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Remarks on AP courses in high school

by mathwonk
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devious_
#19
Mar29-06, 09:02 PM
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EbolaPox:

It depends on the school, but honors-level courses usually cover the same material as their non-honors counterparts but at a much more deeper and enriched level, and sometimes introduce more advanced topics. For example, at my school engineers and scientists have to take their own versions of calculus courses, which are as you would expect very application-oriented; mathematicians have the option of taking honors calculus or advanced honors calculus. The honors-level courses introduced calculus with a very light emphasis on rigor, so you would see things like epsilon-delta arguments, and the proofs of the theorems are more 'intuitive' than 'formal'.

The advanced honors course, however, feels like an analysis course. Every theorem is proved formally, and the material is presented at a much deeper level. For example, the honors course will introduce sequences from the point of view of the real line whereas the advanced honors course will treat them in general metric spaces with the real line as a simple special case.

And usually most first and second year courses have an honors-level equivalent. Of course, as I said, this varies from place to place.

As for Apostol, I think his linear algebra section is lacking (at least in the first volume). I would seriously suggest you look elsewhere (e.g. Friedberg et. al). But that's just my opinion (and keep in mind that I'm not a big fan of his text, so I admit to some bias ).

The introduction of integration before differentiation is historically more accurate, and you can see this in many other, older texts (e.g. Courant and, I think, Hardy). However I also recommend you read another book (like Spivak or Courant) for integration as a supplementary text to Apostol. Because if I recall correctly, he insists on using step functions to introduce the theory of integration, while the others use upper & lower sums. Both approaches are of course equivalent, but you might benefit from sampling both and choosing the one you like more.
mathwonk
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Mar29-06, 09:04 PM
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chicago may count as an "ivy" but they did teach from spivak a few years ago; of course time flies for an old xxxx like me so that may be decades ago now.

but you can always read it yourself.

and when i was a young instructor at central washignton state college in ellensburg washingtom, i taught an honmors calc course from spivak for free on top of my usual load.

so even my students at state college in ellensburg got that course in 1972.

you get whatever people offer wherever they offer it. and you find very interesting people everyuwhere.

at ellensburg i found colleagues very anxious to learn the new stuff they had missed in college and i also ran seminars in the de rham theorem in cohomology and sheaves.

i also taught there from chern's notes on diff geom from berkeley.

of course i got fired eventually for not having a PhD, so cynics may say that my comments abiout knowledge matering more than union dues are wrong.

but the modifier is you must not go too far down the totem pole iof you want to be appreciated. if you are in aplace where people have no idea what you are talking about, then true you will not be appreciated for your knowledge, and then you need a degree.

but if you go somewhere they do understand you, then you can survive on performnace. however there is a balance to be maintained. when you go home where they do not understand what you are sayiong they will ask where are yopur publications, so to survivbe in the real world it is true one should get good grades and publish, but one must not focus exclusively on these trivialities.
moose
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Mar29-06, 09:05 PM
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Chroot, did everyone* in your IB curriculum courses make it out as well as you did?

Oh, and the lazyness thing, AP Physics has 9 students right now, and the average number of them who do their homework on any given night is probably three.
EbolaPox
#22
Mar29-06, 09:06 PM
P: 101
From what I have seen of IB, the only classes worth taking are the HL level courses. I've looked at the material covered in the IB SL Mathematics course, and it is simply algebra II and some Calculus I. I tutored some friends in the class and found the material to be quite simple. The IB HL Chemistry course was on par with what I studied in AP Chemistry (which, according to many universities, is equivalent to Chemistry I and II. I suppose that all depends upon the university.)

Although this is purely anecdotal analysis, IB seems to force students to do large quantities of pointless busy work and internal assesments and other nonsense assignments. When I was tutoring the mathematics students, they had no clue about riemann sums or limits with epsilon-delta definitions. I was rather shocked, as they had claimed IB mathematics was at least on par with AP Calculus AB.

However, if one goes through the IB Diploma program, I'm sure they'll come out with just as much, if not more, work-ethic, experience, and knowledge as one that did an equal amount of AP work. So far, I've got around 45 credits from AP and am taking 6 more AP exams this year.

With respect to the Linear Algebra, my friend happened to have Linear Algebra by Friedberg that he let me borrow (he was too lazy to read it apparently.) I assume it would be best to study LA out of Friedberg and not Apostol? I'll probably end up reading both just for the extra practice.
chroot
#23
Mar29-06, 09:13 PM
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Quote Quote by mathwonk
so there is a huge difference between students whose goal is to get by, and those whose goal is to become mathematicians or scholars. i am there for the latter,
You seem to have this idea that every single student should have some deep, burning passion to become a mathematician before they ever grace the hallowed sanctuary that is your... high school math class. You've indicated this in many, many posts here over the years.

Bull****.

I often think you're more than just a bit presumptuous, man. Most US high school kids don't even go to college (it's a pity, but it's true). Probably fewer than one in a hundred of your students will choose to pursue math as a career. Few, if any of them know enough about mathematics yet to really have developed a passion for it, anyway. Most of them literally do want nothing more than just to pass your class, and that's okay! Some of them don't share your goals or opinions about the One True Path to living their lives.

You're teaching kids, man. Teenagers. People who don't understand how to drive stick shift cars, cook anything with more than two ingredients, or have meaningful interpersonal realtionships. These are not mathematicians, nor should be expected to behave like mathematicians.

Your job is decidedly not to focus on the kids who will one day prove the Riemann hypothesis. Your job, as you are paid by the state, is to provide a fair, reasonable education in basic mathematics that will serve the greater good of society. That's it. You apparently suffer from crippling delusions of grandeur that, in my opinion, probably make you a lousy teacher.

Get off your high horse and stop acting like you're teaching doctoral students at Oxford. You're a high-school teacher. Get a grip on reality, and teach the damn kids (even the dumb ones) some math.

- Warren
chroot
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Mar29-06, 09:15 PM
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Quote Quote by moose
Chroot, did everyone* in your IB curriculum courses make it out as well as you did?
The program started with 400 kids. 42 took the IB exams. 32 got their diplomas.

Oh, and the lazyness thing, AP Physics has 9 students right now, and the average number of them who do their homework on any given night is probably three.
Things really, really must have changed in the last few years.

- Warren
mathwonk
#25
Mar29-06, 09:20 PM
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Chroot: have you mistaken me for someone else? i am not a high school teacher i am a university research mathematician who agrees to teach high school honors classes for free, for those few students who want the best. if they do not want what i am offering they do not belong in my class. in particular i never teach regular classes and i am not paid one cent for what i do in high schools.

As i have stated here before, i taught elite private high school students who applied to be in my course, writing a special NSF grant which paid them a salary to attend my class, while i did it for free.

i like you, but you are wayyy... off base here. In particular everything you have said about me tonight seems to be totally false, and could have been known to be so from the mildest acquantance with my posts "over the years."

i am sure you meant well, but before attacking someone you might bother to get at least some of your facts straight.

please forgive me if i have embarrassed you, as your comments, although totally wrong and misplaced (except that I am an arrogant prick on a high horse, which of course is a given), were obviously meant as sympathetic to students. so you are probably a good teacher.

but, surprize, so am i.
devious_
#26
Mar29-06, 09:24 PM
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chroot:

I think mathwonk is not talking about the average high school "kid" but instead those who DO have the "deep, burning passion to become [mathematicians]." He's talking about offering an advanced class to those who want to take it.

And he's not a high school teacher, either. So yeah...
mathwonk
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Mar29-06, 10:00 PM
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i must admit chroot, i have also written impassioned responses without having read the previous posts. may i suggest you read the earlier posts in this thread? i apologize for their length.
mathwonk
#28
Mar30-06, 01:01 PM
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perhaps it was not clear that these notes were aimed at people who actually want to understand mathematics, e.g. to become mathematicians, or at least to master the subject as well as possible, rather than simply get by in school.

i thought that was apparent by my citing the inspiration of zapper's essays on preparing to be a physicist. i might have titled them "so you want to be a mathemnatician" but it seemed presumptuous to pretend they would have the same value as his notes.

at any rate, i hope the angry outburst they have inspired does not deter anyone from making a comment or asking a question. i have noted over the years that poeple who believe in and care about teaching, almost always argue strongly over how it should be done in theory.

curiously these same people tend not to argue when they actually observe what the other is doing in the classroom. i.e. the difficulty is in understanding what is meant by what is said.

moreover, most of us have buried memories of unpleasant experiences which are set off at times by someone else's comment. we may think "that XXX sounds just like the guy who made my life miserable in junior high." and off we go.

anyway, all comments are welcome here, even contrary or critical ones.
chroot
#29
Mar30-06, 04:02 PM
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Okay, mathwonk, I apologize for my outburst. On the other hand, I *have* read a number of posts from you, over a long period of time, that bemoan the sad state of high-school students -- even the bright ones.

If these high-school classes of yours inspire so much animosity in you, maybe you should just stop teaching them? It doesn't sound like you have (or permit?) many students in the classes anyway, so it wouldn't affect many students. You might be happier for it. Trust me, teenaged kids are generally not all that much fun to work with, even if you have a noble purpose in mind.

- Warren
z-component
#30
Mar30-06, 04:35 PM
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My AP Physics class has about 20 students. I'd say about 70% of them just took the class because they liked Physics I but took it for the purpose of interest only, with no desire to take anything more than Mechanics or something for their major at university. About 10% think that they want to be engineers, myself included, and no more than ten or so students from two AP classes making up a total of about 40 are extraordinary with AP level physics. Those ten or so students are the only ones taking the AP test in May, and that's all. My school district wants to make it manditory for students to take the AP test if they take an AP course, which I highly disagree with. Overall the AP courses in my district are top-notch. I believe AP courses are wonderful, and if it were up to me, I'd make them manditory! It's disgusting how many kids are just plain out stupid. Although, you look at kids who take Calc BC and compare them to schools in the 1970's where you had one algebra class and one geometry class. Now precalculus is manditory, calculus and statistics are optional.

If anything, AP courses teach you problem-solving skills and prepare you for the college workload. I agree with Warren's statement about the IB curriculum preparing him for college, even if the AP curriculum is a bit different.
moose
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Mar30-06, 05:49 PM
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z-component, I do not wish them to be mandatory, at ALL. What I love most about my AP classes is that the students are generally smart.


BTW, if you look at a good high school in europe, almost every single student will be able to pwn almost every student in any AP class in the US.
z-component
#32
Mar30-06, 06:23 PM
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But that, of course, doesn't surprise me at all. It should be easy to pwn American high school students. But really, when will this change? We're a world superpower and the world's policeman, but we have kids dumber than a box of hammers.
bomba923
#33
Mar30-06, 06:59 PM
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That's where my public education reform solution comes into play
mathwonk
#34
Mar30-06, 07:11 PM
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z component, i think i disagree with your comparison between todays AP prepared students and yesteryear's high school students who took only algebra and geometry, at least if you mean that todays students are better prepared.

Namely I think yesterday's kids were much better prepared for college calculus and beyond, by taking algebra and geometry thoroughly, than are todays kids who take AP calculus and learn it shallowly, and most college professors of math I know agree with me.

In fact, high school AP courses were so inferior as preparation for our old college courses that we have had to dumb down our courses significantly to accomodate AP students into them. That may be why students today think they are well prepared, i.e. we have lowered our expectation as to meet their incoming knowledge, so in fact if we are successful, then they ARE well prepared for what we are offering them, because what we are offering is so much easier than it used to be.

I regret if this passes for animosity toward high school students, for it is not, as they are not to blame for this situation. Nonetheless this is the current state of affairs.

Here is a wonderful article interviewing a college math professor Glenn Stevens, who I knew when he was a grad student at Harvard, about his program PROMYS at BU, where bright young high school kids are taught not advanced math, but deep elementary math.

http://www.bu.edu/phpbin/news-cms/news/?dept=4&id=38201

Notice he is reserved in his comments about AP courses, perhaps to avoid sparking controversy, but you can guess at his opinion of them there.

PROMYS is a terrific program, perhaps especially for future mathematicians, which has also been expanded to teachers. it teaches math by actually engaging people with it, not just memorizing it.

Actually I think this program good for anyone wanting to understand math, and I am puzzled by comments that seem to assume only future mathematicians should be encouraged to understand math. I for example would dearly like to understand physics, and also literature and music. I do not think such experiences are only of interest to future writers, critics, physicists, or performers.

When I was a postdoc at Harvard, Glenn and his fellow students ran a student seminar in number theory that was very active. The other programs had faculty seminars, but these kids had the monopoly on stduent run activities. They really learned a lot and were obviously dedicated to their subject and to learning it together in an untheatening and enthusiastic way.

another great summer program is the one at Park City utah, combining activities for undergrads, high school tecahers, grad students and researchers.

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.

and thanks for your note chroot, you are a good egg and i appreciate the slack in spite of my sins.
chroot
#35
Mar30-06, 07:29 PM
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Again, mathwonk, the vast majority of students taking AP Calculus are not actually going to become mathematicians (nor even attempt to become mathematicians).

I will agree with you that a shallow teaching of calculus benefits a future mathematician less than a deep teaching of earlier math -- but most students are not future mathematicians!

The typical AP Cal student, who does not intend to become a mathematician, may well be better served by a shallow teaching of calculus than by a deep teaching of elementary math. After all, even most "technical" professions, like engineering and computer science, require very little mathematical rigor, even though they make enormous use of mathematics. Why teach everyone a very high degree of mathematical rigor when it's wasted on the vast majority? Why not allow them to study the techniques that will directly benefit them in terms of job performance, at the sake of some depth?

Again, you seem to have some tacit belief that the entirety of high-school math education should be designed around the tiny minority of students who will become mathematicians. I (not so tacitly) disagree.

- Warren
moose
#36
Mar30-06, 07:37 PM
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Quote Quote by mathwonk
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

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 knowlege 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.


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