Schools Remarks on AP courses in high school

[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.

 

[moose]

Today I took my AP Calculus exam... The free response section was kind of challenging and fun.

EDIT: I'm dying in physics. In physics, we go over 40% of what will be on Monday's AP Physics exam. I haven't slept more than 5 hours in one night in the past two weeks because of that, biology, calculus, and some other things. I have almost everything for physics down though, I'm excited :)

 

[Pseudo Statistic]

Today I took my AP Calculus exam... The free response section was kind of challenging and fun.

EDIT: I'm dying in physics. In physics, we go over 40% of what will be on Monday's AP Physics exam. I haven't slept more than 5 hours in one night in the past two weeks because of that, biology, calculus, and some other things. I have almost everything for physics down though, I'm excited :)

Hmm... interesting..

A classmate of mine, who's a big math geek by the way, said that the AP Calculus exam screwed him over... makes me wonder what kind of questions came up. :D

What physics exam you doing BTW? Here I'm up for Physics B on Monday (Which I've spent quality time lubricating for...) and Physics C: Mechanics in like... 2 weeks. Physics C is what I'm not so worried about.

 

[mathwonk]

nice example curious. i think now i understand the whole problem with AP courses. The key issue is one raised earlier in posts by Hurkyl, motai and bomba, and also argued in a wordier way in my original essay.

Namely, AP courses have replaced honors courses. AP courses, which used to be meant foir honors level students, are now not challenging or
interesting enough to catch the imagination of the best students.


Maybe what we need to do is pose harder problems in our math couirses, even unsolved ones, to teach thinking and research habits, while it is still useful, i.e. while kids are young and bright and curious.

The worst thing we can do is take young kids and tell them their future depends on their performance on some standardized test, instead of their ability to solve hard problems.


Because then in grad school, we have students in their 20's asking us how to do research, since they have never done any in their studies.

 

[mathwonk]

and moose, I'm glad to hear there was a fun free response section on the calc exam, maybe they are getting better.

 

[EbolaPox]

I took the Calculus BC AP Exam yesterday, and I found the test to be pretty fun. Many of my classmates found it to be quite difficult and challenging, but it's a pretty good judge of computational skill. It is NOT meant to be a test to ask the student to prove anything, however.

Unfortunately, friends of mine in the IB program decided to take the AP Calculus AB (Calc I equivalent) test, and said it was "insanely difficult." Judging from tha tstatement, it appears that IB mathematics is even worse off than AP if the students that are in the top of the IB program where I live found the AB test extremely challenging.

However, what you say is true: many teachers focus purely on the student getting a minimum of a 3 on the test. In fact, there are only two students in my AP Calculus class (me included) that are actually interested in mathematics as a subject and do outside work. I've been reading Volume I of Calculus by Apostol and found it to be completely different from my textbook. Far more interesting and with problems that are actually enthralling, Apostol's seems to be better for a student that is interested in mathematics, as opposed to a student that merely want to exempt themselves from college calculus so they can get ahead in their college career or look better to admissions officers.

It would be nice if High School's offered both an AP Calculus AB/BC course and an "honors" level Calculus course that made the students do proofs and learn more than just the computational skills that are taught in the AP Course. However, I still support the AP program, it is probably the best thing that has happened to me in all of my education, besides my good math teachers.

(BTW: If you're curious, the CollegeBoard posts past AP Calculus BC and AB free response questions online. You can go see them and judge the test for yourself.)

 

[Jameson]

I just took AP Calc BC on Wednesday as well. To comment on a few things:

(1)My Calculus class will not do anything math related for the rest of the year now that the AP exam is over.

(2)The AP Calc test itself is not easy, but because it has to be so comprehensive, it's not rigorous. There are 45 multiple choice questions and 6 free response questions that are supposed to cover everything from limits to infinite series.

I'm attending the University of Florida this fall as a mathematics major and would like some advice. I'm confident I passed the AP Calc BC exam which would give me Calc I and II credit. Should I retake either course at UF?

 

[Cincinnatus]

You should not retake either course. There will probably be some sort of honors linear algebra or multivariable calculus course meant for first year math students. If its anything like where I go to school they will spend a good deal of time in these courses teaching things that you would have gotten in calc 1-2 had you taken them 30 years ago in a college but almost all high schools don't bother teaching.

Professors do understand that almost no students intending to major in mathematics today haven't taken these AP exams prior to coming to college and consequently are unlikely to take calc1-2 in college. So calc1-2 classes typically end up being "math for non-majors" type classes.

 

[G01]

I would say yes to be safe and for good measure to make sure you learned everything. My philosophy is to never test out of courses in your major. I think that can only hurt someone. Take the courses. You won't be behind anyone and it can only help your education. You don't want to chance going into higher level courses without enough prerequisit knowledge.

 

[omagdon7]

I go to UF both classes (calc 1 and 2) are a total waste of time if you know calculus. I retook the Honors version of the series after being in your position, just skip straight to calculus 3 and then start in on linear algebra. If you actually know calculus you will be fine.

 

[mathwonk]

I am not expert on UF, but it reminds me of the course I taught in high school a decade or so ago. The weakest student I had dropped my high school course in several variable calc because basically it was too hard for him.

He enrolled in UF and afterwards argued that he was right to drop my course because he "did fine" in calc at UF after only the basic AP course. So I got the impression that basic UF calc courses are pretty mickey mouse.

Of course we all have this problem in college because we HAVE to water down our courses to suit the weak preparation provided by most AP courses.

But usually, and I assume this is also true at UF, there are better courses available for people who want them, either honors versions, or some kind of higher level course.


(My stronger students, who stayed and worked in my course in high school, went to Harvard and Yale and Duke and UMass afterwards and did fine there also, in real calc courses, not joke ones that were spoiled by trying to please the AP crowd.)

 

[Jameson]

Mathwonk: I see your point. I'm definitely not trying to get by and only get credit. I want to learn mathematics as rigorously as possible. But it appears that UF Calc I and II don't offer the classes at a level where I would learn anything more than I already know on the subject.

omagdon7:Thanks for your advice. I think I'm going to go straight into multivariable and diff eq. Do you know anything else about the program? Anything about the Putnam competition?

 

[mathwonk]

well the fact is that there are college calc courses that are superfluous for someone who has done well in AP calc, I just don't teach too many of them myself.

We are forced to offer courses that our students can handle, and AP courses have forced us to lower the level of standard college courses to what average students can handle.

The point is that AP is a misnomer, i.e. they are not "Advanced placement" courses. One should not jump usually from AP to plain vanilla calc 3.

One should instead go from AP to a beginning course in calc at the elite honors level, i.e. a beginning course from Spivak or Apostol, if you can find one.

To go into ordinary calc 3 wil mean putting yourself in the same class with the weak students from UF or wherever that came up through the ordinary calc 1 and 2.

that way you never get above the non rigorous AL level of course.

 

[jbusc]

This is an interesting discussion for me as I did loads of AP's in high school (most of them self-study) including Calc BC (5) and both Physics C (5 & 5). All told I took 7 AP exams and got 6 fives and 1 four (in english) and got nearly a year's worth of elective credit.

Now that I've gone through 4 semesters of college though I'm definitely willing to say that "Advanced Placement" coursework is more of a "high-school plus" rather than true college level work. Given the option, I would have greatly preferred to take college classes rather than take more AP's - which is another consideration, since many high-schools and universities strongly insist on AP work rather than any regular college credits.

The point is that AP is a misnomer, i.e. they are not "Advanced placement" courses. One should not jump usually from AP to plain vanilla calc 3.

I did this semester, and I think I did well, getting an A (or at minimum an A-minus) in a class of 100 with a grade distribution centered around a B-minus average. *But* I'm not sure how much that's a function of the fact I learned calculus on my own and not through an AP course.

One should instead go from AP to a beginning course in calc at the elite honors level, i.e. a beginning course from Spivak or Apostol, if you can find one.

For physics I'm doing an honors sequence intended for people who did well on the Physics C exams, and it is not easy! I think even the regular 3-semester physics sequence here is more rigorous than the AP physics C, and the honors sequence is far, far more rigorous than anything I've seen before (the text is Ohanian, for those interested)

 

[Lucretius]

Well, on Monday I have my Physics B AP test. Wish me luck!

 

[Pseudo Statistic]

Good luck; I need it too. :P
I've decided not to go to school tomorrow (In the middle east our weekends are Thursday and Friday... don't ask) in order to study for the test... and to go to school at around 11AM on Monday so that I'm not too tired or anything. :)

 

[George Jones]

In the middle east our weekends are Thursday and Friday... don't ask

This makes perfect sense. The Saturday/Sunday weekend originated in the Christian world - the weekend is the day of worship and the day before. In the Islamic world, Friday is the day of worship (Khutbah/sermon and Jumu`ah prayers), so the weekend is Thursday/Friday.

Good luck to everyone!

Regards,
George

 

[Astronuc]

Interesting thread, Mathwonk! Thanks! And you are right on the mark. I could have said the same thing 30+ years ago, and the situation is not any better now than then - which is a sad indictment of general state of education in the US.

I took AP Calculus (BC), Chem, and Physics as a senior, and got 5, 5, 4. I placed out of the fresheman introductory courses, 1st year Math (Calc), Chemistry (but had to the Lab course), and one semester of Physics (I had to take the modern physics part which was intro QM). I did OK in Chem and Physics, and since I was a Physics major, I did not take any more courses in Chem afterward.

In Math, I jumped into an Honors Sophomore course in Linear Algebra and promptly got my a** kicked, so I dropped back to the general sophomore course in multivariable calculus and did OK. What I realized was that as rigourous as our AP Calc (BC) course was, there were certain areas we did not touch, which was unfortunate.

In our high school, the teacher for AP Calc BC was the department head and qualified math teacher with a degree in mathematics. She used a college level textbook, so we actually had the equivalent of a freshman intro calculus course, including ordinary diff EQ and integral calculus. When we learned about the derivative and differential calculus, we did the epsilon-delta proofs, and we did various theorems and proofs in differential and integral calculus. But that seems to be the exception, not the rule.

Similar, our high school chemistry course, was taught by the head of the department, and she had an MS in Chemistry. Again, that seems to be an exception, not the rule.

I would recommend that any student, who takes an AP class, simply go and talk to the appropriate professors and find out about the courses directly. Alternatively, toward the end of the semester, find out what texts are being used at the university one will attend, and then try to compare the college level texts with the one being used in high school.

 

[mathwonk]

jbusc, i did not mean you would never succeed by jumping from AP calc to ordinary calc III.

There are two scenarios likely there: 1) either someone is not too strong and then they will not succeed in calc III (this is the most common one), or else 2) the person is strong and then they are in too easy a course in calc III.

i.e. the weakly prepared AP student is out of their depth in even the ordinary calc II, and the well prepared AP student is in the shallow end of the pool in ordinary calc III.

the real honors student does not belong in the regular calc sequence at all, no matter whether at the I level or the III. Thus an AP course designed to let them skip calc I and go to calc II or II is missing the point of accurate placement for good stduents, and denying them the college course best suited to their needs.

In fact the ideal college course for strong students, the beginning spivak course, does not even exist at Harvard or many other schools today, precisely because so many strong students went the AP route, that the demand for this superior course disappeared.

A few schools, in cluding UGA in Athens, still offer this cousre to a handful of students. E.g. when I teach ordinary calc I or II, I try to identify misplaced strong students in the first few days.

As much as it pains me to lose them, if I have a strong stduent, I immediately advise them to get out of calc II and transfer to the beginning spivak course.

sadly, often they decline, because mommy and daddy want them to save the tuition from the one course that they have earned with AP credit and push on in the non honors track instead.

we could counter this by denying college credit for AP courses, and some coleges are beginning to do this, but we would lose money and talent as many students would go where they are offered that financial incentive.I did nolt always feel this way. In the past i have overheard professors in my department advising students not to take their ordinary calc I course if they have a 4 or 5 on AP but to go to honors. I was surprized as I used to try to teach even ordinary calc I at a higher level than that. I never advised anyone to skip my course. But over the years my course too has slipped down to the level that almost anyone can survive.

When a student comes to me for advice and says they have had AP calc, I may ask them to state the fundamental theorem of calculus, or the mean value theorem, with hypotheses. (it seldom gets as far as asking for the proof.) usually they cannot. If not, then i tell them they will probably learn something in my course, even the first one.

 

[mathwonk]

Last semester I had a very strong student in my ordinary calc II. I tried unsuccessfully to get him to transfer. Since there were also 4 or 5 other good students, I ratcheted up the course level slightly, as it seemed appropriate. I enjoyed it immensely for the talented group of students, but 19 out of 35 dropped out at midpoint, and 5 others got D's at the end.

I am not talking about an impossibly diffcult course since the top student averaged about 107/100. I.e., I set his grade at A+, not A. E.g. no proofs were required at all.

It is not at all unusual for us to have many students in calc II who do not know the product rule or the chain rule, who cannot write an equation for a line, do not know the derivative of sin or tan, cannot graph y = e^x, and even students who factor the 3 out of expressions like cos(3x) "=" 3cos(x) ! Seriously. And not being able to add fractions is very common. I have even had students who literally could not multiply 2 - digit numbers without a calculator.

It turned out my top student had been admitted to a top school but came to UGA for financial reasons. I believe such students belong at better schools for the camaraderie of other fine scholars, but if they choose to do so, they can also challenge themselves at UGA by following our advice.

 

[Astronuc]

It is not at all unusual for us to have many students in calc II who do not know the product rule or the chain rule, who cannot write an equation for a line, do not know the derivative of sin or tan, cannot graph y = e^x, and even students who factor the 3 out of expressions like cos(3x) "=" 3cos(x) ! Seriously. And not being able to add fractions is very common. I have even had students who literally could not multiply 2 - digit numbers without a calculator.

That's worrisome. We necessarily had to master the chain rule in my AP calc class and then use it to prove various identities. So we learned it very well. In fact, IIRC, we had to use the definition of the derivative to prove the chain rule.

The first part of our Calc class was elementary analysis and analytical geometry, and we had to know equations for line, plane, min distance between lines, distance between planes, closest distance between surfaces, conic sections, . . . . Somewhere in there, we did series (infinite series), continuity/discontinuity, then jumped into limits, then into the definition of the derivative.

The Calculus class was developed on the basis of our honor (major works) Algebra II class in which one year of advanced algebra and one year of trigonometry was crammed into one year - i.e. we did the equivalent of the one of algebra in one semester and one year of trig in the other semester during my junior year.

What I didn't see was coordination between Calculus and Physics, and it was not until university that I began to understand better the relationship between Physics and Math.