remarks on AP courses in high school

[mathwonk]

The following remarks were written by me to my sons' high school community some years ago. I believe they are still relevant. They were then considered so heretical that they were not acceptable as publishable.

The idea of posting these remarks was inspired by Zapper Z's extensive and helpful essays on preparation for future physicists, and the response from students requesting guidance for preparing for mathematics and that at the high school level.

Some remarks on high school preparation for a college education:

Some time ago I argued that XXX could not be considered a particularly "hard" school in comparison with many others, because it had so few AP courses. Now that this is changing, I have begun to have some reservations. I had hoped AP courses would strengthen the program by upgrading weaker course offerings, rather than competing against the excellent courses already in place. I believe that in the country as a whole, this may have been a principal result of the proliferation of AP courses, tending to a sort of standardization of advanced instruction, bringing a reduction in quality of education at good high schools, rather than an overall upgrading of the level of the average course offering. I also did not realize that graduates of "Advanced Placement" courses would take the term too literally and try to place out of substantive courses in college which they should have taken. In the case of those AP students who repeat beginning college courses I have also found the problem of trying to teach in depth a college subject to people who think they have already learned it.

I am most familiar with mathematics, which I teach at the University of Georgia, so I use that example. The AP designation in calculus refers to a specific list of "topics" on which one must be prepared to work problems. A year of this AP material coincides with the content of one or two quarters of non honors college calculus at Georgia, but a full year college course, and especially an honors course, not only covers more ground, but treats the material at greater depth. It is ironic that AP calculus courses, which are taken by honors high school students, are comparable at best to non honors college courses, which the best such students would not elect.

As a result many entering AP college students place either into advanced, but less stimulating, non honors courses, or into intermediate honors college mathematics courses for which they are not prepared. Before the AP revolution, students prepared by getting a better grounding in algebra and geometry (and sometimes logic) than is found in high schools today, then took a first year college calculus course which included theory. Introductory college calculus courses for gifted mathematics students which teach theory as well as computation are hard to find today because so many students exempt this course with AP credit. The disappearance of the most outstanding introductory college calculus courses is thus a direct result of the proliferation of significantly inferior AP courses.

In view of its unsuitability, it is ironic that AP credit has begun to be designated as the "prerequisite" to some advanced courses, even though the true prerequisite for advanced work is often just the ability to think in a certain way. This may be the case even when the college catalog says otherwise. At Stanford for example, the prerequisite listed recently for honors intermediate calculus is a certain score on the AP calculus exam, but when asked, the departmental advisor said "Of course that's not the real prerequisite" (his emphasis). The real prerequisite? "To be able to handle proofs, with no apology". The book used in that course is volume 2 of Apostol, an outstanding text treating calculus with theory. Presumably the right preparation is to learn beginning calculus from volume 1 of Apostol, but where can the interested student find such a course? Stanford does not offer it (that's the course that was replaced by the AP courses), and it certainly is not available in most high school AP classes; (books used in the XXX course are ordinarily one or even two levels of sophistication below Apostol). The result of this at Stanford is roughly a 70% attrition rate (after the first week!) in the honors intermediate calculus class, among those students who have the required score on the AP test. Surely many of those students who must drop out are disappointed that they are not in fact prepared for the course, and possibly the career, they had wanted.


Unfortunately AP calculus courses and the standardized testing mentality have helped to eliminate, not just from the college Freshman mathematics curriculum but also from high schools, classes in which theory and proof (i.e. systematic logical reasoning) are taught, since "proofs" are seldom included on AP tests. For example the 1982 and 1987 AP BC calculus tests in my practice book have less than 3% proof questions, whereas the exams in the Stanford course above are said to be100% proofs. This phenomenon has accompanied years of decline, and the current near extinction, of adequate teaching of geometry in high school, which worsens the problem of learning either calculus or deductive reasoning.

I conjecture that these negative effects are not so great in some subjects where AP exemption is less common. For instance my impression is that in the recent past students from XXX's non AP honors English courses have been superbly prepared for beginning college courses in that subject. Presumably the reason is that in these classes, students learn to read, write, and discuss their ideas. I hope these courses are never replaced by ones designed to prepare people to answer multiple choice questions on the correct author of some obscure poem.

Interestingly, although our data at the University of Georgia shows no correlation (and even some negative correlation!), between scores on the quantitative SAT test and performance in our precalculus and basic non honors calculus courses, there does apparently exist a positive correlation (almost a direct one) with scores on the verbal SAT test. My own theory is that the verbal test, as bad as it is, at least measures vocabulary (mathematics is a language), and the ability to comprehend what one has read. Consequently the demise not only of instruction in reasoning in mathematics, but the decline in the ability of the average student to read and write, has steadily tracked the drop in performance also in basic mathematics courses for non honors students. (One might even argue from this that the claim that Saxon's books raise SATQ test scores, also suggests that they may lower average performance in college mathematics courses.)

What is my conclusion? I suggest the school seriously reconsider the practice of creating AP courses in subjects which are already represented by excellent honors courses, since this may well lead to the demise of the superior course, and a decrease in the quality of student preparation. In subjects where AP courses already compete with non AP courses, I strongly urge students to select the course which involves the most writing, and the deepest analysis, without regard to which one boasts a syllabus sanctioned by the Educational Testing Service. In cases where an AP course has already driven a superior course out of existence, I feel there is a strong argument for creating, or recreating, a non AP honors alternative.

I do not oppose taking AP classes in principle, but since (in my experience) they do not play an appropriate role in advanced college placement, I do believe they must justify themselves based simply on their educational merits. I also strongly suggest that a graduate of an AP class consider taking an introductory college honors course in the same subject rather than skipping the introductory course altogether.

The only case in which I see a reason to consider creating an AP class is in a subject where the existing course work is currently on an inappropriately low level. Even in such cases I think it likely that a non AP honors course designed by the teacher may be an even better option. In my opinion such an opportunity exists at XXX in the physics program, which I understand does not ordinarily offer a calculus based course. One possible way to make good use of the existing AP calculus course would be to offer a subsequent or concurrent calculus based physics course, or even a course that combined the two subjects. Since Newton invented calculus precisely to do physics, this is one of the best possible ways to learn both physics and calculus.

From my own perspective I believe there is also a real need for new substantive mathematics courses which are not just oriented towards performance on standardized tests. When I tell my colleagues at the University of Georgia that XXX does not offer a year long course in geometry for example, they do not readily believe me. I would also like to see innovative, faculty - sponsored, courses on other subjects of current or abiding importance in mathematics and related areas, such as linear algebra (an easier and more fundamental subject than calculus), finite mathematics and probabilty, computer programming, algorithms, numerical analysis, or computer aided design.

In general, I believe those of us who are "consumers" of XXX educations, parents and students, should have faith in the knowledge and scholarship of the teachers; these outstanding individuals should be considered at least as qualified to select the content of their courses as the faceless people who write standardized tests. Some of these teachers value and use an AP syllabus in their own courses, which is a recommendation to me of the positive aspects of some AP programs. Others prefer to design their own curricula. Such distinctive courses offer opportunities unique to XXX, and I believe they play a large role in the school's impressively successful identity. Some teachers may even be holding back exciting proposals thinking we want only standardized education from them. I hope such individually conceived courses will continue to be encouraged, and valued for the rare gems that they are.

For the students who must enroll in the courses if they are to survive, I suggest you remember primarily to try to educate yourself. In particular try not to let the quest for a flawless GPA prevent you from studying subjects you find difficult. Even if science courses are hard for you, how much can you understand about our world if you don't know at least something of biology, chemistry, physics, and (yes) mathematics? If you would enjoy going to Paris, or Madrid, it would help to speak French or Spanish. If you think art and music classes are not valuable, you might think about how you are going to create a beautiful environment in your apartment or home, or your life, without such knowledge.

Now what about the "real world" of getting into college or getting a job? Is it practical to just go along learning to read, write, reflect, analyze, discuss, and play, when you fear that college admissions officials are going to judge you based mainly on your standardized test profile? May I respectfully suggest we all try not to hyperventilate over college admission. From my own experience as a college professor, and reader and writer of countless recommendation letters, I recommend to you to be curious, to be diligent, and to pursue activities for which you have real enthusiasm. If you have a genuine enjoyment for learning, if you have thought deeply about any significant topic, if you have worked hard to accomplish something in any area, if you can express yourself well and have practiced discussing your ideas with others, it will come through in your college essay or interview as well as in your letters of recommendation.

I believe too, the tight job market in higher education over the last couple of decades means more and more colleges are now assembling the most qualified faculties they have ever had. Certainly this is true in mathematics. If you honestly embrace your XXX education, I believe you are virtually assured of admission to a college which offers more than any one person can possibly absorb. This is borne out by a glance at recent lists of admissions, and by speaking with recent graduates. Since there is a shortage of well qualified students at most colleges, there may be even a slight danger that you will get into a school which is actually too challenging.

And if after all you find yourself in a situation where you seem to need credentials you don't have? A positive attitude always helps overcome gaps in your vita. I have often been inspired by a story my mother told me about her interview for a secretarial job she needed badly during the great depression. When asked if she had any experience, she said "No, but I can learn to do anything anybody else can do." She got the job. You can too.

Roy Smith
:cool:

[EbolaPox]

As a student that has taken a total of 13 AP courses (currently 7) and received 5s on most of the 6 AP exams I've already taken (I received 4's on two of them, 5s on the rest), I absolutely agree with your statement on AP Calculus. AP Calculus BC is my favorite subject and by far the most fascinating subject I've ever taken, but the actual test itself lacks any requirement for proof or analytic skills. My mathematics teachers for Calculus, however, have attempted to go above the standard AP Calculus BC curriculum and show my class logic behind proofs and always works out proofs for new subjects. (However, a school nearby that offers IB Mathematics, a course that is supposedly equivalent, completely lacks any proof.)

Your analysis of AP courses in mathematics is correct: most lack proofs and rigor. However, my AP Chemistry class seemed to cover exactly what a Chemistry I and II course would cover in a University. Same with my histories and Language and Composition class. Therefore, I feel that it is mostly the Mathematics that is lacking rigor. The others are equivalent to University standards.

I am planning on double majoring in Mathematics and Physics. I love the subjects and I can entertain myself with a multivariable calculus text or a linear algebra book any day. However, I am still rather concerned that I am lacking a rigorous knowledge of Calculus (Although I was able to do an AP Calc BC practice exam with missing but one or two multiple choice questions, I still feel that I lack the rigor that I desire.) I found that my teacher has a copy of Apostol's Calculus Volume I. As I see you mentioned that in your post, I was curious: would that be a sagacious place to start learning a more rigorous basis in Calculus? A Differential Equations teacher at a local College suggested I read a book on Real Analysis if I could find one.

Any suggestions for a young, zealous, math-loving high school student? (Any book suggestions will also be appreciated.) Thank You.

[mathwonk]

apostol is a superb choice.

[Cincinnatus]

Mathwonk is right, some schools that have good honors classes probably shouldn't bother with AP. However, for some schools like the one I went to, AP was essential.

I went to one of those stereotypical urban high schools with a 50% drop-out rate and all that. I took almost all AP courses in high school (including calc BC), did well, and learned a lot. Not everything my college professors assumed I knew coming in, but I wasn't so far behind that I wasn't able to catch up.

One year, when I took chemistry there was no AP version offered and I had to take the regular class. It was quite an experience, I learned literally nothing at all in that class. Nothing was taught! it was all about doing review problems to pass the ridiculously easy state exam. (Which many people failed).

So, if the alternative to the nationally mandated AP curriculum is what I saw, then AP is the only choice worth making really.

[mathwonk]

That is similar to the sad story i have heard from other students, that bad as AP courses are, the non AP courses are often worse, especially in poor schools.

Another problem however is the fact that in math, AP calculus courses have largely displaced more elementary, but more useful courses in algebra and geometry, including proofs.

I.e. there is no AP course in high school Euclidean geometry, so high schools often simply stop teaching classical geometry in favor of AP calculus.


As a college professor I would prefer a student have a good solid 1950's style geometry course, with proofs, than an AP calculus course.

Indeed without geometry it is impossible to understand calculus. And without algebra, it is impossible to do calculus.

Indeed there is no need at all for a student to have calculus before entering many colleges, but much need to master algebra and geometry.

I take this back: physics students would be wells erved to have ahd calculus before taking college physics. Hence it is reasonable to have taken a high school calculus course to use in physics, but one should still take college calculus anyway, to learn calculus more deeply.

I also know many high schools have courses they call algebra and geometry, but the quality of those courses is vastly inferior to what went on in years gone by.

The best high school curriculum is the one pioneered in the 1960's by the SMSG program accompanied by excellent books from Yale. These are no longer in print. They did include calculus, but they preceded calculus by excellent courses in geometry, algebra, statistics, and linear algebra and matrices.

These courses are far more useful and important than calculus for most people, but since they are not sanctioned by the ETS as AP courses they are not offered.

I.e. the SMSG curriculum was designed by experts to actually teach high school students what they needed to know. On the contrary the AP curriculum in math is a very inferior one, whose purpose seems to be to make the ETS rich, to give students bragging rights, and to make college admissions simpler but less effective.

[Cincinnatus]

Its only getting worse too, I have heard that after I left, the high school I went to has decided to start calculus even earliar. They now spend a year and a half doing calculus, at the expense of everything else.

Even, the way it was when I was there, I graduated without having ever seen... induction, matrices, vectors, the definition of a limit and lots of other things. Surely they are even worse now, spending an extra half a year teaching "Pre-AP Calculus"

[EbolaPox]

It is quite true that Non-AP courses are so vastly different from AP courses(at least in my experience), that one cannot even compare the magnitude of work and knowledge required. Therefore, despite the AP Math curriculum's lack of rigor, it is still an excellent choice for any student that desires to learn more than the average high school student. However, as was previously stated, most students I know are merely taking AP classes to look good for admissions into competitive universities.

Just years before I entered high school, my school eliminated a class it had on proofs and logic, a class I would gladly take to further enlighten myself. (Many of the AP mathematics and science teachers argued that a proofs class was integral to doing well in mathematics later in life, but for some reason, the class was removed.)

Many Geometry classes still have an introduction to proofs and require students to write proofs. I was rather deprived of this, unfortunately (my class didn't have time to do proofs, apparently.)

Yes, it is quite true that ETS is making gargantuan quantities of money via its AP program, but it is the best choice for any high school student in the United States. (Some may argue IB, however.) Despite its lack of mathematical rigor, it is very rare that one would recieve a more intellectually augmenting class in a non-AP class. Although, of course, it is impossible to prove something via anecdote, I shall provide an example. My school offers a set of Engineering courses. As an excited future physics and math major, I gleefully joined and witnessed the most dull and uninteresting class. Algebra I was the most "difficult" math required, and many students groaned when they heard it would require any form of math. The fact remains that many students despise mathematics, and for that very reason, the number of students desiring advanced mathematics decreases and therefore the supply of upper-level rigorous mathematical courses is low (at least in my areas and from those I've spoken to in several different states.)

Therefore, it comes down to the fact that the best choice for most students is to take a curriculum that maximizes AP courses and the student self-studies what s/he misses. Many high schools offer "Dual Enrollment" programs where students may go to a university or community college nearby and take some courses. Perhaps that would offer a more rigorous mathematics courses as opposed to AP calculus.

I highly doubt the ETS will be editing their AP Calculus exams any time soon (although, they did recently add Slope Fields and separable differential equations onto the AB Exam.) It would be beneficial for students if teachers would encourage them to read more advanced books or work out proofs on their own (as my teachers have tried.) However, I highly doubt the ETS will augment their AP Exams to require students to write proofs. It would be nice if ETS made an AP Linear Algebra Exam. Personally, I think that would be good for students. If it is not up to par with a universities curriculum, said university should NOT offer credit for the course. Simple as that. If the university feels that the AP exam credit will exempt the student from a class worth taking, the university should simply not accept the credit.</rant on AP>

[scorpa]

My school didn't offer AP or IB courses, so as I result we were forced to take courses designed with the abilities of the average students in mind. I still did ok once I got into my first set of university courses, it just took a lot of hard work. However, here the people who took AP or IB courses for the most part elect to take the courses that they technically could qualify to be exempt from. I think it's a good idea on their part, but unfortunately it really screws over the rest of us who didn't have the same opportunities as they did, it really throws off the curve when you have people who have basically done the course already against those who are seeing the material for the first time.

[mathwonk]

I have never seen an AP calculus course on a par with the non honors calculus course I teach, and I have never seen even the best AP student who should exempt my honors calculus course.

The only reason we offer credit for it, is not because it compares well to our own courses, but because parents expect it.

And in the case of my own childrens school, then AP revolution ahd a decidely negative effect on quality. That was an excellent private school, whose own courses, in English and History and sociology especially, were greatly superior to AP courses.

But college admissions officials look for AP courses so they phased out vastly superior courses for them. As to math, I myself taught vector calculus at this high school, well beyond any AP course, buit the school did ,not appreciate it because they were mroe interested in preparing stduents for the AP courses than for some higher level courses that would not be appreciated by the admissions office.


Of 6 in my class, one of my ex students is now full professor of math at brown, and 2 of them got science PhD's, one in math one in physics.

but i do not offer that course any more, so no other students have that opportunity.

[EbolaPox]

Out of curiousity, what does an Honors-level Calculus course cover? Is there an Honors version of Differential, Integral, and Multivariable Calculus? What topics does it cover in addition? Is it a proof-based class as opposed to the simple working out exercises that is offered in an AP course?

I've begun reading Apostol's Volume I Calculus from the very beginning. So far, it's an excellent text and has introduced me to sets and I've been working all of the exercises and proofs. One thing I find odd is that it introduces Integration before differentiation. Is there any particular reason the book seems to do it?

I'm aiming for a strong, rigorous knowledge of Calculus, Linear Algebra, and any other topic of mathematics. I've gone through differential equation books and was entertained (seemed easy enough), but I realize that I have never had a proof-based mathematics course. A question: I see that Apostol's Volume I has Linear Algebra towards the end of the book. Is the Linear Algebra section equivalent to what would be taught in a standard LA course?

Thanks for answering any questions, and thank you for suggesting Apostol. So far, an excellent text.

[Dr Transport]

I have often wondered why the schools and parents are pushing these AP courses. When I was in grad school, I had kids who claimed to have taken all of the AP courses in their high school, many times they were not the brightest students out there and for the most part were amongst the laziest. A dose of reality was imparted on them after the first mid-term exam where they scored in the low half of the curve.

[mathwonk]

to dr transport:

when i taught "beyond calculus" in a private high school, i found i had several lazy and ungifted students. it turned out their parents insisted they take all "honors" classes offered. after my class their parents attacked me for "poor teaching:. nonetheless my gifted students achieved the results i have described above.

to be fair those ungifted stduents cruised in their non honors state college calc courses. 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, i.e. that is why i do this hard work for below market wages (my teaching in high school was voluntary), but am willing to teach anyone who will work hard.

i do not wish to waste my time on students who do not want to understand. i could care less what college they want to get into to please mom and dad.

to ebolapox: i cover whatever i damn well want to in my honors calc course, guided by my professional standards, which means what I think is in line with the syllabus, the needs of the students, and their gifts, and my interests.

last time i taught honors integral calc, i taught the generalized fundamental theorem of calculus often mentioned here, that a lipschitz continuous function which is differentiable almost everywhere, with derivative equal to the value of a riemann integrable function at points where that function is continuous, does equal the indefinite integral of that function.

in particular we studied the contrast between continuity, uniform continuity, and lipschitz continuity.

i also treated power series, and their differentiability and integrability, as a subtopic of convergence of continuous functions in the uniform norm. this was for honors freshmen in college. i ask you to find me a high school AP course anywhere near this in level, outside of the bronx high school of science.

my notes are available, perhaps i can put them here.

[courtrigrad]

I want to major in applied math. I have gone through the texts of Courant and Apostol a few years ago. But I have never read Spivak. Do any universities besides the "Ivies" teach calculus from these books?

[mathwonk]

my notes will not reproduce here until I scan them it seems. but i covered also the cauchy criterion, and pointwise convergence of functions as compared to uniform convergence.

this was a shock to my entering freshmen with theoir measly AP background in calculus, and I advised the universit to stop giving honors credit for AP courses, since my scholars knew little or nothing of rigorous convergence criteria or continuity a la epsilon and delta.

[mathwonk]

if you have actually mastered most of apostol and couranmt you have no need of any other books on calculus. it is all there in courant, and more rigorously and in modern language in apostol.

the "super honors" freshman course sequence at university of georgia, math 2400-2410 is taught every year from spivak, but NOT at harvard or stanford, whose students think they do not need it.

[moose]

I'm currently taking AP Calculus AB. A few days ago, I was looking over some example problems from a Calc AB AP test and was shocked at how completely trivial the questions were. I'm just glad that my teacher doesn't simply try to get the class to do well on the test, but actually wants us to know something...

The best thing I see about AP courses is that they almost force the teacher to teach something. I have taken non AP courses which cover absolutely nothing, like my programming class. It's mid semester and the teacher hasn't yet spoken a word about programming, ever. He tells us to take his code and play with it. There are only about three people in the class who have any idea what they are doing. Someone could take a good programming book and learn far more in a day than if you sat in every programming class the teacher has ever taught. The teacher gives no initiative whatsoever, people can get by with an A without ever touching a single program, they can just turn in what he gave them. I have talked to him about this and he simply says that he once tried to teach a class something a few years ago, and it never led to anything so he prefers to just let the class have fun with his programs... What a horrible teacher!


EDIT: That was weird, three new posts since I began typing this O_o
Many teachers don't seem to realize that if they expect more, they will get more, if they expect less, they will get far less. My earth/space science teacher freshmen year expect tons from the students. Every concept which was covered in physics last semester, was covered in more detail in earth/space science. In earth/space science, people understood the concepts instantly as well. In physics, the teacher didn't expect people to understand things and went through things incredibly slow. Wow I got off track....

In most AP courses however, the teacher is pressured a lot more to actually do something and teach something....

I'm still looking for a good program/class to truly need to work and think in. None of the classes I have ever taken are truly challenging.

[EbolaPox]

I do not claim that an AP Course is anywhere near the level of mathematical rigor that you have taught in your Calculus course. If I have made such a statement, I have made it in error and apologize.

Furthermore, Dr. Transport's statement that many students in AP are incredibly lazy is quite true. All of my friends are either in AP or IB(International Baccaleurate) and they all adhere to the idea that "through laziness, we shall succeed." I'm quite different. I take classes such as AP Physics, Calculus, Differential Equations, etc. out of pure love for the subject and a desire to understand. I do wish to become a mathematician and I truly hope I have the intellect and capability to become one. Therefore, I have tried to take the best math classes my school offers, and that would happen to be AP Calculus BC. Do to the fact that I have a 104 in the class and it is nothing but a review for me, I was enthralled when you mentioned that a more rigorous version of calculus existed and I may be able to learn it through Apostol and hard work.

If you happen to have any notes that you could post, I would be very thankful. As you may have guessed, we have neither covered nor mentioned Lipschitz continuity (Although I did read a post of yours in which you mentioned it).

[chroot]

I dunno guys. I took the IB curriculum, placed out of 56 hours (almost two years) of undergraduate courses in everything from math to foreign language. And I had no problem going on to graduate cum laude with a degree in computer engineering and two minors in math and astrophysics.

Perhaps AP and IB are not as comparable as I thought, or perhaps I was lucky. I busted my ass in high school. In fact, I worked harder in high school than I ever did in any but my senior year of college. It was an incredible amount of work, and I gained study and time-management skills unheard of for a high-school student.

I think the IB program prepared me decidedly better than the non-IB courses offered at my high school.

- Warren

[devious_]

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 :tongue2:).

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]

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]

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]

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]

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]

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. :uhh:

- Warren

[mathwonk]

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_]

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... :rolleyes:

[mathwonk]

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]

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]

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]

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]

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]

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]

That's where my public education reform (http://www.physicsforums.com/showthread.php?t=104494) solution comes into play :approve:

[mathwonk]

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.:smile:

[chroot]

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]

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 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.

[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 transfered 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 dont 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 wont 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 occured 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 http://www.physicsforums.com/showthread.php?t=104494 :approve:

[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.:smile:

[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, highschool 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 highschool 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 highschool 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" irregardless 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! :wink:

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 ! :surprised 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.:smile:

[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 wont be behind anyone and it can only help your education. You dont 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.

[Astronuc]

. . . . the generalized fundamental theorem of calculus often mentioned here, that a lipschitz continuous function which is differentiable almost everywhere, with derivative equal to the value of a riemann integrable function at points where that function is continuous, does equal the indefinite integral of that function.

in particular we studied the contrast between continuity, uniform continuity, and lipschitz continuity. I think this would blow most high school math teachers away. :biggrin:

[mathwonk]

well i only learned it recently myself, but it is amazing how simple you can make something seem, once you understand it yourself.

the benefit of my courses, i hope, is that I learn something new every year and put it in my courses. i am still infected with the bug i caught at harvard as a student, i.e. we expected our professors to teach us things that they knew that others did not. we did not expect our courses to be as easy as possible, but to be as valuable as possible. we wanted to learn things that students at easier schools did not learn.

i am a beginner at diff eq and have never liked or understood it before, so i worked hard this semester and came to love the stuff. I consulted extensively in v.i.arnol'd's works, and when he recommended the 100 year old treatise of edouard goursat i bought a used copy of that and consulted that too. I also used as references: braun, guterman - nitecki, boyce - di prima, edwards - penney, coddington, waltman, hurewicz, tenenbaum - pollard, henry helson, jerry kazdan's harvard notes, thats all i can remember.

[mathwonk]

here is the list of topics from my course:
math 2700 sp2006 Topics

first order equations, review of integration from 2210
the inverse function theorem and separable variables.
linear equations, exactness, integrating factors. why not every vector field is a gradient, hence exactness cannot always be achieved.

exponential functions, problems involving exponential growth,
population problems, mortgage problems, radioactive dating of paintings

systems, vector fields, predator prey models, equilibrium points, intro to linearization and stability.

second order equations, linearity, characteristic polynomial of a second order equation. factoring second order constant coefficient equations using linear constant coefficient operators, and uniqueness of solutions.

method of annihilators, method of power series, for finding formal inverse of operators, partial fractions.

method of judicious guessing, when the non homogeneous function is simple, e^x or sin or cos (i.e. has a known annihilator).

power series methods for solving homogeneous linear equations with anlytic coefficients, especially simple ones, like constants or polynomials.

[mathwonk]

Euler’s equation - factoring it using the linear operators (xD-a).

variation of parameters, for solving non homogeneous equations when general solution of homogweneous equation is known.

linear systems, matrix exponentials, eigenvalues, eigenvectors, determinants for 2x2 matrices, diagonalizing matrices (2x2 case), dealing with non diagonalizable matrices as diagonalizable + nilpotent ones. how to compute matrix exponentials in that case (2x2 “jordan form”).
application to electrical circuits, harmonic motion, simple or complex pendulum. stability in terms of eigenvalues, attractors, repellers, centers, hyperbolic points.

non - linear systems, linearization, Hartman Grobman theorem on when stability agrees with linearization.

approximation by Euler’s method related to exponentiation of vector fields. Connection with existence and uniqueness theorem.

[courtrigrad]

I hated my AP Calculus BC class. The teacher was inexperienced, and the course was not rigorous. It seemed that all we did was memorize formulas, and 'plug and chug' to get ready for the "exam of our life." I honestly feel that I did not learn anything in this class. Right now, I am starting from scratch, reading Spivak's book. As I will be a math major, I want to learn the subject properly.

[mgiddy911]

I am currently a High School Senior (woohoo only 5 more days!) from a private Jesuit high school.
My junior year I opted to take regular Physics B, the non-AP course, however taught by the same teacher as the AP class who was also my water polo coach.
I regret not taking the AP course because it would have been more rigorous and I would have learned more problem solving techniques. However as for the AP test, I feel I was still prepared to take the AP test even though I didn't take the class.(note I actually decided to not take the test, but prepared for it for a while before i made the decision, I opted not to take it only because I plan to major in maths/physics and decided that AP credit in algebra based physics would not really mean i should skip out on any college physics.)
I bring this up for two reasons. One is to show that I certainly agree with previous posters who mention that the merit of one AP class over another at various schools often comes down to the teachers. I felt prepared for the AP test having not taken the AP class because my teacher was amazing. Yet I do not think that this means in any way I was prepared to be placed in advanced college physics classes.
Another point I bring up, and I apologize if i skimmed the 7 pages of this topic and didn't see anyone else address this. My chemistry teacher, and a few members of my class, previously this year, discussed the old way our school did certain AP subjects. My school used to teach Accelerated Chemistry, where the teacher taught how and what he felt necessary. And student took the AP exam if they wanted to if they felt prepared. However now that our school offers AP Chemistry, our teacher is forced to teach in a specific manner, because now students expect to be able to do well on the AP exam, so he has to make sure he covers what The College Board wants him to cover including lab work.
I also recently took the AP AB Calculus exam. I think our class may be a little more in depth than those of some posters. I feel confident going into Calc 2 for college (though if i receive a 5 , my college could grant me credit for calc 1 and 2 for some reason). The only exception i have is proofs, and some definitions. My AB class did almost no proofs, and we skipped or glanced over some definitions.

I also took the AP Computers AB exam ( the computers AB is the harder of the two exams A and AB, possibly equivalent to the relation of BC to AB in calculus, as the computers AB would count as credit in you intro computers class as well as Data Structures)
I am wondering if anyone else has taken the exam and gone into college with advanced placement form it? Do you feel like you missed out on anythign big? were you prepared to take whatever the third computer class would generally be at your university?
I guess I should note that the AP Computers exam/class is now completely Java Based.

I am not certain where I stand on the AP issue as a whole, as it seems to differ form school to school and particularly from teachers. One thing I do think is still good is that Students who have some knowledge of what they want to major in are able to get elective credit out of the way. So someone who wants to major in math can get history credit out of the way, giving them more time to broaden their scope of mathematics. And depending on their school, they may get some value as far as writing or problem solving or study habits go, from taking an AP course.
I don't see AP as a good way for students who want to pursue maths to challenege themselves unless the AP class happens to be the highest math they can take at their high school. I wish my high school had offered me dual enrollment for things like math. I would have rather dual enrolled Calc 1 at a local university than taken the AP Calc exam where The College Board determines what I should know to be on par with college level.

I'd liek to hear back on anyone's comments, especially on the AP Computer topic, or my poor grammar / typing ability.
thanks

[Hurkyl]

I do think is still good is that Students who have some knowledge of what they want to major in are able to get elective credit out of the way.
Only to some extent.

I got a 3 on the music theory AP exam, but still had to take a music general ed class. :frown: (Actually, it wasn't so bad, but still!)

[jbusc]

It is very difficult to say whether a computer science AB exam will substitute well for a "intro to programming" or "data structures" class at a university. For example, both classes at my university are waived for the AB exam, and it is recommended to skip "intro to programming" if you passed the AP. If you are good at programming "data structures" can be skipped, but some do not, opting for better preparation for later classes.

However, at some universities Computer Science is taught from the beginning in a higher, more theoretical fashion and "intro to programming" classes are taught in functional languages such as Lisp or Scheme. Students should _not_ skip those classes unless they are familiar with Lisp or Scheme and the AP computer science test provides little preparation for those languages and functional programming.

[Pseudo Statistic]

I took the AP Physics B exam earlier on today. (I figure people in the US are taking it as we speak :))
After taking it, I have to say, I was pretty surprised at it..
I mean, I expected a harder test-- not to say that I'm 100% certain I got a 5. (I guessed a lot of the multiple choice questions and, from what I predict I got in the free-response (71/80), I have to get somewhere in the region of 23-27 wrong to pull off a 5 (I skipped 7 questions), according to my predictions)
However, the test, from my perspective, didn't do a good job of properly measuring people's thinking skills, in my opinion.
Everything was so plug-and-chug... with the exception of one part of one of the questions....
And the multiple choice was so..... well, it wasn't that bad, but it would have been VERY straightforward had I shifted my focus in the topics I studied.
Overall it was pretty OK. However, I don't think I'll be able to say how I did for certain, lest I get <5 meaning I would have spoken too soon. :)

[0rthodontist]

I think that the AP exams when I took them were far too easy. A 60% or higher raw score on the CompSci AB was a 5. Maybe if the test scores were given as a straight percentile, the AP tests would be held in higher regard.

[mgiddy911]

On the note of the scores possibly being reported as a straight percentile. Could anyone explain to me why they are scored the way they are? besdies the fact that I guess it spreads out the scores neatly on most tests. And on some tests give you alot of leway with your mistakes. I think for some tests a percentage grade would make more sense. For example tomorrow I take the AP chemistry exam. Now in theory tht class, and exam tests your knowledge over that would generally be covered in what most colleges call General Chemistry I and II plus thier respective labs. Now that is alot of material. It seems like it would be easier for a school to determine whetehr you are qualified to skip thier class based on seing your overall percent score and the test itself. I don't know if they have access to those materials

[jbusc]

I think, it's for two reasons:

1. The actual grades are supposedly determined(in part) by giving the same questions in college classes, and comparing the raw scores from those students with their final grade in the class.
2. Free response changes in difficulty from year to year, so percentiles change also. Standardised grading makes scores comparable year to year.

[0rthodontist]

I said percentile, not percentage. A percentile is a number that indicates what percent of other test-takers scored lower or higher than you, such as "80th percentile."

Obviously not all courses at all colleges are equal in difficulty or have students who are equal in aptitude. Therefore normalizing the test to a "standard college course" is not legitimate if indeed that is what they do. There is no reason to deliberately toss away data.

[Lucretius]

Well, I found the Physics B test somewhat difficult. Let me clarify:

The multiple choice I breezed through — left only about 5 blank, and I was fairly sure of the ones I did fill out, so that wasn't a problem.

The free response questions, a few of them at least, I had to leave totally blank. I cannot discuss the nature of these questions yet, or else the AP guys will hunt me down and burn my test — suffice to say, I had a fair amount of trouble with some of them. Still, I'm fairly sure my score will be high enough to wave a physics 101 class in college next year.

[Pseudo Statistic]

Well, I found the Physics B test somewhat difficult. Let me clarify:

The multiple choice I breezed through — left only about 5 blank, and I was fairly sure of the ones I did fill out, so that wasn't a problem.

The free response questions, a few of them at least, I had to leave totally blank. I cannot discuss the nature of these questions yet, or else the AP guys will hunt me down and burn my test — suffice to say, I had a fair amount of trouble with some of them. Still, I'm fairly sure my score will be high enough to wave a physics 101 class in college next year.
48 hours have passed for me.
So DUDE, wait a sec.
The free-response... the ones I got were like super easy. The first one was like, measuring g and you had a table of d vs. t. The second one was on electricity I think. The third was on reflection/refraction and slit stuff. The fourth was on Thermodynamics. The mechanics question was somewhere in there and one on an electron and a positron and stuff.
If that's the paper you had, I seriously think you were mistaken if you thought it was difficult and the multiple choice was easy... 'cause... well... the multiple choice pretty much raped me. :(

[Lucretius]

48 hours have passed for me.
So DUDE, wait a sec.
The free-response... the ones I got were like super easy. The first one was like, measuring g and you had a table of d vs. t. The second one was on electricity I think. The third was on reflection/refraction and slit stuff. The fourth was on Thermodynamics. The mechanics question was somewhere in there and one on an electron and a positron and stuff.
If that's the paper you had, I seriously think you were mistaken if you thought it was difficult and the multiple choice was easy... 'cause... well... the multiple choice pretty much raped me. :(

You got Form B of the free-response questions, from your description. I didn't have any sort of table for section 1. Was this (http://collegeboard.com/prod_downloads/ap/students/physics/ap06_frq_physics_b_form_b.pdf) the test you had? Here (http://collegeboard.com/prod_downloads/ap/students/physics/ap06_frq_physics_b.pdf) is the test I took. I didn't think the multiple choice was very hard; the practice multiple choice exams I had taken in class were much more difficult than this years'.

Look at question 3 on my free response section and see if you can tell what they wanted there. I left that whole page blank.

[jbusc]

You got Form B of the free-response questions, from your description. I didn't have any sort of table for section 1. Was this (http://collegeboard.com/prod_downloads/ap/students/physics/ap06_frq_physics_b_form_b.pdf) the test you had? Here (http://collegeboard.com/prod_downloads/ap/students/physics/ap06_frq_physics_b.pdf) is the test I took. I didn't think the multiple choice was very hard; the practice multiple choice exams I had taken in class were much more difficult than this years'.

Look at question 3 on my free response section and see if you can tell what they wanted there. I left that whole page blank.

on question #3, basically they are asking questions about the electric field generated by those two point charges. Each point charge has an electric field associated with it; the total electric field is the sum of the two fields. Given the charge on one point, you can calculate its contribution to the field; since you know the total field at that point is zero, then you can calculate the field due to the other point and from that calculate the charge.

[Pseudo Statistic]

You got Form B of the free-response questions, from your description. I didn't have any sort of table for section 1. Was this (http://collegeboard.com/prod_downloads/ap/students/physics/ap06_frq_physics_b_form_b.pdf) the test you had? Here (http://collegeboard.com/prod_downloads/ap/students/physics/ap06_frq_physics_b.pdf) is the test I took. I didn't think the multiple choice was very hard; the practice multiple choice exams I had taken in class were much more difficult than this years'.

Look at question 3 on my free response section and see if you can tell what they wanted there. I left that whole page blank.
Yup, turns out I did have form B..
On the subject of tests on the website, they released the AP Physics C Mechanics test!!!
I mean... err... I have the test this Friday, and according to records there has never been an AP Physics C form B. I hope I get lucky. :D
So question 3 from your paper...
Since it says the net electric field at point P is zero, The electric field produced by q1 plus the electric field produced by q2 = 0.... so, from there, you can plug in your known values and solve for q2.
Chances are, however, q2 is positive because since q1 is negative, a test charge 1C at point P would be attracted to it, (to the right), so it needs a positive charge, q2, to repel it to the left enough to keep the net electric field 0 at that point I guess.
For part c, the electric force on q2 would be the force on q2 caused by q1, Coulomb's law...
d.... Electric potential would be 0 when
q1/4pi*e*d = q2/ 4pi*e*(0.3-d)
And solving for d.. (d would be the distance from q1 to the point)
e.... I would integrate the coulomb force from infinity to the point, but I'm not sure if that's acceptable on physics B. You could say, instead, that it's = to the potential energy. I'm not sure why, though.

[Lucretius]

Hmm I musta looked at it incorrectly. I meant the one with the graph and they wanted the two pieces of information to be put in. The optics one.

[Pseudo Statistic]

Oh, then you meant question 4.
For part a) you could basically use snell's law... n1 sin t1 = n2 sin t2.. and since n1 = 1 (air), you're left with:
sint1 = n2 sin t2 or n2 = sin t1 / sin t2
That's what you have to graph to get a linear relationship and thus, from the slope, calculate the index of refraction you need.
For part c, I'm not sure, but I'd guess "The air-oil interface only". (Not sure why though)
For part e there's a formula... 2t = something....

[jbusc]

e.... I would integrate the coulomb force from infinity to the point, but I'm not sure if that's acceptable on physics B. You could say, instead, that it's = to the potential energy. I'm not sure why, though.

By the work-energy theorem, the work done on a particle is the change in the energy of the particle (potential energy, in this case) The potential energy of a particle at infinity is zero (because it's infinitely far away from the charges) So bringing a particle in from infinity, the potential energy is equal to the work done.

In AP physics, this is assumed to be true; you can show this using a little calculus 3, stating that since electric fields are conservative, the fundamental theorem of line integrals states the line integral of the field over the smooth curve from infinity to the final point (which is the definition of work) is just the difference between the potential function on the field at the endpoints.