Heigh Ho's post reminds me that Robin Hartshorne, author of the guide to Euclid I modeled my course on, made polyhedra, (3 dimensional geometry models), from toothpicks and jellybeans, with his college students at Berkeley! So these visual and tactile ideas appeal to lots of ages. (And they cost far less than prefab versions targeted at the education market.)

Thank you quark. I hope these examples can help someone learn to get back on the horse after falling off. I was often aided by challenges or encouragement from older mathematicians. When I returned to college after a year off I mentioned to my high school math teacher's husband, a college math professor, that I had fallen out of the honors math program, and he just said "get back in it!" I entered a non honors course at first determined to do well. Having forgotten or never learned a lot, even with regular attendance and study I got a D on test 1. When I objected to the teacher the material I missed was from the previous course he just said: "math is cumulative". I bought a book of worked problems to practice, and went to the library for one hour every day after class to review and fill gaps in the class notes. I also studied with another student taking a more advanced class with the same prof, and learned proofs that were omitted from my course. The last week the class elevated from computational to theoretical, and I loved it. On the final he said "work any seven problems". I worked them all, including adding material he had not given in the course, and started to write "grade any seven". Prudence won out and I crossed out all but seven I was surest and proudest of. I received the highest score in the class "head and shoulders above the other perfect papers". For a week I celebrated, than realized I was in the wrong class. I remembered the advice I had received, went to the professor of the next honors class and asked him what I needed to do to prepare for it. (In those days essentially nothing could be repeated so you had to take the next class even if you did poorly in the previous one.) I got an advanced calculus book, read the topic recommended, took the honors class in the Fall, got a B+, continued to A- in the spring. (I had a D- in the previous honors class.) Later, after leaving grad school with only a masters, faced with losing my teaching job with a family, I began to commute from central Washington to a course in Seattle, 110 miles each way, 3 times a week all Fall. At a dinner for a famous mathematician he asked my goals. I said "do good mathematics with or without a PhD". He responded "That's a cop out!" I became angry, and motivated, took leave, returned to grad school and got my PhD. Because I had a masters, I was allowed only three years to finish, and my leave from my job was good only for two. After two years still with weak results, I had a choice of taking a "minimal PhD" back to my modest job for the rest of my life, or staying one more year hoping for better results. I resigned my job with a wife and two kids, "stepped out on faith", and stayed to complete a better degree. After that, I went up a level, i.e. I began to reach my goal. That was 35 years ago. In retirement the most rewarding experience and fun I've had was learning from Kathy in Richmond and the epsilon team, how to teach Euclid to children (see post #39 above). I essentially taught the same course I offered undergrads and grads in college, since Euclid has so few prerequisites, but how to present it to kids was something I needed expert help to approach. Kathy and George and the MathPath staff were superb role models. Most people here also know more about teaching children and others than I do, having been primarily a lecturer, so I prefer to advise primarily on math content, and quality of various resources, or maybe how to navigate college administration. But I am a student of teaching well, and aspirant.

I would let him do whatever he wants with that colored euclid book. drawing the diagrams sounds great! I cannot recall when i did not enjoy math. my dad used to read to me in his lap and i wanted to read too. he was an inspector of railroads and had a booklet of numbers meaning i don't know what, and he would let me help him by adding up the columns of numbers. (My dad died when I was 15.) In school math (and reading) was rather boring as they went at the speed of the slowest essentially. Hence the faster ones had nothing to do. I used to sleep with my head on my desk in class. Fortunately I had a teacher who introduced math competitions into our activities and we began to have fun taking competitive tests. We met after school and practiced taking tests, like a sports team. From the time I began participating, over a period of 5 years, our coach progressed from telling us we had no likelihood of actually winning, to the point where she assumed we should always win. We accumulated a large collection of trophies, and our school was perceptive enough to display them publicly in the same case used for sports trophies. One year I took no math, because there were not enough courses available for me, having already taken algebra in 8th grade, and in a curriculum that ended with solid geometry. My enjoyment of math in high school was thus mostly from the competitions. I had learned a few formulas like the sum of the first n integers, 1+2+3+....+n = (1/2)n(n+1). Finally in college I had a good course, which used the calculus book of Courant. I still remember that on page 27, in a footnote, there was a complete proof explaining how to derive a formula for the sum of the rth powers of the first n integers, for any r! This result, which went far past anything I had learned in whole courses in high school, was tossed off as a trivial aside. At that moment I said to myself: at last! this is a book from which one can learn something. I have finally come to the right school for me! I still had a lengthy journey to realize my dream, however, due to poor study habits from high school. After a long struggle to graduate from Harvard, I had learned to survive in school, but was losing my love of math due to courses where I merely memorized theorems and proofs. I almost left the subject to become a student of French literature or something else, but in my senior year, I detrmined that I should not leave mathematics befoe finding out what is was really about. So I went grad school at Brandeis, and found myself in the algebra class of a brilliant mathematician and teacher, Maurice Auslander. At last I began not only to learn some math, but how to understand and do math myself. Politics intervened (Vietnam) and I again had a hiatus from the subject. Eventually, with a family and a teaching job, I was threatened with losing it without a PhD so I again returned to school. It was embarrassing to be in a profession where a degree meant more than expertise, since everyone in my department agreed I knew much more math than anyone else. This time, as a breadwinner, I had enough motivation to complete my PhD, a very hard challenge at age 35. After that I entered the Valhalla I had always aspired to. I have spent 35 more years since then on the" inside" of the subject, interacting with some of the most brilliant people in the world. What fun! I even returned to Harvard as a postdoc and finally took advantage of the wonderful intellectual riches there. For me, Harvard was much to high powered as an undergraduate or even graduate school, and about right as a postdoctoral site.

I think you are right, and this intimidation or reluctance to seek help I think is very crucial. When as students we do not succeed we have a feeling something is wrong with us, that we are not intelligent enough. But my impression is that this is not as crucial as the way we study, how willing we are to ask for help, to go to class and to office hours. I was even afraid to go to class sometimes for fear the professor was angry with me for missing a previous class or assignment. I never went to office hours for help. My time at Harvard persuades me that although many Harvard students are very intelligent, their key distinction is a strong self confidence and aggressiveness in seeking their goals. They know what they want and they go after it to a degree that differs greatly from students at the typical state school familiar to me. I.e. these elite students do better mainly because they try harder, are disciplined, and are not shy. Even many of the questions asked on this forum should better be asked of the administrators at colleges, but some of us are too reluctant to ask there, and prefer to ask our peers here, even though we do not know the answers.

My impression is also that MCAT scores mean a lot. My wife got her college degree in math from a minor state college at age 22, then raised our two kids primarily for the next 6 years before realizing she wanted to be a doctor. We were then at Harvard on a postdoc, so she took several chem and bio courses in the Harvard night school, an open admissions school whose mission is to serve the general public. When we returned to UGA she took some other courses there, and their excellent premed prep course, and then nailed the MCAT's. She only applied to one med school at age 33, Emory in Atlanta, and got in. Her MCAT scores were 20-40% higher than those of some optimistic applicants I spoke with, so I assumed they played a big role. After getting consideration, I felt the interview was also important and deserves thoughtful preparation. I could ask her for more authoritative info if desired.

Those are good ideas. Another thing I tried recently was scheduling office hours in a class room rather than in my office. I noticed few students coming to office hours and conjectured that it was intimidating. A classroom was more familiar and attracted a larger group who then solved problems together. I also scheduled them at different times to accommodate people with different schedules.

Possibly fun for math, here's an 1847 version of euclid geometry in color! and free. http://www.math.ubc.ca/~cass/Euclid/byrne.html

In our family, I had a preferred college for my son, (my alma mater), but he had another one in mind, even without ever visiting there. He got into mine but chose his and loved it. He still lives near there. So just suggesting to consider whether ds has an idea of a favorite place, no matter how unlikely it seems?

Another idea occurred to me that may be of independent interest for home schoolers, possibly in a separate thread. I.e. in some sense the well intended questions I posed above, concerning what are the right prerequisites for a college course, are the wrong questions. I.e. it may be that success in a college course may be more closely related to how a person studies than to whether he/she has specific prerequisites. The data is from a study by Uri Treisman who tried to increase the success rate of certain ethnic groups of students. To summarize briefly, he discovered that students who came in with all the right scores and background and motivation, but were too independent and studied alone, tended to do less well than students who studied in groups, helping, pushing and encouraging each other. However at the top of p. 368 he says the real core was not just group study, but "the problem sets that drove the group interactions". My theory is that students who excelled in poor high schools learn to avoid other students and work alone. When their success gets them into a good college this habit backfires. This may explain my failure in college after excelling at a mediocre southern high school working entirely on my own. It occurred to me that some home school students may have the same problem I had, of not realizing that in a good college the students can help each other, and those who do not take advantage of this may fall behind. I.e. challenging study groups may be advised. Here he is now: http://www.utdanacenter.org/staff/uri-treisman.php And here is the summary of the original study that started it all. The key results are stated on page 366, but the previous (and later) parts are also entertaining and useful background. (Is this study well known?) some dramatic results of the program as well as frustrations (when the funding ran out) are expressed on page 369. There is also a statistic there, possibly from the 1980's that over 40% of all the 600,000 incoming calculus students nationwide fail each year. So my 50% "unsuccessful" figure was fairly typical. http://www.utdanacenter.org/downloads/articles/studying_students.pdf

As a partial answer to question 4) in my previous response, i.e. what does it take, beyond AP scores, to succeed in a particular course, we have Kathy's testimony about math 51h at stanford. Here's what her daughter studied prior to a course whose only stated prerequisite when my son took it was a high AP score. posted by Kathy: "I love Marsden and Tromba and used it with both my kids for multivariable calculus at home. My PhD advisor worked with Jerry Marsden, so I knew it would be full of good stuff. It's one of the few books that L insisted on taking to college with her!" [Marsden and Tromba was a several variables calc book used at Berkeley! This is miles beyond any AP syllabus I know of.] Kathy again: "Unfortunately, as homeschoolers, that class had no 'documentation' in the eyes of college admissions officers - just a mommy grade and course description. .......,.,That doesn't mean that ...... I can't add a heavy dose of proofs to my version of AP Calc." Notice again that Kathy greatly enhanced her home AP calc course, even though Stanford gave her no AP credit for this. Nonetheless it seems to have benefited L. I am guessing she also used a better than average book for single variable AP calc, maybe Spivak or Apostol? By the way, when my son took the course it was also taught by Leon Simon, and used Apostol vol. 2. Unfortunately my son had not been prepared as well as Kathy's, even though he had taken intro to calculus at rival GaTech, while in high school. Here is the stanford course description, which still does not give any specific requirement beyond an AP BC score of 5, quite inadequate, in my view. "MATH 51H: Honors Multivariable Mathematics For prospective Mathematics majors in the honors program and students from other areas of science or engineering who have a strong mathematics background. Three quarter sequence covers the material of 51, 52, 53, and additional advanced calculus and ordinary and partial differential equations. Unified treatment of multivariable calculus, linear algebra, and differential equations with a different order of topics and emphasis from standard courses. Students should know one-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on BC Advanced Placement exam, or consent of instructor."

The way this question is asked, the answer be that it is not worse. But the premise is a rather narrow and optimistic one. I.e. the reason a prescription is required for this drug is its potential for abuse. Thus using it "responsibly" a few times may lead to increased likelihood of abuse. Also excluding the fact that it is illegal to buy informally also skews the answer, since that one fact makes it a heck of a bad choice.

Next I want to discuss the specific topic 8fillthe Heart asked about at UGA, BC scores and placement in math 3500H, the several variable elite honors course. The website says students with a 5 on the BC are advised to “consider” taking this course as opposed to the first year (Spivak) courses 2400H and 2410H, which I myself would recommend. The available (5 year old) data seemed to lump 4 and 5 scores together. So I asked what percentage of the class in 3500H is made up of BC”4/5” students as opposed those who are graduates of UGA’s own “Spivak course”, math 2400H-2410H. It turns out that the majority of the 3500H is indeed composed of entering students with a 4/5 on the BC exam, to my surprise. However, it is also true that this group comprises a minority of the entire group of entering students with 4/5 on the BC. Thus the students entering with a 4/5 on the BC outnumber the entire class size of math 3500, and most of the 4/5 BC students take something else. It is still not clear what the answer is to the natural question: “If my child has a 4/5 on the BC exam, should he/she take math 3500H at UGA?” I.e. according to the professor I spoke with, he did not know immediately how many such students were actually being personally advised to take math 3500, as his impression was that they were essentially self selected, presumably after reading the website. I also did not get an answer to my next question, which was “how successful are those 4/5 BC students in math 3500H, among those who opt to take it?”. I wanted to know if it was the same situation as at Stanford, where 65% drop out after one test, or whether UGA does a better job of placement. I.e. presumably one wants to know the answers to several questions: 1) if my child has a certain score on the AP test, which courses can he/she exempt with credit? (a version of the title question of this thread.) 2) if my child has a certain score on the AP test, which course do you recommend for him/her academically, if different from 1)? 3) if my child has a certain score on the AP test, but nothing else, and if he/she opts for such and such a course, what are the chances of success, based on past experience? 4) If my child wants to take such and such course, for which AP preparation is not necessarily sufficient, what additional preparation has been found adequate in the past? maybe even, at a school like Stanford: 5) What level of withdrawals from, or failures in, this course, among people having the stated prerequisite, is usual?

I want to thank 8filltheHeart for fact - checking my statements about AP and UGA math. This inspired me to call them and try to update my data. I reached one very helpful professor, a friend who kept such data up until 5 years ago. I will try again later to get more recent data. One thing I learned that has happened since I have left, is that the course I would have advised most entering students with AP preparation to take, “honors” math 2300H, i.e. beginning differential calculus for “honors” students, no longer exists. This has several reasons. One is financial. UGA apparently did not offer AP credit to students taking math 2300H, and presumably they or their parents were not willing to pay for a course they could exempt even if (in my opinion) they really belonged in it academically. So this, in my personal opinion the academically best option for most such students, no longer exists. I consider this a direct result of the proliferation of AP courses, and I lament it. Basically schools are forced to offer courses for which there is an audience rather than those which I personally think make academic sense. However, it does not mean there are no reasonable options for entering AP students. I.e. we are always adapting our courses to fit the audience we have, to the best of our ability. The next course, 2310H, still exists and presumably is the one many entering AP students take. This is the course that fueled my frustration in the first place, since I taught it regularly, and most entering students with only AP background, did not do well in it, at least with me teaching it. (This is the one the student with a 93 on test 1 dropped out of.) I asked whether I had been an “outlier”, and whether such students did succeed with other teachers, but that information was not available immediately. I suspect that may be true to some extent, and this is one reason it is useful for old people to retire. Some of us cling rigidly to standards as they were when we were in school, or when we started teaching and have trouble adjusting the level of our courses continually downwards, as it seems to us. I kept telling myself my students deserved the same level preparation I assumed was available to students at other good schools, since they would have to compete for the same jobs. Probably I was not aware of what was happening also at other schools. Today however, younger teachers who also have high standards, but probably are better in touch with the preparation current students have, and what one can reasonably expect, are likely tailoring the courses to them more skillfully than I did. Even curmudgeons like me were constantly tinkering with our curriculum and test questions to try to teach effectively the people who were actually in the classroom. I.e. since AP tests and credits are effectively the norm today for entering students, colleges have been forced to make the courses more or less suitable for these students. Thus as old hands like me retire who did not do this as much, the problem lessens. Still the data posted by Kathy tells me some colleges, such as Stanford, are still not doing a good job of calculus placement, i.e. its not just me. Originally Posted by Kathy in Richmond "Over 100 kids started in Math 51h in September; after the first midterm almost 2/3 of the class dropped out."

@8fillThe Heart: "(From my reading, it does not appear that UGA requires its math majors to take Math 2400-2410 which is open to non-honors students and students w/o a calculus background.)" This is correct to my knowledge. Math majors are not the same as future mathematicians. We will take all the math majors we can get, from almost any background. But I believe we do recommend 2400 to the ones who want to get started right away with theory and can handle it. Just because a course is designed "for" someone does not mean he is required to take it. math 2400 is apparently modeled somewhat on the course math 11, taught at Harvard in the 1960's, which I took, and which Michael Spivak, the author of the standard book for math 2400 apparently also took. When I took that I had not had calculus in high school nor studied it on my own. Maybe you are (rightly) confused by my use of the word "honors". At UGA, math 2400 is certainly an honors course in the intrinsic sense of high quality and expectations. However at UGA "honors" courses also has a technical meaning, namely courses offered through the official honors program. You should check me on this, but this program may be set up to prevent people from taking honors courses unless they are in the official honors program, and maybe all honors program members are expected to take only "honors" courses. That does not make sense in our discipline where often there are people who are outstandingly gifted in math but do not belong to the honors program. There are also many people in the honors program who cannot possibly handle math 2400. That's why we have two "honors" courses, 2300 and 2400. All this is somewhat complicated and possibly political. Again this is why it is crucial to get current expert faculty advice. I am not sure what the exact requirement is for math majors. Since increasing the number of math majors is a perennial goal, presumably it is not too strict. When I was in college it was something like "advanced calculus and any other 6 semester courses from the department. Again this should be checked but I believe at UGA it is even weaker, i.e. not even advanced calculus was required when I was there.

In regard to doing ones homework on the internet, this next remark does not apply to the link we were referring to, which looks excellent. But it did remind me that in the past, when I was at UGA, I complained repeatedly that we had three different conflicting web based descriptions of the same course posted, and it was impossible for a student to find out what the actual expectations were if he did not know which one was correct, nor for that matter for a new teacher to know exactly what to teach. Hopefully that glitch is fixed by now, but I suggest always phoning and/or visiting and verifying things with the same people who will implement them.

I'm not sure which of my statements you are referring to. You have done your homework well, and my statements were based on my memory of what the associate department head told me years ago. It is probably me that is missing something rather than you. Things do change and I had not read that catalog advice lately. As I recall he said earlier that those students wanting AP credit were not allowed to take introductory calculus unless they took the honors level course with theory, the 2400 sequence. They might take people in 2400 with 4's and 5's, but I am not sure of the exact requirements. My impression was also that the audience for math 3500 and 3510, the several variables honors calculus, was mostly made up of graduates of math 2400. The page linked says that entering students even with only a 5 on the AP (BC) exam are advised to "consider" taking the more advanced course. That surprises me, but I do not read it as a recommendation to take it. Rather it seems to be a recognition that such a step may be suitable for some very strong students, and to give them further options. Each student in this situation is advised personally by the instructor and associate department head to determine proper placement. To me, making that jump would mean a student who has strong computational skills and a grasp of basic theorems, but possibly without proof experience is being asked to consider a very high powered proof based course. That 3500, 3510 course is really advanced, and jumping into it with only a 5 on the AP is analogous to jumping into the super honors courses at Stanford and Harvard. I don't recommend it myself, but I didn't write that blurb, and there are always exceptions. Dr. Shifrin who often teaches it is very helpful however, and if a student who tried this course will attend office hours faithfully, he has a fighting chance. Any student who refuses to go to office hours however and tries to go it entirely on his own, may be in trouble. The advisors at UGA that I know really know their stuff however, and I think you can trust them. I have been gone for two years and they are there now teaching the course. I am sure e.g. that if a student got into math 3500 and did not find it working out, they would certainly arrange for her/him to go back to 2400 or even to the non honors 2500. The teacher who wrote that also knows what the audience and the current expectations are for the courses involved, so I would put more faith in that blurb than in my statements, which are less current and less precise. I would not choose a course without talking to the associate head, possibly the undergraduate coordinator, and the instructor. So as always, I have my opinions, but one needs to do homework, consult with the professor who will teach the course and make a decision based on that. There is no one simple rule that fits all cases. Does this help? I will also make a recommendation specifically for a student attending UGA. The Associate Department Head, Dr. Theodore Shifrin, is one of the best advisors (and instructors) I have ever known, and he probably wrote that blurb on advanced placement. I would talk to him for specific advice on that program. He is a tremendous instructor, especially for the highly motivated math student, and often teaches the advanced honors math 3500 course. In fact he wrote a book for it. http://www.amazon.com/Multivariable-Mathematics-Algebra-Calculus-Manifolds/dp/047152638X/ref=sr_1_3?s=books&ie=UTF8&qid=1341876566&sr=1-3&keywords=theodore+shifrin

Wow! I guess I lucked out big time. My freshman comp instructor was distinguished historian Nathan Huggins. http://dubois.fas.harvard.edu/nathan-i-huggins-lectures http://en.wikipedia.org/wiki/Nathan_Huggins My introductory calculus class was taught by John Torrence Tate. (And i still didn't understand it! ) http://en.wikipedia.org/wiki/John_Tate http://www.utexas.edu/news/2010/03/24/john_tate_abel_prize/ I had fun a few years back communicating again with Professor Tate over a homework assignment he had given. He had asked us in that freshman year class in 1960 to prove that the expansion of the famous constant e begins 2.718281828.... but I never did it. After telling the story to my own class in Fall 2006 while teaching the topic, I felt guilty, since I always expect hard work from my classes. So I sat down and cranked it out. I emailed Professor Tate to tell him I had finally done it, and he wrote a nice message back saying how it made him smile to get a homework assignment 46 years late!

I am sorry you had that experience, and I have certainly encountered ridiculous rules that will not go away! When logical argument does not help, I do encourage enlisting an advocate among the faculty. Unfortunately you were off campus at the time. I was always shocked at the way I was received when i called some office on campus and identified myself as a professor. I felt a little like Danny Kaye in the "Inspector General". If you have seen that movie, you know he was a fraud who entered town in a fake uniform and was mistaken for someone of importance to whom everyone showed respect! I felt an obligation to use that uniform on behalf of deserving cases. And thank you, we had a great time in WA!

Forgive my clumsy expression. What I meant to say is that there exist good freshman comp courses, and if you find yourself at a school where the one you are in stinks, you should get out of it. I don't think I said freshman comp should never be skipped. I just said it helped me. The blanket recommendation that it should routinely be skipped, and that an AP class is always better, was what I was reacting to. I think no specific rule here applies to everyone. I also did not mean to imply at all that the professionals do it better. Rather I believe and have tried to say before, that home schooling offers the chance to adapt a class as closely as possible to the needs of the student. I am just hopeful of sharing what I have learned about teaching math, and about navigating formal school systems so home schoolers can use it both at home and when they get into a regular school. I greatly prefer small personal classes to the ones I had to teach in university. It is also obvious from the comments here that home school courses can be greatly superior to what is available in schools. Indeed that is the stimulus for home schooling, to do a better job. The significantly home schooled group of children I met last summer were certainly the best prepared class I have ever met in over 40 years of teaching. I think if I had been home schooled I would not have needed that freshman writing course. However I also admit that my son's outstanding high school English and history teachers did a better job of teaching him writing than I could have done. Math? maybe not so much. That does not mean other home schoolers could not have beat my son's professional teachers also in English. We all have different experiences, but probably few of them yield either universally valid or universally invalid lessons, at least in my opinion. I fully support using AP credit to exempt courses equal to or inferior to the AP course. I am just trying to help clarify that this judgment needs to be made in individual cases. In my experience it has certainly not been the case that all AP courses are equivalent to the college courses they substitute for. My hope is to help people decide what level they want to shoot for at least in math and how to get there, whether the AP really satisfies their goals or not, and if not how to upgrade past that level. I am motivated by the fact that in my experience AP courses and tests stop short of what it is desirable to know, but this may be overly influenced by the math tests, and my faith in, and enjoyment of, the value of creative reasoning as compared to mechanical computation. If we try to say that a college course should always, or should never, be replaced by an AP course, we may be falling into the same trap as the trig student who does not want to understand the concepts and just wants to know which buttons to push. One needs to do some homework and make a judgment in each case. So thank you for giving the counterpoints to my own arguments which reflect my bias and lack of experience.

You are so welcome! I enjoy crafting explanations for individuals. Whenever I try to offer my old notes to anyone I always feel they aren't quite suitable for that particular person and prefer to write a fresh version. This sequence of explanations here started at least as a response to signals picked up from questions here. When there is a little back and forth, one gets a better sense of what to say. Of course I usually go off the deep end after a while and say whatever comes to mind, but even that is inspired by the conversation, and may become useful later. This last stuff e.g. is essentially an intro to vector algebra, usually taught in 3rd semester calculus, but without using those words. I think I was in the second or third grade (??) when some ambitious student teacher showed us the law of cosines, but I probably had my PhD before I realized it is just the distributivity of the dot product. And it was only recently that I found out it had an area formulation in Euclid! I think young kids should be shown these connections as soon as feasible, independent of the artificial separation of topics in usual curricula.

This advice is probably excellent in some settings, i.e. some colleges but it does give me pause. I never took any AP courses in high school at all, had all A's in everything, and was forced into a freshman writing course at Harvard that taught me to write, to some extent, for the first time ever. In fact it taught me that I hadn't yet learned how. It apparently didn't hurt people who wrote well either. On our first assignment, my more literate friend who got one of the few A's, was simply kicked out and put into honors writing class instead. My older son, who went to a high school where literature and exposition were highly valued, went to Stanford and dominated in his freshman writing class, but not on the strength of an AP English class, but from the spectacular honors writing classes in his high school, created by the outstanding teachers there. Even at a school where the class is inadequate for your needs, I believe the problem of being forced into a class that bores you is one that usually can be dealt with by being insistent and persistent. No one at UGA ever has to take a math class that they do not need, if they come to the attention of the department. It helps a lot if the student knows this and advocates for herself. As they put it in the student guide for incoming undergraduates at Harvard "Never take a low level 'NO!' for an answer." Your advisor, instead of laughing, should have picked up the phone and lobbied to get you into the right class. I have done that many times at University of Georgia, where there are also many classes that are inappropriate for strong students. In fact I have essentially never been unsuccessful at getting rules waived. Students don't seem to realize how much clout professors can have in academic matters. On the other hand if it is a financial matter, and you want credit for the class without taking it, that takes more work. However, even there, we have exemption tests that can be taken by students to award them credit in many cases. Always ask for help when things don't make sense. I have even had deadlines waived for application to a program or a degree that were missed by a semester or a year or even several years. (By the way, we spent July 4th on Vashon, in beautiful Puget Sound, to escape 100+ degree heat in Atlanta! :001_smile:)

Another simple question that cannot be avoided in a calculus class is: what does it mean to say the area under a given graph is a well defined finite number, i.e. that a given function has a definite integral over an interval [a,b]? Any student who wants to understand calculus should have some grasp of this question and its answer. Here are a couple of little tests: 1) Define what it means for the integral of f to exist on [a,b]. 2) Using that definition, prove, i.e. explain convincingly, why a strictly increasing function on [0,1] always has an integral. 3) Give an example of a function defined on [0,1] whose integral does not exist. And for students who know the "fundamental theorem of calculus: 4) What can be done about calculating or estimating the integral of an explicitly given function defined and continuous on [a,b], but for which you do not know any explicit antiderivative? (without using a calculator). (Hint: what good is a Riemann sum if you can't take its limit?) To me such questions are more fundamental than "what is the antiderivative of sin^2(x)?". Do they ever occur on AP tests? And I'm not restricting myself to math. When I was in college, almost every test was "essay style", written in "blue books", in virtually every subject. Do AP tests ever ask one to explain what the various concepts are good for? E.g. in integration, proofs are usually done using upper and lower sums, while "Riemann" sums are better for estimates, and "precise" (sometimes only theoretical) calculations come from antiderivatives. Do kids learn this?

Here is a question I always thought was interesting, but one seldom sees asked (not in AP syllabus?): what is a real number? if the answer is that they are finite or infinite decimals, as stated on page 1 of Thomas and Finney, 9th edition, then how do you add them? I.e. addition usually starts at the "right end" of a decimal, so where do you start with two infinite decimals? I taught a precalculus course to high school students in which we studied such questions. Without knowing what a real number is, how much can one say he understands about calculus? Here is another property of real numbers referred to on page 2 of Thomas Finney as "hard to make precise": what does the "completeness" of the real line mean? They say it means there are no "holes" in the real number line and leave it at that. This gives a false impression that there is something unusually difficult about it, something beyond the ken of a calculus student. However on page 3 they discuss open and closed intervals as something basic and simple. Why don't they just go on to say that "no holes" means there is no separation of the real numbers into two disjoint open intervals? I.e. given any two disjoint (non empty) open intervals of the real line, there must be some more real numbers in between the two intervals. This property gives us all the "existence" results in calculus. E.g. if no real square root of two existed, then the real line could be separated into the disjoint two open intervals, consisting of numbers x with x^2 > 2, and those x with x^2 < 2. How hard is that? I dislike giving someone the impression there are "hard" topics that are reserved for some other course you have to pay extra for and spend years preparing for, especially if it is not true. By the way this tendency is widespread, even at good schools. The following free calculus notes from MIT open courseware (see pages 202-204), although pretty good in some explanations, give slightly the wrong definition of the Riemann integral (an integrable function may not have a maximum or minimum on a subinterval), and they also skip over the easy fact that monotone functions are integrable, giving the impression, in remark 4 p.204, that after the trivial rectangular cases, one is naturally led next to the continuous case. Remark 3 should logically have been the easy monotone case. http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/MITRES_18_001_strang_5558.pdf When I teach a class I try to adapt it to the needs of the members, their background, their goals, and their mood from day to day. The first day I ask everyone his major, so I can try to throw in references and applications to those subjects. If they do poorly on one test, the next test is easier to reduce discouragement, and I may throw out the previous one altogether. Home schooling can do the same, even better, but it is hard to get that from a canned curriculum or standardized tests. I will try to put the notes from my high school precalculus class on real numbers on my webpage, in case anyone is interested in them.

As to the kinds of questions I like, and to show my way of thinking about math, I enjoy especially finding a problem or statement in the textbook which is incorrect, where the author has made a mistake. I will sometimes assign that problem and see if the students get it. I want them to learn to believe the data rather than the false statement by an "authority". This has only happened significantly a couple of times, and only in graduate courses, and even graduate students often find this difficult, but when they get it, boy are they excited and proud!

@snowbeltmom: Of course you want to do both, but indeed you must do something also after you get in. I love Kathy's daughter's story too. I have a different one about myself, illustrating a different point, but one that helps form my own perspective on teaching and learning. I was one of the larger percentage of students who did not make it to the 3rd semester of the tough math series at my college. I was forced to drop down a level to a less demanding and less interesting course, in which I again did poorly, and by the 4th semester I had been dismissed from school, and allowed to reapply only after a year of work in a factory, in construction, and elsewhere. If you saw Paul Newman in "The Hustler" last night, it was a question of "talent" versus "character". This experience hardened my resolve to recover my lost situation, and within one semester of my return, I had re entered the elite program and gradually began to climb back up the ladder, first with a B+, then an A- in the elite advanced calculus course. I had another forced hiatus in graduate school, but ultimately gained enough "character", with help of my family, to attain my goal of becoming a mathematician. My point is also that even failing in a hard course can help set one's sights higher, and determine to someday reach the higher level exhibited in that course. So my first trip through school showed me where I wanted to be, and it took longer to get there. So reaching ones ultimate goal in life is not always a sequence of unbroken successes, but perseverance plays a role as well. I try to teach this to my students but mostly they think one bad grade dooms them. I sometimes show them my checkered transcript to encourage them, but it does not always have the desired effect.