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Courses Remarks on AP courses in high school

  1. Mar 24, 2006 #1


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    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
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  3. Mar 24, 2006 #2
    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.
  4. Mar 24, 2006 #3


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    apostol is a superb choice.
  5. Mar 25, 2006 #4
    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.
  6. Mar 25, 2006 #5


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    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.
  7. Mar 26, 2006 #6
    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"
  8. Mar 26, 2006 #7
    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>
  9. Mar 26, 2006 #8
    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.
  10. Mar 29, 2006 #9


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    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.
  11. Mar 29, 2006 #10
    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.
  12. Mar 29, 2006 #11

    Dr Transport

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    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.
  13. Mar 29, 2006 #12


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    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.
    Last edited: Mar 29, 2006
  14. Mar 29, 2006 #13
    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?
  15. Mar 29, 2006 #14


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    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.
  16. Mar 29, 2006 #15


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    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.
  17. Mar 29, 2006 #16
    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.
  18. Mar 29, 2006 #17
    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).
  19. Mar 29, 2006 #18


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    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
  20. Mar 29, 2006 #19

    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.
    Last edited: Mar 29, 2006
  21. Mar 29, 2006 #20


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    chicago may count as an "ivy" but they did teach from spivak a few years ago; of course time flies for an old xxxx like me so that may be decades ago now.

    but you can always read it yourself.

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

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

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

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

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

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

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

    but if you go somewhere they do understand you, then you can survive on performnace. however there is a balance to be maintained. when you go home where they do not understand what you are sayiong they will ask where are yopur publications, so to survivbe in the real world it is true one should get good grades and publish, but one must not focus exclusively on these trivialities.
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