teaching algebra deeply enough so they understand it, and might even be able to apply it in the odd circumstance.

(2) Breezing through algebra so there is time to put them in a calculus class. (Which they won't understand because it's not taught in depth either, and their poor algebra skills are an obstacle too)

then I would advocate choice (1).

Again, for most technical professions (engineering, computer science, etc.) breadth is much more important than depth.

It would certainly be ideal if we could tailor each and every student's education to the his/her future career goals, but psychics aren't real.

Note also that we're not talking about "breezing through" algebra, or putting the students in a calc class they can't follow. We're talking about teaching mathematician-level rigor to every student, regardless of whether not they might ever be in a position to use it.

Industry is going to prefer graduates who know a little about a lot of subjects, rather than graduates who know a lot about a few subjects.

- Warren

 

[0rthodontist]

I have been reading this for a while. I have a few comments:

I read somewhere that European high schools introduce material sooner, rather than going the simple-but-deep route. Of course maybe the coverage is also deeper, since they spend more time on it.

Depending on what type of computer science you do, it can require a lot of mathematical rigor.

Math is more important than most other study. It is nothing less than how to think about carefully defined things in a sound manner. In my opinion logic should be a core curriculum in kindergarten.

 

[moose]

Also, everything covered in a non honors, normal physics course in high school was also covered in 7th grade, but nobody remembers it. They also don't remember that they understood it then and did well with it then, and now they fail physics. People get dumber or they start thinking they can't do it because the teachers make it seem like it can happen that people won't know what to do.

 

[Hurkyl]

I wasn't equating "depth" with "rigor" -- the two aren't synonymous.

Note also that we're not talking about "breezing through" algebra, or putting the students in a calc class they can't follow.

You see, that's exactly what I thought we were talking about! I've often heard gripes from teachers about how students manage to get into their calculus classes with completely inadequate algebra skills. When I tutored, I was astonished how often I had to tell someone in a second semester calculus class how to do arithmetic with fractions! And heaven forbid they wanted to solve for something that appeared twice in an equation, or in a denominator!



And I recall being told that, historically, mathematics courses were where students were supposed to learn and refine their problem solving skills. If that is supposed to be true, then there should be rigor in primary school mathematics courses.

 

[bomba923]

When I tutored, I was astonished how often I had to tell someone in a second semester calculus class how to do arithmetic with fractions! And heaven forbid they wanted to solve for something that appeared twice in an equation, or in a denominator!

Same things occurred when I was tutoring others--they lacked certain prerequisite knowledge and skill. However, I am a high school student and I tutored other high school students; but Hurkyl...were your tutees college students?

Grade inflation (coupled with low standards...and/or, teachers with low standards) allows students to pass into courses that their level of competence would otherwise proscribe (not permit).

And I recall being told that, historically, mathematics courses were where students were supposed to learn and refine their problem solving skills. If that is supposed to be true, then there should be rigor in primary school mathematics courses.

Right, though there is some debate on how "much" rigor should be enforced in those primary courses. And even then...you'll always teachers with their own perspectives (regardless of accepted/decided standards) and independent methods of grading (so those debates...would not really affect much for the teacher).

However, I do have a plan to improve this, and more...see https://www.physicsforums.com/showthread.php?t=104494

 

[Manchot]

You know what I think? A lot of times when older people say things like, "Today's students aren't as good as yesterday's," they're just showing a cognitive bias. It's easy to forget what you knew at a certain age, because it all blends together. For example, when I think about taking algebra-based physics in high school (just three years ago), I find that I have a tendency to insert calculus into the mix, when I didn't really learn it until the following school year. The further back that you go, the worse it gets. Can I tell you precisely when I learned what a prime number is? No. All I can say is that it was sometime between 3rd and 6th grade. However, even one year can make a huge difference when it comes to a subject like math. I can imagine how easy it would be for a group of older people to decide that today's students aren't up to par, because a sort of mob mentality sets in as well.

 

[chroot]

Manchot,

There are studies going back to the early 1800's that have shown that people have always felt things are going downhill with the younger generation. It's human nature.

- Warren

 

[moose]

The sad thing is, studies at my high school show that the incoming students are far worse off than we were three years ago. It isn't a small change either, it's drastic. Nearly all the freshmen talk about drugs *all the time* and talk about "the last time that a cop searched them" or "the last time I was arrested...blah blah blah".

Aside from that, did you know that there are online ap calculus courses available to take?...

EDIT: And online PE, how does that work!??!?

 

[gravenewworld]

The biggest factor on the quality of an AP course is WHO IS TEACHING IT. The AP calc class I had in high school was, hands down, the hardest math class I have ever taken in my life (and I just graduated with a math BS). I am not ashamed to admit that on my AP calc tests in high school I would score in the 30s (out of 100) on tests. I got a D for the 3rd quarter when we started covered integration. Luckily though I had a teacher who won awards from the White House for excellence in math education, and when it came time to take the AP I scored a 5. As a matter of fact, every student who took his AP calc class scored a 5 for 10 years straight. My teacher had taught AP calc for well over 20 years, had every AP exam from all the way back to like the 40s, and even was a grader for teh AP exam.

I skipped Calc I and II in college and did fine. As a matter of fact, Calc III and diff eqs. were joke courses compared the AP calc class I had in high school. I wish I had some of the tests I took in high school to show you how rediculously hard they were. My calc teacher in high school taught us many tricks and "black magic math" as he would call it. Heck I even knew some tricks that none of my professors even knew in college. LOL i was surprised that some of my professors didn't know Horner's algorithm or some of the tricks I learned for synthetic division which i learned in high school.

 

[mathwonk]

what puzzles me is why my opinions make you angry, chroot.

why not just state your own opinions without attacking me personally? as a lousy teacher and so on. or claiming that merely having my opinions makes me presumptuous, shows my hubris, etc. i don't attack you for differing with me.

when you do that you leave the realm of logic and also stop offering useful advice to readers here. they don't care of you disagree with me, they just want useful information. it also suggests that you need heat as opposed to light to make your point.

lets try to keep the discussion on the topic instead of making it personal.

in answer to moose, who asked about summer programs, the one i plan to teach will take place in summer 2007 if it is funded, and i will try to announce it here when we get news.

in the meantime i strongly support the PROMYS program at BU going on this summer, as a great place to learn to understand and do number theory.

but in case there can be any misunderstanding, it is not a program for people who just want to put another entry on their college application form, it is for people who are actually curious and want to learn something from excellent teachers.

 

[mathwonk]

there is another argument used here which is common to people i tend to disagree with, and it goes " i myself did not have such and such, or i did have such and such, and i did fine in college" or life, or whatever.

this is not at all an argument that someone else, or even the speaker himself, might not benefit from having a different experience.

for example some of the posters here has used this sort of argument and i have deduced some of them are from certain colleges and so i have looked at the website of those colleges to discern some information about their math offerings in which the posters "did fine".

some of these colleges list course offerings which are literally teaching grammar school math and junior high math, some for college credit, in the first 10 or 20 or 30 of their courses listings.

But the course offered at harvard in 1960 to the best prepared freshmen, is listed as a 400 or 4000 level course at many of these schools.

well of course a student with AP preparation does fine in courses of the current freshman or even sophomore level of difficulty, but years ago, before the AP revolution destroyed the better algebra and geometry course preparation in high schools, (some) freshmen were prepared for real calculus courses in college, courses that almost do not exist anymore.


one might also ask for the definition of "does fine". If it just means got an A in a shallow course, this is not my definition of fine. more like "suffered deception and consequent delusions of success".

as i have stated here before, the only time i ever got an A+ in a college math course i celebrated for one day and then concluded that I must have been in too easy a course, and transferred into another level of course work. I am much prouder of the sucessive B+ and A- from that harder course sequence than of the A+ in the easier course.

but to respond to another recent plaint again, the low level of preparation of young people is not the fault of the young people themselves, as they are not responsible for setting the standards low. it is their elders who are doing so.

We need to inspire young people, and older people too, to want to understand the world around them, not just in math, but in chemistry, English, history and all other fields. We need to remind people of the joy of thinking deeply, as opposed to treating all learning as a utilitarian pursuit.

we are not asking people to be mathematicians when we ask them to try to understand a mathematical argument the way a mathematician would. to say a car salesman does not "need" to understand algebra, is beside the point. he might enjoy doing so, and might have a more colorful and happy and productive life.

as i recall fermat was a jurist, not a mathematician by trade.

 

[mathwonk]

in spite of my explanation now repeated at least twice, that this thread was aimed at people who are planning on becoming mathematicians, several readers may be forgiven, because of my overly general title, for interpreting them as meant for every rough beast now slouching toward bethelem to be born. hence i propose to start over with a more blunt and restrictive title in a new thread, with invaluable tips for that tiny minority of misfits who either wish to be mathematicians, or at least learn to think like one. it ahs been noticed by astute observers that indeed all 2,686 of my previous essays seem to have the same goal, but i shall announce it this time plainly, so the modest or timid may avert their eyes.

 

[gravenewworld]

there is another argument used here which is common to people i tend to disagree with, and it goes " i myself did not have such and such, or i did have such and such, and i did fine in college" or life, or whatever.

this is not at all an argument that someone else, or even the speaker himself, might not benefit from having a different experience.

for example some of the posters here has used this sort of argument and i have deduced some of them are from certain colleges and so i have looked at the website of those colleges to discern some information about their math offerings in which the posters "did fine".

some of these colleges list course offerings which are literally teaching grammar school math and junior high math, some for college credit, in the first 10 or 20 or 30 of their courses listings.

But the course offered at harvard in 1960 to the best prepared freshmen, is listed as a 400 or 4000 level course at many of these schools.

well of course a student with AP preparation does fine in courses of the current freshman or even sophomore level of difficulty, but years ago, before the AP revolution destroyed the better algebra and geometry course preparation in high schools, (some) freshmen were prepared for real calculus courses in college, courses that almost do not exist anymore.


one might also ask for the definition of "does fine". If it just means got an A in a shallow course, this is not my definition of fine. more like "suffered deception and consequent delusions of success".

as i have stated here before, the only time i ever got an A+ in a college math course i celebrated for one day and then concluded that I must have been in too easy a course, and transferred into another level of course work. I am much prouder of the sucessive B+ and A- from that harder course sequence than of the A+ in the easier course.

but to respond to another recent plaint again, the low level of preparation of young people is not the fault of the young people themselves, as they are not responsible for setting the standards low. it is their elders who are doing so.

We need to inspire young people, and older people too, to want to understand the world around them, not just in math, but in chemistry, English, history and all other fields. We need to remind people of the joy of thinking deeply, as opposed to treating all learning as a utilitarian pursuit.

we are not asking people to be mathematicians when we ask them to try to understand a mathematical argument the way a mathematician would. to say a car salesman does not "need" to understand algebra, is beside the point. he might enjoy doing so, and might have a more colorful and happy and productive life.

as i recall fermat was a jurist, not a mathematician by trade.

Yes I didn't take Calc I or II in college and "did fine" in all of my math classes. Hell, I haven't even ever taken a class on trigonometry and did fine. I only took Algebra I & II and geometry and then took AP Calc.

And I "did fine" in classes such as

-real and complex analysis
-hilbert spaces
-formal/mathematical logic
-group/ring/field theory
-combinatorics
-number theory
-game theory
and even dominated in the graduate level courses I took.

I said it before and I will say it again. It all depends on the background you come from. If you actually have quality teachers teaching AP Calc, high school students would actuallly be better prepared for college and could easily skip calc I and II. Demanding a rigor on the level of epsilon-delta proofs in an AP calc class is just absurd. AP calc and even Calc I and II are just supposed to be introductions to calculus. The theory is suposed to be saved for later. There are colleges that offer high school math for credit like "descrete math" and "college algebra", but I can pretty much guarantee you that those classes would not count for a math/engineering/science major at those schools.

why is it that older people are always doom and gloom? f

 

[gracetodd]

I have never heard such honesty about our AP idealogy. At my high school, I have heard some teachers complain that the idea of an AP course was to make students think. They are in the minority. It is understood that we are to get the students to pass so we look good. The students are mostly average, some above average, a couple real scholars.
We are teaching them awful habits. They get the idea that they can take a slew of practice MC tests all year until April. Then they cram for hours upon hours, day and night, over Easter Break, weekends, etc. Then they take the test, then they watch movies the rest of the year.
I asked to teach calculus next year, an AP course. Then I talked with the current calc teacher, with whom I would be working. I said I wanted induction on the Algebra II test. He said it was a waste of time. He also refuses to teach the entire section on matrices erquired in Algebra II.
I am hoping to teach stats now. There won't be the pressure of having to take the AP test or of working with those AP teachers whose idealogies are opposite of mine.
Grace

 

[gracetodd]

I did not interpret mathwonks message to be anything about young people at all. It was about education taking the wrong direction. As a math teacher I am constantly asked why do we have to learn this...which I no longer answer. I tell them it is to make them think, it does for your brain what weight lifting does for your body. It grows connections. It makes you reason better. I don't care if they like math anymore, only that they learn to repect it for what it's worth.
The NCLB is making us teach 5 year olds to read. Then in middle school we don't careif they can multiply without a calculator. `The problem is not the students, it's those in charge of education, and that certainly is not us teachers anymore.
Grace

 

[0rthodontist]

There are colleges that offer high school math for credit like "descrete math" and "college algebra", but I can pretty much guarantee you that those classes would not count for a math/engineering/science major at those schools.

College level discrete math is central to a computer science degree to the extent that my computer science department teaches it within the department. The math department version of it is also a requirement for a math degree. Discrete math is taught at the high school level but I really doubt that any reasonable college discrete math course is the same course.

 

[tmc]

discrete math can be a very hard course, and would definitely count for credit at any university.

Saying the "discrete math" course is a joke because you are taught discrete math in high school, is like saying the university's "stats" course, or "algebra" courses are pointless.

 

[bomba923]

We are teaching them awful habits. They get the idea that they can take a slew of practice MC tests all year until April. Then they cram for hours upon hours, day and night, over Easter Break, weekends, etc.

And that is they're problem, not the problem of "AP Ideology".

However an individual wishes to prepare for a known test date (in this case, known a year in advance) is UP TO THEM. It is his/her own individual responsibility to prepare for the test.

You must understand that studying is a CHOICE. When and how a person decides to study...is their choice. If he/she decides to study steadily throughout the year, so be it! If he/she decides to cram the night before...so be it! The students are responsible for their own preparation, especially if they CHOSE to take the course.

If students wish to study steadily throughout the year...good for them. If they wish to cram the night before...well, good for them. They CHOSE to take the course, and it is THEIR responsibility to be prepared for the test! Just as people who post in PF's IR must be respective of the guidelines they agree to, so must students be prepared to take the exams they have agreed to (a year in advance, btw).

It always annoys me how modern education always seeks excuses and ways to absolve the student of any individual academic responsibility.

Then they take the test, then they watch movies the rest of the year.

:uhh: Ok...good for them. The purpose of an AP course is to prepare students for the AP test. It's no small coincidence that such incentives result in a more rigorous approach to teaching and educating students (which is what we need).

I asked to teach calculus next year, an AP course. Then I talked with the current calc teacher, with whom I would be working. I said I wanted induction on the Algebra II test. He said it was a waste of time.
He also refuses to teach the entire section on matrices required in Algebra II.

Induction on the AlgebraII test seems like a good idea. But how would you implement it? IIRC, it is MC-only.

 

[Hurkyl]

Ok...good for them. The purpose of an AP course is to prepare students for the AP test.

No, it's not! It's this very perception that people are rejecting.

The purpose of the AP course is to teach the students about a subject. The problem arises when people forget that, and start thinking that the AP course is merely supposed to teach them how to pass an AP test.

 

[mathwonk]

my son's AP calc teacher, in an expensive private school in Atlanta, stopped teaching math altogether the last 2-3 weeks of the course because the AP test had already been given, and she showed movies of Jane Austen novels instead, because she said, "no one wants to study math now that the test is over". gosh silly me, i thought maybe they might want to learn the subject, since they would need it in college and we were paying her to teach it to them.

My son is not to blame for this, the school thinks AP courses are just badges on the arm for college admissions, and that we want them to prepare our children for prestigious colllege admissions rather than to learn.


Hurkyl's attitude is of cousre the ideal one, but to my knowledge it is almost non existent in high schools today.

 

[shmoe]

discrete math can be a very hard course, and would definitely count for credit at any university.

"discrete math" can also be a very simple class. The title conveys nothing at all about the value of such a course, a blanket statement like it would "definitely count for credit" is very wrong- it really depends on the individual course.


I took AP calculus. Most of the class only cared about getting a university credit or getting a head start on their university class (that is they planned to take first year calculus as an "easy class" regardless of their AP score). My teacher was good, but didn't stray much beyond the curriculum, though he did make himself available for questions not directly relating to the course.

My fist year calculus was a common one for all undergrads, not an honours one. I can't say this really built a deep understanding of anything. It wasn't until 3rd year analysis that the goods came to the table. After doing things 'properly' in the "analysis" stream, every other math course seemed so much simpler after the skills that were needed to make it through analysis. I would have loved it if this had come earlier in my education, but I can't blame anyone for this oversight. I knew where the library was.

I do believe that forcing this kind of deeper learning on non-math majors would have advantages over a broader but less thorough exposure to math. Most of the math my friends in engineering had to cope with seemed trivial in comparison with what I had done in analysis, and I didn't have much problems learning what I needed to in order to help them out with the math parts of their studies (provided they could distill their problems into math ones).

 

[bomba923]

No, it's not! It's this very perception that people are rejecting.

The purpose of the AP course is to teach the students about a subject. The problem arises when people forget that, and start thinking that the AP course is merely supposed to teach them how to pass an AP test.

~NO, that's what HONORS classes are for!

AP classes are designed to prepare students for the AP exam.

 

[0rthodontist]

Discrete math is an entire branch of mathematics. If not every course that offers an introduction to it is for science/math credit, then 99% must be.

 

[motai]

AP classes are designed to prepare students for the AP exam.

That's the problem with AP courses. They teach you how to play the game, to resubstitute answers back into questions, to learn the tricks that would shift only a few students towards the right side of the bell curve. Thats not my only gripe about them though.

AP courses (I have seen anyway) benefit affluent schools who are able to hire phenomenal teachers. Having come from a small, rural school, there wasn't much selection of AP courses, and what ones we had, the quality was only marginally better than the non-AP courses. Our best course at our high school was AP American History, which had a 75% passing rate with most of those scores being a 3.

In comparison, some of my classmates in college had AP classes with pass rates of 100%, some of them with 4s and 5s (with a few rare ones getting 3s).

Are those students who did extremely well necessarily more intelligent, or was it the result of coming from a well-funded, excellent school?

 

[mathwonk]

the point of an academic education is to learn to think critically, imagine creatively, and express yourself persuasively and clearly. perhaps to formulate and solve problems.

any course that teaches these things is useful. others are not. courses designed to prepare you to pass a canned test written by trained monkeys is useless.

 

[mathwonk]

A bit of data: I was just looking at my grade roll from fall, to write a letter of recommendation for a good student in integral calc, and noticed that 19 out of 35 entering students, all with AP credit for differential calc, had been forced to withdraw from the course at midpoint, failing.

Of the 16 who remained, 5 earned D's. Grades would have been worse but I dropped 2 low test scores out of 4 tests, and gave extra points on tests, so the 4 best students scored over 100.

That gives you some idea of the value of an AP course as preparation for college calculus, and for skipping a college version of beginning calculus.

Those were decent high school students, who were misled by this whole AP system into thinking they already understood college calculus. The falsehood that AP courses substitute for good college courses, is a disservice to most students, and that fact needs to be better understood.

Think about it: a course is roughly as valuable as the expertise of the teacher. As a state school professor, I am a researcher, with over 30 published research papers totalling several hundred pages, some in top journals, over 50 national and international speaking invitations, and over 35 years university teaching experience.

My own former calculus professor at an Ivy league school, is a legendary and still internationally famous researcher, a member of the National Academy of Sciences, and a Wolf prize holder.

There are exceptions, but normally a high school AP calculus teacher is just someone who took calculus in college. That's it. That teacher may be willing, bright, and experienced, but to expect that course to substitute for a good university course is optimistic at best. It is very unlikely the teacher will know much more than is in a standard book, and probably a good deal less.If you want to master a subject, find the best qualified teachers you can to study with. This is the opposite of the AP philosophy. Don't be a sucker.

As Opus said to the lady who didn't want to renew her subscription to the paper because she got all her news from Bill O'Reilly, "yes, and I get all my nutrition from deep fried ding dongs!"

Get your math education from someone who understands math.

 

[mathwonk]

i am beginning to regret the last post, as it appears nothing quiets the crowd like pulling rank.

for months, even years, I have patiently posted my personal "wisdom" anonymously, with implicit faith in the power of logic, only to be countered repeatedly by ridiculous arguments from people undeterred by having little information or data.

then i say, "hey i am a big time (or medium time) professor", and suddenly some people seem to think, "well gosh maybe he does know something. I can't respond to that."how depressing. or maybe they just noticed i had at last gone round the bend, and gave up on me. my apologies to all.

remember, you do not have to have a PhD or publish papers to be correct. after all Galois was a punk kid with a table knife.peace.

 

[mathwonk]

let me give you an example of the difference between a typical high school AP calculus course, and the beginning calculus course I had in college.

On the first homework assignment, after we had been told the definition of a least upper bound, the professor gave us a bunch of sets of real numbers to compute the least upper bounds of.

one of them was the set of all prime numbers n such that n+2 is also prime.

In how many high school classes do they assign homework problems whose answers are unknown?

Unfortunately for me I knew this was a famous open problem, and I had never been challenged in high school to believe I might one day do something new, so I did not attempt it.

Probably I will never again be as creative and intelligent as I was then, and it would have been better had I tried it at the time. Who knows, someone might get it someday.

But if all you aspire to is a 5 on the AP test, then of course you do not want a course like this where they actually expect you to think.

 

[Curious3141]

let me give you an example of the difference between a typical high school AP calculus course, and the beginning calculus course I had in college.

On the first homework assignment, after we had been told the definition of a least upper bound, the professor gave us a bunch of sets of real numbers to compute the least upper bounds of.

one of them was the set of all prime numbers n such that n+2 is also prime.

In how many high school classes do they assign homework problems whose answers are unknown?

Unfortunately for me I knew this was a famous open problem, and I had never been challenged in high school to believe I might one day do something new, so I did not attempt it.

Probably I will never again be as creative and intelligent as I was then, and it would have been better had I tried it at the time. Who knows, someone might get it someday.

But if all you aspire to is a 5 on the AP test, then of course you do not want a course like this where they actually expect you to think.

Interesting you should mention that. Once when I was around 18 (and doing conscripted military service, meaning loads of free time), I tutored a schoolkid around the age of 12 or so. There was a "starred" problem in his Math workbook couched in simple language :

Here's a simple algorithm : If a number is even, divide it by two. If it's odd, multiply by three and add one. Start again with the new number you get.

Determine if the cycle goes to one when you start with (a few numbers are given as examples here) ?

The first two numbers reduced to the trivial cycle (4,2,1) easily. The last number they gave (27) seemed to be getting nowhere fast.

So I left it there and returned to my military post (which was at the Defence Ministry). I grabbed hold of the nearest computer (AT 286s on Windows 3.1, at the time) and wrote a short C program that proved the third number went to one, but took 111 iterations to do so ! I printed out the path of numbers and faxed it to the student (these were the days before email or the Internet had really taken off).

I tried and tried to figure out a "simple" way to prove the cycle always reduced to (4,2,1) but couldn't.

Of course, later on (when the Internet was better established), I discovered that this problem was actually the Collatz conjecture, a famous open problem.

So, it is not unheard of for open problems to be posed, even at an elementary level. I agree with you that it is nice to have unclouded and fresh insights looking into these problems.

 

[franznietzsche]

In comparison, some of my classmates in college had AP classes with pass rates of 100%, some of them with 4s and 5s (with a few rare ones getting 3s).

Are those students who did extremely well necessarily more intelligent, or was it the result of coming from a well-funded, excellent school?

Some of it may just be the position of a school within the district. My high school was the only one in the district which had the IB program, and as such had a much larger concentration of honors students than any of the other schools in the district. The school was not better funded than the other schools (at least not outside of the honors curricula), but it had amassed almost all of the exceptional students from the entire district. Such a thing is not uncommon with IB high schools in their districts, in california anyway, as there isn't often more than one or two per district of seven or eight schools.