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Which AP exams earn credits at the most colleges?

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Which AP exams are most widely recognized by colleges in terms of getting college credits? There is a list at http://about.myedu.com/data-and-infographs/2011/3/22/top-10-ap-classes-most-widely-accepted-at-colleges-and-unive.html , but I don't know how they arrived at the numbers.

 

My guess is that high scores on the older, single-discipline exams, such as calculus, physics, or U.S. History are more likely to earn credit than high scores on newer, inter-disciplinary subjects such as environment science of human geography.

 

The College Board has online data on how many students take each exam, which is likely correlated to widely accepted the exam is accepted by colleges.

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What is also tricky about lists like this is that some schools will give credit for 5's only, others for 4's and 5's (but maybe 3's in math). Policies vary widely.

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A more generic question for you along the same lines.

 

What does AP US History accomplish? In prehistoric times when I took AP English (no idea if it was Lang. or Lit), I received credits and I did not have to take Writing 121 and it seems like something else which I no longer remember.

 

I understand the value of an AP science, language, or English course, but what do history and Human Geography courses net the student?

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A more generic question for you along the same lines.

 

What does AP US History accomplish?

 

About 20 years I graduated from college in 3 years because I was given "sophomore standing", based on 5's on 3 AP exams, one of them being U.S. History.

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I understand the value of an AP science, language, or English course, but what do history and Human Geography courses net the student?

 

College credit!

At our state university (and in other states probably as well), it is mandated by law that all students must take a US history course (not sure about private unis). APUSH can get the student out of this requirement.

 

AP Human geography or World History gives college credit for Human geography or world history (3 credits for a 4, 6 credits for a 5). This means that the student has to take fewer humanities electives.

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About 20 years I graduated from college in 3 years because I was given "sophomore standing", based on 5's on 3 AP exams, one of them being U.S. History.

 

College credit!

At our state university (and in other states probably as well), it is mandated by law that all students must take a US history course (not sure about private unis). APUSH can get the student out of this requirement.

 

AP Human geography or World History gives college credit for Human geography or world history (3 credits for a 4, 6 credits for a 5). This means that the student has to take fewer humanities electives.

 

This makes sense. We don't have a state requirement for history of any kind, so the AP history and human geography courses would count as electives, but would not enable the student to waive a general requirement. I can work with that. Thank you!

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Policies do vary widely and often in ways that don't seem really logical or don't seem to fit a pattern.

 

As a really general rule though the biggest bang for your buck are going to be some of the ones you mentioned: BC calc, bio, chem. Foreign languages can earn quite a lot of credit but that is typically reflecting many years of study. History tends to be a mixed bag based on the requirements of the college. AP lang and lit again are really variable. Some colleges roll their English into seminars or interdisciplinary courses so they aren't interested in granting credit for AP lang or lit.

 

Environmental science and human geography are two of the APs that are least likely to earn credit because they are seen as easier one semester courses.

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Yep, AP credit policies are too variable to categorize neatly.

 

My kids both had about a dozen APs each (mostly from the same list).

 

The only courses that they both got credit for were Physics C and Calc BC.

 

Stanford gave my daughter additional credit for AP Chemistry & Computer Science AB.

 

MIT gave my son additional credit for AP Biology (they no longer do, though) and a boatload of "general elective credit" that didn't really do him any good.

 

Neither got any credit for their Latin, US Government, Statistics, or Economics APs. Dd did get Latin placement, but it was based on SAT II scores instead of AP.

 

MIT would have given credit for English Comp (but ds didn't take it); dd got no credit for her 5 in AP English; everyone at Stanford is required to take the freshman writing and humanities classes regardless.

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My daughter took four AP courses during high school and now attends a selective liberal arts college.

 

She scored fours on her AP tests in US History and Comparative Government & Politics. She would have needed to score fives in order for her to obtain credit at her college.

 

She scored a five on AP Latin which would have fulfilled the foreign language requirement; however, since she is majoring in Latin/Classics this is a moot point. A score of four would have netted the same result.

 

Her five on the AP Statistics exam enabled her to fulfill one of the two requirements for a (hmm, drawing a blank on the correct terminology) data manipulation class. Once again, a score of four would have garnered the same result.

 

Regards,

Kareni

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AP credit is a mixed bag. In over 30 years, I have almost never seen a high school student who should have been advised to skip college calculus based on his/her high school class, because those classes, and AP tests, are vastly inferior to what I consider a decent college class.

 

But this raises a financial problem for the parent. If the student goes back and takes calculus over in college to learn it well, the course costs the parents money. Moreover it causes a recruiting problem for the school. If we refuse credit because we know they don't really know calculus, we lose the student to a college that will give credit.

 

By and large colleges that refuse credit at least in math, are doing so in my opinion not to gouge tuition, but because experience shows the inferior quality of these high school AP courses and tests. However this is a lost war, AP courses are too entrenched to eradicate, and only the most elite schools can afford to tell the truth about their less than collegiate quality.

 

Since AP prepared students are usually not really well prepared, colleges also have a placement problem. The result at public schools I know of has been a lowering of the quality of college offerings. Thus we give credit for AP courses, but to prevent the AP students from being swamped by the next class, we have lowered the quality of our college courses down to what AP high school classes are.

 

One compromise we maintain at UGA is our "super honors" calculus course, the one for future mathematicians. If a student has a 5, or at least a 4 on the AP calc test, and permission from the instructor, and wishes to take our "Spivak" style (very high quality) introductory class, we will still give AP credit.

 

Another more serious problem occurs at elite schools like Stanford and Harvard. They have so many AP students that they no longer offer even a Spivak style introductory class. Thus top students with AP credit who want the best honors course are thrown into the second year super honors class, from Apostol volume 2, for which most, no matter how strong, are not prepared. (Even the BC calc test has few proofs, whereas the Stanford honors class test is ALL proofs. As the professor at Stanford told me, "the technical prerequisite is a 5 on the AP, but that's not the real prerequisite. The real prerequisite is to be able to handle proofs, no apology.")

 

These elite courses, like math 55 at Harvard, are now populated mostly by students lucky enough to have taken a Spivak style class while in high school, either from a college like UGA, or at an elite private school like maybe Andover. One of the few top colleges that still offers Spivak intro calc is University of Chicago, long famous for good high level instruction. In my experience, Harvard and Stanford are more in the "sink or swim" category in honors math.

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AP credit is a mixed bag. In over 30 years, I have almost never seen a high school student who should have been advised to skip college calculus based on his/her high school class, because those classes, and AP tests, are vastly inferior to what I consider a decent college class.

 

.

 

Yes, this can be a real problem for traditionally schooled students who don't have a choice in curriculum. Fortunately, homeschoolers can choose their own curriculum, their own pace, and their own approach to the study. We used self study (using Spivak by the way).

 

As a practical matter for homeschooling students, the AP can be important. It provides some outside proof of the student's mastery and as a practical matter for many students it is a savings of money or time. The AP standard doesn't have to be the primary focus of a student's study - just a practical tool to help them get what they need for admissions and placement.

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I am very impressed that you used Spivak at home! My impression is that this level of preparation is unusual also in home schoolers, but I may be wrong.

 

If home school teachers do not have the information to design a course at the level of yours, I am afraid they will assume that the AP is the accepted standard of excellence, as traditional schools often do, rather than a degradation of the previous level of calculus instruction.

 

Do you think it is well known in home schooling circles that typical AP courses do not in fact match up with strong college courses?

 

It has been a painful experience for me for decades to watch students with only an AP high school preparation, struggle even in second semester non honors calculus. The failure or withdrawal rate was perhaps 50%, and that was with generous evaluation. The savings in tuition by skipping first semester calculus became somewhat illusory when ultimately I had to downgrade the content of the course for them, and half of them still failed and were forced to repeat the course.

 

It does seem that home schooling offers the potential to avoid this, and I hope this information helps some to do so.

 

In one regard at least, I would expect home schooled students to excel at a key skill for college success which is independent of curriculum,

namely I would expect that home schooled students have learned how to learn. This one thing can atone for a great many omissions.

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My impression is that this level of preparation is unusual also in home schoolers, but I may be wrong.

 

...

 

In one regard at least, I would expect home schooled students to excel at a key skill for college success which is independent of curriculum,

namely I would expect that home schooled students have learned how to learn. This one thing can atone for a great many omissions.

 

You are correct that this level of preparation is unusual in homeschoolers... but the Hive tends to attract a segment with higher levels than normal. ;) We have a few on here who have successfully used AP credit and gone on and did well in math heavy majors, but like you, it's not something I tend to recommend.

 

I've also seen many go from our high school college course (which is community college, not AP) on to college and struggle. It's not a pretty picture. Because of that, I never intended my guys to take the AP test in Calc even though we did Calc at home. Now, at school, the general recommendation is to retake Calc at college if one is in a math dependent major.

 

For our situation, oldest wishes he had taken the AP test as he would have tested out of math (for his major) at college. Instead, he got an easy A in the Business Calc class he needed, but he was bored and would have preferred to take something else. In hindsight, I wish I'd had him take the AP for his path.

 

Middle just got his college Calc book in the mail yesterday (Stewart). He looked through it and said it looks like stuff he's done and can handle. He's heading pre-med, so doesn't need the top Calc course his college offers (heavy on proofs) - just regular Calc. He needs an A, and an "easy" class, so redoing Calc is what we'd planned all along. "Easy" is in quotes because we're both aware that many of his fellow students will be at the same level prep he is at, so he's still expecting a challenge - just not as much of a challenge if he were to skip the course or not had Calc already.

 

As guidance counselors, it's a good idea to research carefully before sending our students out on their paths. The more we learn from the experience of others, the fewer mistakes we'll hopefully make...

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Do you think it is well known in home schooling circles that typical AP courses do not in fact match up with strong college courses?

 

No. Unfortunately AP is considered the Jewel in the Crown.

 

 

It has been a painful experience for me for decades to watch students with only an AP high school preparation, struggle even in second semester non honors calculus. The failure or withdrawal rate was perhaps 50%, and that was with generous evaluation. The savings in tuition by skipping first semester calculus became somewhat illusory when ultimately I had to downgrade the content of the course for them, and half of them still failed and were forced to repeat the course.

 

It does seem that home schooling offers the potential to avoid this, and I hope this information helps some to do so.

 

Frankly I was stunned to learn that AP required calculator based algorithms for estimation but did not have a single epsilon/delta limit problem. Talk about watering down the material!

 

Spivak and Apostol live on my shelves but I used neither with my son who does not share in his mother's love of mathematics. After using Dolciani (through the Intro Analysis text) we chose a more conventional book...for the sake of AP.

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Do you think it is well known in home schooling circles that typical AP courses do not in fact match up with strong college courses?

 

 

Dealing with this thought alone, it's not just AP. There are often questions wondering why upper level colleges won't give credit for community college courses too. ;) The "general" belief among parents and even some teachers at my high schools is that all courses with the same name are equal (eg Bio 101 = Bio 101 no matter where you take it and all course credits should transfer). It's almost impossible to convince them otherwise. They'll tell you that colleges who refuse to grant credit are doing it "just for the money."

 

I'm not in that camp BTW. I've even done my part to share my thoughts with the teachers at school who insist differently ("once you've taken my cc class you'll have credit anywhere that matters because it's all the same course"), but, I doubt I've been very successful. For some of the lower level schools students go to, they do get credit and the courses could, indeed, be the same. But for those who choose higher level schools... it doesn't work that way, nor should it.

 

On another note, MANY of our grads test into remedial classes at college... but that's getting off the topic I selected. It does, however, show that an A in Course A at one school does not necessarily = an A in Course A at another school or they should be testing OUT of Course A.

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I am very impressed that you used Spivak at home! My impression is that this level of preparation is unusual also in home schoolers, but I may be wrong.

 

Oh I'm sure it's unusual. We have a future mathematician. Most kids don't plan to be mathematicians so it makes sense there would be less depth there. That's the beauty of homeschooling is that it allows for that kind of individuality and flexibility. What we could do at home far exceeded the rest of the options available. One segment of the homeschooling community are kids who would be in the most selective gifted programs in public schools. Often they are doing quite a bit of college level work (both in the classroom and out) during high school, and often middle school as well.

 

If home school teachers do not have the information to design a course at the level of yours, I am afraid they will assume that the AP is the accepted standard of excellence, as traditional schools often do, rather than a degradation of the previous level of calculus instruction.

 

And, I guess then the worst that has happened is that the level of calculus instruction didn't exceed what the student would have gotten in their local public school. There are resources, including the Hive, where motivated homeschoolers can tap into the many options available.

 

Perhaps the responsibility needs to fall to some extent on the shoulders of universities as they decide their policies for granting credit. So, for example, perhaps they should not grant two semesters of credit for a 3 on the BC calc exam (as many still do). More honest and open advising of incoming freshman would also be in order. If half of the AP students fail Calc II that's information students should receive in advising.

 

Also, I've noticed a trend that some colleges are offering "boot camps" or special support programs to help bridge the gap between high school math instruction and college level courses. The need for this type of program is certainly evidence that something in the system is broken, but at least these colleges are trying to offer greater possibility that students will be successful.

 

Do you think it is well known in home schooling circles that typical AP courses do not in fact match up with strong college courses?.

 

Any efforts to generalize about the homeschooling population are doomed to be unsuccessful! Contrary to popular stereotypes homeschoolers are an incredibly diverse and complex collection of individuals representing every extreme - left, right, and center. There are families who don't know homeschoolers can take APs. There are families who are homeschooling in part because they hate the public school focus on APs. There are families who build their high school strategy primarily around APs.

 

And, there are quite a lot of us who have a pragmatic view of APs. Don't make it the sole focus of your homeschooling, but be open to using it as a tool that will help with admissions and placement. Particularly for students planning to attend their state u, APs can reflect a huge cost savings and make it easier for students to be placed in courses they will find more challenging. Entering college with a lot of credit can clear the way to a lot of freedom and flexibility. It can get the student out of classes with 250 people faster and it can clear the way to finish a dual degree or double major. Or, for students who need to save money, it can open up the possibility of graduating early or beginning a master's degree.

 

It has been a painful experience for me for decades to watch students with only an AP high school preparation, struggle even in second semester non honors calculus. The failure or withdrawal rate was perhaps 50%, and that was with generous evaluation. The savings in tuition by skipping first semester calculus became somewhat illusory when ultimately I had to downgrade the content of the course for them, and half of them still failed and were forced to repeat the course.

 

And, the College Board gets richer.... It is a frustrating situation. What I see here is that so many traditionally schooled kids are pushed into APs (not calc, but other APs) when they really aren't ready for that level work. The pass rates on a lot of the exams are quite low. Our district sounds the trumpet that they are so great because so many students are taking APs, and they gloss over the fact that they aren't passing!

 

Do you think that this failure of calc students at your university also reflects something about the quality or approach to K-12 math education in your state?

 

In one regard at least, I would expect home schooled students to excel at a key skill for college success which is independent of curriculum,

namely I would expect that home schooled students have learned how to learn. This one thing can atone for a great many omissions.

 

Yes, and I'll add to that a very common compliment of homeschoolers receive is that they've retained that childlike sense of curiosity and love of learning. Far too many kids lose that in schools that don't fit their needs. If you are curious and you know how to learn - that's the golden combination.

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And, the College Board gets richer.... It is a frustrating situation. What I see here is that so many traditionally schooled kids are pushed into APs (not calc, but other APs) when they really aren't ready for that level work. The pass rates on a lot of the exams are quite low. Our district sounds the trumpet that they are so great because so many students are taking APs, and they gloss over the fact that they aren't passing!

 

This is true at our public school as well. Many Ap's offered, but not much success on the actual test. I requested the AP results one year: Only 18 kids took Calc (each graduating class is around 225 students) - 16 students received a "1", 1 student received a "2" and the remaining student received a "3." Yet Newsweek considers our public school one of the best public schools in the nation.

 

Do you think that this failure of calc students at your university also reflects something about the quality or approach to K-12 math education in your state?

:bigear:

 

The math program used at the elementary level at our public school actually gave me the confidence to homeschool. I figured I could not possibly make educational decisions for my kids that would be worse than what the "experts" had chosen.

 

Our elementary school adopted Everyday Math. Within two years of using the program, the tests results on the state tests had fallen from 97% being proficient in math, to only 80% of the same kids being proficient on the State tests. However, since our state only requires that 75% of the kids achieve the proficient status, our public school still meets the state's highest standard for "excellence."

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This is true at our public school as well. Many Ap's offered, but not much success on the actual test. I requested the AP results one year: Only 18 kids took Calc (each graduating class is around 225 students) - 16 students received a "1", 1 student received a "2" and the remaining student received a "3." Yet Newsweek considers our public school one of the best public schools in the nation.

 

Wow, I could have written this back in my school's (the one where I work) AP days except our graduating class has >300 generally and fewer took the Calc AP test. The scores were similar though. Our school don't rank with anything nationally and is below average in our state in state rankings.

 

The math program used at the elementary level at our public school actually gave me the confidence to homeschool. I figured I could not possibly make educational decisions for my kids that would be worse than what the "experts" had chosen.

 

Our elementary school adopted Everyday Math. Within two years of using the program, the tests results on the state tests had fallen from 97% being proficient in math, to only 80% of the same kids being proficient on the State tests. However, since our state only requires that 75% of the kids achieve the proficient status, our public school still meets the state's highest standard for "excellence."

 

Ditto this too. When our school switched to fuzzy math I switched, but not soon enough for youngest. He was behind 2 years by the end of 4th grade. We spent middle school catching up to where he could do Alg 1 in 8th successfully (as normal kids do here). He's in the top group of kids now in high school, but like kids at our high school, this means roughly average on the SAT/ACT. I wish I could convince him to keep homeschooling. He admits the education at home is superior, but he prefers ps for the socializing and NOT liking a more rigorous education. I have to keep reminding myself I'm training him for his path, not my preferred path for him. He'll do ok for what he wants (not math or English dependent), but I wish our school were better so he could be more academically educated.

 

Most parents don't see the inside of their ps in high school and just assume they are doing well (the school is educating students well). It's not always true. I'm glad the two of mine who were more academically inclined were able to homeschool. They had far more than this ps peers (unless you count gym or such things).

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Most parents don't see the inside of their ps in high school and just assume they are doing well (the school is educating students well). It's not always true. I'm glad the two of mine who were more academically inclined were able to homeschool. They had far more than this ps peers (unless you count gym or such things).

 

:iagree:

I am so glad that I made the decision to homeschool my kids and that my kids are thankful to be homeschooled.

 

I was once one of the naive parents that just looked at the school's published information (and my school system is great with public relations). Each fall, every household receives a glossy calendar in the mail that promotes the state ranking and various other accolades, such as Blue Ribbon and Newsweeks list.

 

It wasn't until I realized how horrid the math program was that I started to dig deeper into the actual stats. The vast majority of parents don't even know that one can request AP scores. In fact, initially the guidance counselor refused to provide me with the information when I requested it.

 

It would be more meaningful if these rankings were based on actual AP results, but I don't ever see that happening.

 

I am just thankful that I figured it out while my kids were still young enough to make a difference.

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This is obviously a very sophisticated group, much more than my average entering student population. For what it's worth, I reproduce a letter (edited) that I wrote in frustration to my sons' private school in 1997. My main interest here is to suggest being very careful about using AP tests either to measure competency or to determine college placement.

 

A question about the value of AP courses

At a parent- teacher meeting some years ago I argued that our school was not particularly difficult, observing that it had only one AP course

(then calculus). The headmaster patiently observed that there were a number of courses which, although not technically AP courses, were quite difficult and advanced. In spite of the simple truth of this argument I did not give in.

 

Over the years I have gotten my wish as AP courses proliferated. When our older son went off to college an odd thing happened: he had difficulty in his advanced mathematics course, into which he placed by virtue of his AP preparation, whereas he "dominated" in his English course, for which he had prepared by taking traditional advanced courses in English literature.

 

Suddenly, I regretted not listening to the headmaster’s argument more closely. I now feel that this whole AP revolution is regrettable, in fact that it is actually hurting the cause not only of good education but of good college preparation.

 

I believe there are two reasons for this: first, AP courses are designed to prepare people to answer multiple choice questions on chosen topics,

while the traditional courses, especially the honors seminars, are designed simply to teach people to read closely, analyze deeply, and to discuss and write effectively. These latter skills are much more useful in college and elsewhere, than is familiarity with a particular AP syllabus.

 

Second, students do not realize that the name AP is often a complete misnomer, and that AP courses are not at all equivalent to college courses. Consequently a student coming out of a traditional honors high school course is likely to take a beginning college course in the same subject (possibly an honors section) for which he is well prepared, while the AP student often tries to skip the introductory college course in his subject and enter an intermediate course, for which, in my experience, he is seldom even adequately prepared. There may be some miscommunication between admissions officials and professors, but the professors I know actually prefer to teach beginning calculus to people who are well versed in algebra and geometry, but who have not had calculus.

 

To some extent colleges are accommodating the situation by gradually making college courses easier, in response to the weaker preparation

students have today, but this is hard to do perpetually. Our difficulty is that students today have a shallow grasp of more and more advanced

subjects, when we would prefer them to have a deeper grasp of basic subjects. In my opinion AP courses are a primary cause of this problem,

and I hope something can be done to retard their advance at the expense of outstanding and unique honors courses at this school, before it is too late.

 

 

[i received the following response from a friend who is the author of a famous calculus text.]

 

It may be heretical, but it is 100% accurate. You have expressed the failure of the AP program and the success of your son’s and similar schools as well as it could be expressed. My simplistic version---students who take AP calculus merely learn a superficial calculus course without deep understanding, and thereby waste a year forgetting algebra---is not as well put as your analysis. Congratulations and send it to every editorial page in the country!

 

 

That last sentence encourages me to post it here.

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@ Creekland: I think one placement problem is that AP tests can cause an honors high school student to place into the second semester of a non honors college course, which is the wrong population for a bright kid. They belong in the honors course, which they would have been in had they not gotten "advanced" placement. At UGA we have three or 4 levels of intro calculus, terminal, 1st semester continuing quality, honors level, and future mathematician level. This is unusual today, since AP courses have reduced the audience for many of these courses, or at least reduced the audience that realizes they should be in one of these.

 

Most entering AP students do not take any of these but enter in, and struggle in, some second semester course.

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AP credit allowed me to graduate Stanford in 3 1/3 years and my DH to graduate with dual majors in electrical engineering and history in 4. I could've graduated in 3 except there was one course required for my major that was only offered in the fall and the scheduling didn't work out to take it earlier. :glare: The only thing that we got completely out of was the foreign language requirement but we were able to skip the intro courses for many subjects.

 

Contrary to what some of the PP have said here, neither of us struggled in the higher-level courses after APing out of the intro ones. Now neither of us were interested in taking the math major track so I can't say how well prepared we would've been for those classes (honestly, I don't think that even the most rigorous math classes at an elite prep or public exam school could've gotten me through those as I'm not gifted mathematically). But for the sciences we AP'd out we did well in the subsequent courses. DH actually did better in the chem class that he placed into after his AP chem course than I did after taking the intro class at Stanford (my H.S. didn't offer AP chem when I attended it).

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It may be heretical, but it is 100% accurate. You have expressed the failure of the AP program and the success of your son’s and similar schools as well as it could be expressed. My simplistic version---students who take AP calculus merely learn a superficial calculus course without deep understanding, and thereby waste a year forgetting algebra---is not as well put as your analysis. Congratulations and send it to every editorial page in the country!

 

That last sentence encourages me to post it here.

 

Yikes! I am starting to get the pit in my stomach feeling regarding the choice I have made for my oldest. Part of me wishes I had the courage to do what 8filtheheart did, but another part of me feels that I don't have the energy or expertise at this point to help ds prepare for the AP exam. Hopefully, some of you can chime in and offer advice:

 

I am a huge AoPS fan. My oldest has taken all of their online courses through pre-calc. Unlike most kids who use AoPS, he does not love math, he just barely qualified for AIME and scored average on the test, he does not always enjoy working on proofs, etc. He plans to major in engineering and then hopefully go on to medical school.

 

My current plan that I am now seriously questioning, was to have him take BC calc with PA Homeschoolers this year, and then take AoPS calc next year after he had the "mechanics" down.

 

Any advice? Is this a poor idea? Sometimes I wish I did not worry so much about outside validation, but I think scoring a "5" on the AP exam is the language that admissions officers understand and I am not sure that a "5" would be possible going the AoPS route if he didn't spend time prepping for the exam.

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AP credit allowed me to graduate Stanford in 3 1/3 years and my DH to graduate with dual majors in electrical engineering and history in 4. I could've graduated in 3 except there was one course required for my major that was only offered in the fall and the scheduling didn't work out to take it earlier. :glare: The only thing that we got completely out of was the foreign language requirement but we were able to skip the intro courses for many subjects.

 

Contrary to what some of the PP have said here, neither of us struggled in the higher-level courses after APing out of the intro ones. Now neither of us were interested in taking the math major track so I can't say how well prepared we would've been for those classes (honestly, I don't think that even the most rigorous math classes at an elite prep or public exam school could've gotten me through those as I'm not gifted mathematically). But for the sciences we AP'd out we did well in the subsequent courses. DH actually did better in the chem class that he placed into after his AP chem course than I did after taking the intro class at Stanford (my H.S. didn't offer AP chem when I attended it).

 

I'm not anti-AP at all. I wish my guys had done more of them in the past. But I do think parents should think through taking credit for the courses (if available) wisely. Middle son is accepting credit for Psych and Stats - will start with a higher level Psych class and will still do Calc based Stats when he needs it. There's no reason for him to start at entry level Psych IMO.

 

We skipped taking Bio and Chem tests, but I now wish I'd had him take Bio as a 5 on that test is the PRE-REQ for an intro Bio class he'd have enjoyed taking this coming year, but tough luck. He doesn't have it since he didn't take the test. We never would have taken credit for either exam since he's going pre-med (possibly).

 

We skipped History at the AP level completely, but could have done it. I'm not sure if he'd have used credits or not.

 

Oldest would have loved to have had AP Calc credits fulfill his math requirements at college. I regret not letting him take the test as it just made him retake a class he was bored in (but it was an easy A). He wouldn't need more math with his major.

 

These are things that need to be looked at wisely - not with a broad blanket that supposedly covers every student. Like anything else, there is no one right answer IMO.

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These are wise comments. Some bright high school students to whom I have told my concerns have said that their AP courses really were an improvement in many cases over the ones that were in place before.

 

I myself once taught a vector calculus class called "beyond calculus" to students in a private high school who already had AP math, from a book by Marsden and Tromba that was used at Berkeley, and some of my students protested that they did not expect or need such a tough class. One of them who dropped out, later did "just fine" in a presumably non honors calculus class at a state school and claimed that justified his objections to my hair - shirt standards.

 

Another student however who went to Harvard and took second year non honors calculus there from a world famous mathematician, said it was so advanced he "could not have survived" without our prior introduction to such topics as integration of differential forms. Of the 5 finishing students, one graduated Yale and then got a PhD in math, and another graduated Harvard and then got a PhD in physics. Another high school student who only took Spivak style precalculus from me for one summer month (rigorous development of real numbers from Spivak's appendix) went to U of Chicago and survived the honors intro calc class.

 

One has to be very clear about what ones goals are, whether ones goals are to pass a certain test or a certain course, or whether one wants to understand a subject deeply.

 

E.g. if one wants to be a pre med student and does not need a deep grasp of calculus, but just needs a credit in it, or a good grade, then an AP exemption may be just the ticket. (But hopefully she/he learns biology/chemistry as deeply as possible.) The same holds for people using math courses only to satisfy a formal requirement for further study, even in scientific fields such as chemistry and engineering.

 

But if one wants to be in a stimulating, challenging math course with the best other students, and possibly become a mathematician or theoretical physicist, one can probably not afford to allow an AP score to be the only measure of ones math understanding. AP courses are not inherently bad, but using them for more than they deserve is hazardous to ones education.

 

They may have many uses, but in my own experience the one thing they are not good at is the one thing they claim to do, namely justify advanced college placement. If they were called "standardized short answer tests of minimal computational proficiency", then they might be less misleading.

 

Even this however is changing, as I have said. I.e. AP tests are so ubiquitous that they have indeed become the standard in many places. Thus in some schools and some courses, a certain AP score is indeed the de facto correct criterion for advanced placement, because the quality of the college course has declined to allow this to be true. Even in the same department there may be professors who teach at the level of AP preparation and some who vastly exceed it. Uniformity is hard to enforce. So do some homework.

 

But if a student goes to Stanford and enrolls in the second year honors course from Apostol, or Harvard's math 55, with preparation equivalent to only a 5 on the AP test, after learning from a book like Stewart, he/she better be ready to be shocked by what will be expected. One should always ask the person teaching the course what that is. E.g. when I called the Stanford professor teaching the Apostol class he freely told me that the formal AP prerequisite was not the real prerequisite. Unfortunately I did not call him 2 years in advance. One recent success I know in math 55 prepared by taking essentially a complete math major including graduate courses at UGA before enrolling in college.

 

So there is no simple answer. The point is to take advantage of whatever the AP courses and tests have to offer, but not to assume they are always what they claim to be, namely proof of suitability of advanced placement. Always consult with the professor. A basic rule may be that AP scores are useful for impressing admissions, but professors want real knowledge and understanding.

 

Caveat: I am getting much too preachy. Fortunately you guys are wise enough to separate wheat from chaff! In my defense I have been frustrated for decades because AP courses have made much harder my life's goal of trying to do a good job of high quality college teaching .

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Someone asked about K-12 preparation in my state. I will try to keep this briefer, but I was once charged with evaluating the testing materials that were used to prepare the high school math teachers in my state for certification, so I know something of this topic as well.

 

They were so flawed, (wrong sample answers, wrong explanations, huge gaps in test coverage plus enormously unrealistic syllabus, and so on....), that I eventually conjectured that these materials and tests had been prepared not by experts but by (then) currently certified teachers, in sort of a grandfathering system - i.e. if you are certified, then not only are you qualified but you also determine future certifications. This was indeed confirmed.

 

Everyone in the world knows a more recent news story about K-12 preparation in our state.

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Caveat: I am getting much too preachy.

 

Shall I get you started on math texts written by math educators--not mathematicians? :lol::lol::lol:

 

Preach it, Mathwonk!!

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possibly become a mathematician or theoretical physicist, one can probably not afford to allow an AP score to be the only measure of ones math understanding.

 

But you're only talking about a very small percentage of students even at a school like Stanford. Only 2% of Stanford's undergraduates major in math (16th most popular) and only 1.8% major in physics (18th most popular) compared to 12.6% who major in human biology and 8% who major in economics (the first and second most popular majors).

 

The overwhelming majority of students who take a math course at Stanford are doing so not because they are passionate about the field but because it's a requirement for pre-med or other health profession, engineering, or pre-MBA.

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Another more serious problem occurs at elite schools like Stanford and Harvard. They have so many AP students that they no longer offer even a Spivak style introductory class. Thus top students with AP credit who want the best honors course are thrown into the second year super honors class, from Apostol volume 2, for which most, no matter how strong, are not prepared. (Even the BC calc test has few proofs, whereas the Stanford honors class test is ALL proofs. As the professor at Stanford told me, "the technical prerequisite is a 5 on the AP, but that's not the real prerequisite. The real prerequisite is to be able to handle proofs, no apology.")

 

 

Interesting thread!

 

I went straight into 3rd-quarter calculus (multivariable) at Stanford, thanks to a 5 on the BC, had no problems, it saved me time, etc. but this was just the standard engineering-track calculus. More relevant to this thread is my 15yo son's future ... I was already planning to incorporate more proofs into his high-school coursework, because he is a future math (or perhaps computer science) major, and this thread reinforced the importance of that. He received a 5 on the BC test in 9th grade (having skipped a grade), and TA'd an AP Calc AB/BC class last year with over 40 students (and will again next year) ... none of which will help him in a high-level class at an 'elite' school (should he be fortunate enough to attend one)! He has had an introduction to writing proofs at two summer math camps, and he is planning to take the WOOT class offered by Art of Problem Solving. Without math camp and AoPS I can see that his preparation for college math-major classes would be woefully inadequate -- the courses he has taken at our local community college (ODEs, etc.) just manage to cover the basics. Oh, and the many math competitions he participates in always include a "power round," in which teams of students collaborate to write proofs (for which they are given the better part of an hour). I am left wondering if all this will be enough ... :confused:

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Interesting thread!

Yes, it is! I've been glancing at it all weekend, but my son was visiting, and I didn't want to take time away from him. Now he's safely on his plane back to CA, though.:)

 

But you're only talking about a very small percentage of students even at a school like Stanford. Only 2% of Stanford's undergraduates major in math (16th most popular)

 

My daughter is a rising junior majoring in math at Stanford, and during her freshman year she took that set of honors math classes that mathwonk is talking about in this thread.

 

It should be noted that this math sequence we're discussing (Math 51h, 52h, 53h) is not even required for the math major at Stanford, only for the honors track.

 

Over 100 kids started in Math 51h in September; after the first midterm almost 2/3 of the class dropped out. By the third quarter Math 53h class, only about 30 kids remained (so about 2% of the freshman class, the figure quoted by Crimson Wife). My daughter was one of four women to make it that far in her year (and the only non-Asian, non-Russian woman).

 

Some bright high school students to whom I have told my concerns have said that their AP courses really were an improvement in many cases over the ones that were in place before.

 

Here in suburban Richmond, I can't imagine any better alternatives for the public school kids, even in my county's math-sci magnet school. While the AP courses certainly have their faults, my county has done away with most 11th and 12th grade honors courses. so the kids are left with AP English (or whatever subject) or regular mainstream English. I can't even imagine what they'd learn in that regular class!

 

I myself once taught a vector calculus class called "beyond calculus" to students in a private high school who already had AP math, from a book by Marsden and Tromba that was used at Berkeley, and some of my students protested that they did not expect or need such a tough class.

 

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!

 

Unfortunately, as homeschoolers, that class had no 'documentation' in the eyes of college admissions officers - just a mommy grade and course description. We homeschoolers have to work twice as hard as schooled kids to accumulate a hard record of accomplishments if our kids want to go to competitive colleges. So in addition, my kids did do lots of APs and SAT 2 tests and math contests, but they were a tool, not an end in themselves. Like Jane in NC, I dislike the Calc AP test in particular, due to its heavy reliance on calculators and lack of proof. That doesn't mean that I have to spend more than a day or two teaching the basics of how to use the TI calculator functions, or that I can't add a heavy dose of proofs to my version of AP Calc.

 

Whether homeschool, public school, elite private school, gifted program, or community college, a student's learning experience is much more likely to be tied to the skill of his or her teacher than to any other factor. An AP course can be outstanding or a complete waste of time.

 

But if a student goes to Stanford and enrolls in the second year honors course from Apostol, or Harvard's math 55, with preparation equivalent to only a 5 on the AP test, after learning from a book like Stewart, he/she better be ready to be shocked by what will be expected. One should always ask the person teaching the course what that is. E.g. when I called the Stanford professor teaching the Apostol class he freely told me that the formal AP prerequisite was not the real prerequisite. Unfortunately I did not call him 2 years in advance. One recent success I know in math 55 prepared by taking essentially a complete math major including graduate courses at UGA before enrolling in college.

 

Roy, I know we talked about Stanford math last summer at camp, but I don't think that we ever talked about the textbooks used in the 50h series. Did Stanford use Apostol v2 when your son took the class?

 

They don't use Apostol any more ( I wish they had! I think it would have been more accessible). Leon Simon teaches 51h now, and he used this book, which is just a compendium of his lecture notes on linear algebra and real analysis, according to this outline. I used to have a link to the final exam given to L's class, but it's been taken off-line. As you said, it was almost 100% proofs (and given from 7 to 10 pm at night -- I don't know how those kids had the mental stamina to make it through).

 

The second quarter class, 52h, (topics list)is now taught by Yakov Eliashberg, using online lecture notes rather than any textbook. This was L's favorite course of the trio.

 

Math 53h, also taught by Prof. Eliashberg, uses Ordinary Differential Equations by V I Arnold. Judging by one of the reviews on Amazon, I would guess that you like this text?

 

The 51h course page has several difficulty warnings these days, and most of the students are well aware of what they're getting into (there are links to a "Survival guide" and a "how to drop this class, even after the drop date" advice pages). But face it, most of the kids attempting 51h were high school superstars, and they come to Stanford believing that advice doesn't apply to them...I know for a fact that many are sad that they can't keep up...it really is a shocking experience for many smart kids). I personally felt that while there was a ton of good math material, that each of these 10-week classes could have been done more justice at a slower pace. Maybe it's just my age showing!

 

My daughter took many classes (epgy, AoPS, home-brewed) far more advanced than AP calculus, but she knew from years of attending Mathcamp and traveling to math tournaments like PUMaC and HMMT that many kids out there are better than she is when it comes to the really hard stuff. So she went in with that attitude, didn't expect to earn straight A's, and just tried her darnedest to *learn* something. During the process, and even after the first year ended, she felt a bit frustrated at the difficulty of it all, but in retrospect (as a really wise sophomore, LOL), she's glad that she did stick it out.:001_smile: And as an aside, we see that while it's easy to earn an A/B grade in the humanities and social sciences at Stanford, the grades given in math and science (esp the honors level - she did the honors level computer science classes this year) aren't given out easily and really have to be earned. She finally received an A in her last comp sci class this past year, and she's proud as heck (as are her parents).

 

Someone asked about K-12 preparation in my state. I will try to keep this briefer, but I was once charged with evaluating the testing materials that were used to prepare the high school math teachers in my state for certification, so I know something of this topic as well.

 

I taught in the math department at your rival cross-state tech university in the mid 80s. Yeah, I'll agree that the preparation of the kids coming from the GA public schools left a lot to be desired.

 

More relevant to this thread is my 15yo son's future ... I was already planning to incorporate more proofs into his high-school coursework, because he is a future math (or perhaps computer science) major, and this thread reinforced the importance of that. He received a 5 on the BC test in 9th grade (having skipped a grade), and TA'd an AP Calc AB/BC class last year with over 40 students (and will again next year) ... none of which will help him in a high-level class at an 'elite' school (should he be fortunate enough to attend one)! He has had an introduction to writing proofs at two summer math camps, and he is planning to take the WOOT class offered by Art of Problem Solving. Without math camp and AoPS I can see that his preparation for college math-major classes would be woefully inadequate -- the courses he has taken at our local community college (ODEs, etc.) just manage to cover the basics. Oh, and the many math competitions he participates in always include a "power round," in which teams of students collaborate to write proofs (for which they are given the better part of an hour). I am left wondering if all this will be enough ... :confused:

 

Laura,this is all very good! Has your son considered participating in USAMTS next year? That alone taught my kids a lot about proof writing. It's free, they have at least a month to complete each round of proofs, and their proofs are read & commented on by real mathematicians at the NSA as well as by AoPS grading groups of university math majors who know their stuff. The kids can learn a lot if they compare their proofs to the solutions posted online after each round. It was motivating to my kids at least.:) And I always recommend thinking deeply about a few problems and working hard to get past the roadblocks.

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A question about the value of AP courses

At a parent- teacher meeting some years ago I argued that our school was not particularly difficult, observing that it had only one AP course

(then calculus). The headmaster patiently observed that there were a number of courses which, although not technically AP courses, were quite difficult and advanced. In spite of the simple truth of this argument I did not give in.

 

Over the years I have gotten my wish as AP courses proliferated. When our older son went off to college an odd thing happened: he had difficulty in his advanced mathematics course, into which he placed by virtue of his AP preparation, whereas he "dominated" in his English course, for which he had prepared by taking traditional advanced courses in English literature.

 

Suddenly, I regretted not listening to the headmaster’s argument more closely. I now feel that this whole AP revolution is regrettable, in fact that it is actually hurting the cause not only of good education but of good college preparation.

 

I believe there are two reasons for this: first, AP courses are designed to prepare people to answer multiple choice questions on chosen topics,

while the traditional courses, especially the honors seminars, are designed simply to teach people to read closely, analyze deeply, and to discuss and write effectively. These latter skills are much more useful in college and elsewhere, than is familiarity with a particular AP syllabus.

 

Second, students do not realize that the name AP is often a complete misnomer, and that AP courses are not at all equivalent to college courses. Consequently a student coming out of a traditional honors high school course is likely to take a beginning college course in the same subject (possibly an honors section) for which he is well prepared, while the AP student often tries to skip the introductory college course in his subject and enter an intermediate course, for which, in my experience, he is seldom even adequately prepared.

 

:bigear: to all of you.

 

While I have a long while before dealing with this topic in practice with my own children, this was my experience going through high school as well. My high school offered AP tests only as an option, no AP classes, prep was mostly on your own, and went the Honors route instead because they felt there was more latitude for challenge and that it was better preparation for college. I think they have been pushed into going the AP route now due to parental demand, but I agree with you 100% and felt very well prepared for my "public ivy" college experience. In fact, had I realized how easy college would be after my high school, I would have chosen a different university.

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I am really enjoying the comments here! To add some data to the philosophy I have espoused, I looked up a BC calculus test online and took part of it tonight. I assumed I would get 100% but did not. To make it "fair" I did it in my head without pencil or paper while watching a couple of Matt Damon/Jason Bourne movies and having dinner with wine, but I missed two of the first 14 computational problems. One was a minus sign and one was a constant I overlooked, but I felt somewhat downcast and defensive. I also cheated and used a pencil to draw the graph of r = 1 + 2cos(t) for 0 ≤ t ≤ 2pi.

 

I did this also partly so those who wish to assess my comments about these tests may have more in formation as to what the comments mean. Indeed if anyone gets all these problems right he/she is a strong calculator who knows a lot of basic formulas. Few of my typical students could have done these at all accurately. I.e. a good performance on this test is something to be proud of. I just like and recommend other types of questions more. These questions are all straightforward, if you know the method, but the computations were still tedious. I usually ask questions that are fairly easy for someone who knows the method.

 

As a mathematician I also found the test uninteresting. To work hard on a problem, it helps if there is some challenge to it or some interest. Thus to me the most interesting ones were the ones that required a calculator, since I did not have one. (By the way, calculators are never allowed in my courses and not in most other college math courses.) E.g. to use calculus to maximize the area of a rectangle based on the x axis, and with upper vertices on the graph of y = cos(x), with -pi/2 ≤ x ≤ pi/2, one needs to solve the equation x.sin(x) = cos(x), not so easy in ones head.

 

But I just tried x = pi/4 and x = pi/3, and found that the maximum must be greater than 1.1 and apparently lies somewhere in between those two points. This ruled out all multiple choice answers except roughly .8 and .9. As to which one, I guessed wrong, but then I looked more closely and by taking the value halfway between those two found the right answer to be indeed more likely.

 

Another easier question not needing a calculator that I also got wrong was to recognize which of 5 or 6 possible sums could be a Riemann sum for the volume of a solid lying over the rectangle between x=0 and x=2 and y=0 and y=1, and with height over (x,y) equals to 1+3x. The first thing that occurs is that only a very compulsive person would use calculus to compute this volume since the shape is so simple it is easily computed as half a certain rectangular parallelepiped.

 

Secondly the question is annoying because there are at least three directions in which to "slice" the volume up into areas, and so one has to try all of them to see which one has been chosen in the problem. What is being tested here, patience? Finally I got it wrong because the answer was written deceptively with a constant factor pulled out that I did not multiply back quite correctly, so i got it wrong for a trivial arithmetic reason. What is being tested here, arithmetic?

This is a question that has no use in real life. In real life one might want to write a Riemann sum for such an integral, but not to recognize someone else's.

 

I missed a differential equation question by being lazy and trying all 5 answers but made a mistake with a constant as mentioned above. After seeing my error, I easily solved the equation more easily directly in my head.

 

I eventually lost interest and stopped. There were no definitions to give, no theorems to state, and no proofs. My tests usually involve all 4 aspects, definitions of concepts, statements of theorems (sometimes via true false questions, which is also done on the AP), computations, and proofs.

 

Some picky remarks about the AP: inconsistent use of the terminology for "function", confusing a function (which has a precise domain) with the formula defining it, also deviating from the notation f and f(x) used in the instructions, by using "variable" expressions such as x = t^2 - t +6, instead of x(t) or f(t). Use of terminology not entirely standard in instructions, such as "use Euler's method to approximate a solution to this differential equation" instead of "use differential approximation" or just "approximate and explain your method". Hence to do as well as possible on such a test one should practice beforehand, not just on the math, but on old examples of AP tests, to learn the language and style of questions.

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Here for comparison, are some questions from my second semester honors course. Unlike those on an AP test these are meant to be easy for someone who understands the concepts. There are however concepts that are not usually found in high school courses. E.g. it is common to define integrals and then to state but not prove, that continuous functions have integrals. Thus continuous functions are a subclass of integrable functions. Then the fundamental theorem of calculus is stated for continuous functions. A curious person (e.g. potential mathematician) might ask what the fundamental theorem would say for the more general class of integrable functions. Here the concept of "Lipschitz continuity" occurs naturally. Another type of functions for which integrability is easier to prove, and which suffices for almost all practical cases, is the class of monotone functions. Thus one can easily treat this case in more detail.

 

 

2310H Test 2 Fall 2004, Smith NAME:

no calculators, good luck!

1. (a) Give the definition of "Lipschitz continuity" for a function f on an interval I.

 

(b) State a criterion for recognizing Lipschitz continuity in the case of a differentiable function f on an interval I.

 

© Determine which of the following functions is or is not Lipschitz continuous, and explain briefly why in each case.

(i) The function is f(x) = x^1/3, on the interval (0, infinity ).

 

(ii) The function is G(x) = [t] , on the interval [0,10], (where [t] = "the greatest integer not greater than t", i.e. [t] = 0 for 0≤t<1, [t] = 1 for

1 ≤ t < 2, [t] = 2 for 2 ≤ t < 3, etc....[t] = 9 for 9 ≤ t < 10, [10] = 10.)

 

(iii) The function is h(x) = x + cos(x) on the interval (- infinity, infinity ).

 

2. (i) State the "fundamental theorem of calculus", i.e. state the key properties of the indefinite integral function G(x) associated to an integrable function f on a closed bounded interval [a,b]. You may assume f is continuous everywhere on [a,b] if you wish.

 

(ii) Explain carefully why the definite integral of a continuous function f on [a,b], equals H(b)-H(a), whenever H is any "antiderivative" of f, i.e. whenever H'(x) = f(x) for all x in [a,b]. Justify the use of any theorems to which you appeal by verifying their hypotheses.

 

(iii) Is there a differentiable function G(x) with G'(x) = cos(x^2)?

If so, give one, if not say why not.

 

3. Let S be the solid obtained by revolving the graph of y = e^x around the x axis between x=0 and x=3. Define the moving volume function V(x) = that part of the volume of S lying between 0 and x. (draw a picture.)

 

(i) What is dV/dx = ?

 

(ii) Write an integral for the volume of S, and compute that volume.

 

4. Consider a pyramid of height H, with base a square of side B. Define a moving volume function V(x) = that part of the volume of the pyramid lying between the top of the pyramid, and a plane which is parallel to the base and at a distance x from the top.

 

(i) Find the derivative dV/dx. [Hint: By similarity, b/B = x/H.]

(ii) Find the volume V(H).

(iii) Make a conjecture about the volume of a pyramid of height H with base of any planar shape whatsoever, and base area B.

 

EXTRA: Either: Prove the FTC stated part 2(i); you may draw pictures and assume your f is monotone and continuous if you like.

Or: ask and answer your own question.

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If anyone wants to teach proof to their child, I strongly recommend the Green Lion edition of Euclid's Elements for under $20. One will also benefit from using it together with Hartshorne's companion book, Geometry: Euclid and beyond (about $50), or the more modest, but free, notes on my webpage at UGA math dept, from epsilon camp.

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These are wise comments. Some bright high school students to whom I have told my concerns have said that their AP courses really were an improvement in many cases over the ones that were in place before.

 

AP would be a vast improvement over what our school offers (a cc class taught exclusively in the high school). We used to offer AP (AB only) and rarely did someone get greater than a 2. Most got 1s - that's if they took the test. Most opted out. Now they don't have to worry about the test at all. They get credit based upon classwork alone, but their knowledge isn't any better. They go to college and test into remedial math. Even our school doesn't recommend most accept the credit they "have" from the class unless it totally fulfills a math requirement.

 

 

One has to be very clear about what ones goals are, whether ones goals are to pass a certain test or a certain course, or whether one wants to understand a subject deeply.

 

...

 

So there is no simple answer. The point is to take advantage of whatever the AP courses and tests have to offer, but not to assume they are always what they claim to be, namely proof of suitability of advanced placement. Always consult with the professor. A basic rule may be that AP scores are useful for impressing admissions, but professors want real knowledge and understanding.

 

 

:iagree: The absolute vast majority of students do not need Calc all that deeply and many paths do not need Calc at all. For those who do, I almost always suggest taking the whole series at their 4 year school to be certain they are where they need to be with knowledge of the subject.

 

The most important thing is to choose each track based upon each student and their path. My youngest will probably never need Calc at all (nor Physics). I'm ok with that 100% even though I'm a Physics major so it almost "kills" me to allow him his path. His niche is is flora and fauna and he can go far deeper into those subjects than I've ever been able to even though he's just a rising junior in high school. We're all created for our niche and do best when in it.

 

But you're only talking about a very small percentage of students even at a school like Stanford. Only 2% of Stanford's undergraduates major in math (16th most popular) and only 1.8% major in physics (18th most popular) compared to 12.6% who major in human biology and 8% who major in economics (the first and second most popular majors).

 

The overwhelming majority of students who take a math course at Stanford are doing so not because they are passionate about the field but because it's a requirement for pre-med or other health profession, engineering, or pre-MBA.

 

I appreciate schools that have different calibers of math classes to allow students to choose how deeply they want to delve. My pre-med or med researcher wannabe needs Calc, but not as a be all, end all. A super tough class could kill his med school dreams if he ended up not being able to handle it or spent too much time on it and hurt his other classes or extra curricular stuff (ALL important for med school admission). I'd never recommend a pre-med teacher take a tough math course unless they really, really, wanted to. The world could lose out on some great future physicians.

 

 

Over 100 kids started in Math 51h in September; after the first midterm almost 2/3 of the class dropped out. By the third quarter Math 53h class, only about 30 kids remained (so about 2% of the freshman class, the figure quoted by Crimson Wife). My daughter was one of four women to make it that far in her year (and the only non-Asian, non-Russian woman).

 

Major congratulations to your daughter!!!

 

 

 

Whether homeschool, public school, elite private school, gifted program, or community college, a student's learning experience is much more likely to be tied to the skill of his or her teacher than to any other factor. An AP course can be outstanding or a complete waste of time.

 

:iagree:

 

 

I taught in the math department at your rival cross-state tech university in the mid 80s. Yeah, I'll agree that the preparation of the kids coming from the GA public schools left a lot to be desired.

 

And this is depressing, but not unexpected. There's a reason our school is only slightly below average for our state and nation. Here we are discussing the top of the top and most kids really can't even do Alg 1 or 2 effectively if we're honest.

 

these are meant to be easy for someone who understands the concepts.

 

This brought back a memory. Many years ago I was doing a long term sub assignment for one of our Calc teachers who had been sent overseas by the National Guard. This was when we were still doing AP Calc. One student had a question about the period change for sin/cos waves. I started to explain it and he stopped me by saying, "Look, you're trying to teach me the concept. I don't want to know the concept because that's confusing. Just give me a formula I can remember. I can memorize those." It was incredibly frustrating and he was adamant about not even trying to get the concept. FORTUNATELY, most kids actually do prefer to learn the concepts when I teach them... but few teachers at our school bother trying to teach concepts. I'm kind of convinced that many don't understand the concepts themselves. I had to correct some of the notes from that Calc teacher. Even today I find myself still correcting notes here and there for most teachers (and they use the same notes year after year for the most part).

 

Like I said before, substitute teaching in my public high school is what led me to homeschooling... and our school is only slightly below average as per SAT scores.

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Yikes! I am starting to get the pit in my stomach feeling regarding the choice I have made for my oldest. Part of me wishes I had the courage to do what 8filtheheart did, but another part of me feels that I don't have the energy or expertise at this point to help ds prepare for the AP exam. Hopefully, some of you can chime in and offer advice:

 

I am a huge AoPS fan. My oldest has taken all of their online courses through pre-calc. Unlike most kids who use AoPS, he does not love math, he just barely qualified for AIME and scored average on the test, he does not always enjoy working on proofs, etc. He plans to major in engineering and then hopefully go on to medical school.

 

My current plan that I am now seriously questioning, was to have him take BC calc with PA Homeschoolers this year, and then take AoPS calc next year after he had the "mechanics" down.

 

Any advice? Is this a poor idea? Sometimes I wish I did not worry so much about outside validation, but I think scoring a "5" on the AP exam is the language that admissions officers understand and I am not sure that a "5" would be possible going the AoPS route if he didn't spend time prepping for the exam.

 

snowbeltmom - the PA Homeschooler AP calculus and the AoPS calculus are two very different approaches. Does your son express a strong preference? As a future engineer/doctor, either approach would be fine IMO; I'm not sure that I'd do both courses, though. THe PA HS class will prepare him for the 5 on the AP exam, and the AoPS class will teach him the proof-based version. The former will give him lots and lots of practice problems of the AP type; the latter will assign just a few thought-provokers at a time.

 

If you choose the AoPS path, I'd be glad to advise you on how to add in AP prep (I worked with 8Fill's son this past spring doing just that), and I'm available for that sort of part-time tutoring, too. I'm also toying with the idea of running an online AP calc prep group next spring.

 

AP would be a vast improvement over what our school offers (a cc class taught exclusively in the high school). We used to offer AP (AB only) and rarely did someone get greater than a 2. Most got 1s - that's if they took the test. Most opted out. Now they don't have to worry about the test at all.

 

Our county math science magnet school in this upper middle class suburban area doesn't do much better. I tutored one of my daughter's friends who earned a 104% in AP calc AB at the magnet, but scored a 1 on the actual exam. She repeated the course the next year (when I tutored her) at William and Mary & was totally clueless. It was like she'd never understood a thing from her high school class.

 

I appreciate schools that have different calibers of math classes to allow students to choose how deeply they want to delve. My pre-med or med researcher wannabe needs Calc, but not as a be all, end all. A super tough class could kill his med school dreams if he ended up not being able to handle it or spent too much time on it and hurt his other classes or extra curricular stuff (ALL important for med school admission). I'd never recommend a pre-med teacher take a tough math course unless they really, really, wanted to. The world could lose out on some great future physicians.

 

:iagree:

Btw, the honors math sequence at U of R is one of those hidden gems for future mathematicians. They used Apostol when I took it, and I learned a LOT that year. but, yeah, that sort of math is certainly for future mathematicians, not kids whose passion is neurobiology. :) Each kid has to find his or her own area of passion, for sure!

 

 

This brought back a memory. Many years ago I was doing a long term sub assignment for one of our Calc teachers who had been sent overseas by the National Guard. This was when we were still doing AP Calc. One student had a question about the period change for sin/cos waves. I started to explain it and he stopped me by saying, "Look, you're trying to teach me the concept. I don't want to know the concept because that's confusing. Just give me a formula I can remember. I can memorize those." It was incredibly frustrating and he was adamant about not even trying to get the concept.

 

My husband once was hired to tutor trig to a public schooled student in New Orleans (and this student attended the well-respected gifted magnet high school there). The student was adamant about not wanting to know the meaning of sine and cosine. He demanded, "Just tell me how to use the buttons on my calculator to get the answer!" over & over to my husband, who finally threw up his hands in frustration. Turned out that's how his math teacher had presented trig functions in class: as calculator buttons, and nothing else! No wonder the poor kid was completely befuddled. Trig was a deep, dark mystery!

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snowbeltmom - the PA Homeschooler AP calculus and the AoPS calculus are two very different approaches. Does your son express a strong preference? As a future engineer/doctor, either approach would be fine IMO; I'm not sure that I'd do both courses, though. THe PA HS class will prepare him for the 5 on the AP exam, and the AoPS class will teach him the proof-based version. The former will give him lots and lots of practice problems of the AP type; the latter will assign just a few thought-provokers at a time.

 

If you choose the AoPS path, I'd be glad to advise you on how to add in AP prep (I worked with 8Fill's son this past spring doing just that), and I'm available for that sort of part-time tutoring, too. I'm also toying with the idea of running an online AP calc prep group next spring.

 

Thanks, Kathy, for your advice and offering to provide guidance if we decide on the AoPS route. My son doesn't always enjoy the proofs, but I am not sure how he is going to feel about doing lots of practice problems after 4 years of AoPS.

 

I had started the kids in Saxon when we first started homeschooling. My oldest completed every problem in the Alg. I book in 2 months without a single complaint. Middle son, who very rarely complains, was very vocal about hating to do the "same type of problem, over and over again." So while I know for sure which approach I will use with middle son - he is responsible for me finding AoPS - I am not sure about oldest.

 

I guess at this point, I will go ahead with the initial plans and make a switch if the PA Homeschoolers approach is not working out and just hope that the AoPS class will still have openings if son wants to change direction.

 

Thanks.

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@creekland:

"I appreciate schools that have different calibers of math classes to allow students to choose how deeply they want to delve. My pre-med or med researcher wannabe needs Calc, but not as a be all, end all. A super tough class could kill his med school dreams if he ended up not being able to handle it or spent too much time on it and hurt his other classes or extra curricular stuff (ALL important for med school admission). I'd never recommend a pre-med teacher take a tough math course unless they really, really, wanted to. The world could lose out on some great future physicians."

 

Actually it is my conjecture that a math course in which creative thinking is encouraged would be better preparation for a medical student who intends to make diagnoses from limited information, than one which is all computation. Again I am not talking about impressing an admissions committee looking for all A's, but impressing a medical professor (think "House") who is looking for depth of understanding and reasoning ability. Hard math courses also help teach students how to learn. (My wife is a math major and a physician - I think her favorite math course was abstract algebra, but she also took logic and proof, complex variables, and advanced calculus. After finding out public school teaching was not her thing, she went to night school (with two children), took the pre med courses and nailed the MCAT's. Her scores were quite high and she was admitted to the only med school she applied to, Emory.)

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Actually it is my conjecture that a math course in which creative thinking is encouraged would be better preparation for a medical student who intends to make diagnoses from limited information, than one which is all computation. Again I am not talking about impressing an admissions committee looking for all A's, but impressing a medical professor (think "House") who is looking for depth of understanding and reasoning ability. Hard math courses also help teach students how to learn.

 

Unfortunately, if you don't impress the admissions committee, you will never get that chance to impress the medical professor.

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I love these conversations, though they fill me with regret for what might have been, had I actually learned some math.

 

Carry on :lurk5:

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Actually it is my conjecture that a math course in which creative thinking is encouraged would be better preparation for a medical student who intends to make diagnoses from limited information, than one which is all computation. Again I am not talking about impressing an admissions committee looking for all A's, but impressing a medical professor (think "House") who is looking for depth of understanding and reasoning ability. Hard math courses also help teach students how to learn. (My wife is a math major and a physician - I think her favorite math course was abstract algebra, but she also took logic and proof, complex variables, and advanced calculus. After finding out public school teaching was not her thing, she went to night school (with two children), took the pre med courses and nailed the MCAT's. Her scores were quite high and she was admitted to the only med school she applied to, Emory.)

 

I agree. I'm convinced that good math programs develop creative problem solving. My son wants to major in math. He feels that his study of math has been very helpful with his science and even humanities courses. It seems the benefit has less to do with content and more to do with creativite problem solving and logic.

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@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.

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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!

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My husband once was hired to tutor trig to a public schooled student in New Orleans (and this student attended the well-respected gifted magnet high school there). The student was adamant about not wanting to know the meaning of sine and cosine. He demanded, "Just tell me how to use the buttons on my calculator to get the answer!" over & over to my husband, who finally threw up his hands in frustration. Turned out that's how his math teacher had presented trig functions in class: as calculator buttons, and nothing else! No wonder the poor kid was completely befuddled. Trig was a deep, dark mystery!

 

:banghead: This brings to mind the time that I did a two week sub in a Precalc class for an instructor who had a family emergency. She had given the students a seven page packet of trig junk and told them to memorize it. When I reveled that all one really needs to "memorize" are a few relationships that could be penned on an index card and the rest is easily derived, I had people in the class cheering. The class was in a corporate setting and populated by "non-traditional" students. They had no interest in memorizing and spitting out material--they were there to learn.

 

And that is really my problem with the algorithmic approach to teaching mathematics. If one memorizes methods without asking "what does it mean?", does one retain much let alone understand what is going on? I recognize the fact that engineers do not necessarily need to learn the proofs of mathematical theorems, but I will posit that a good engineer will do more than plug and chug. Unfortunately there are a number of students (and possibly teachers) who feel that this is what mathematics is, just a series of problems that are cranked out to get "the answer".

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Our county math science magnet school in this upper middle class suburban area doesn't do much better. I tutored one of my daughter's friends who earned a 104% in AP calc AB at the magnet, but scored a 1 on the actual exam. She repeated the course the next year (when I tutored her) at William and Mary & was totally clueless. It was like she'd never understood a thing from her high school class.

 

Ok, I won't be as disappointed in my school now... that's definitely got mine beat, but it sure is sad...

 

 

Btw, the honors math sequence at U of R is one of those hidden gems for future mathematicians. They used Apostol when I took it, and I learned a LOT that year. but, yeah, that sort of math is certainly for future mathematicians, not kids whose passion is neurobiology. :) Each kid has to find his or her own area of passion, for sure!

 

My guy probably could take it and do well. I just don't want to chance it. ;) He'll have enough classes to delve into that I'm sure he'll enjoy. Then he's going to be part of the Chess Club, have a work study job, and wants to get into research - oh yes, - and have some sort of life outside of all of this (maybe!). He's thought about picking up his violin again. He had to end that when we pulled out to homeschool as we're too rural to have options outside of ps for that (and they don't let homeschoolers participate in orchestra). Since lessons are free there... I told him to go for it.

 

 

 

Turned out that's how his math teacher had presented trig functions in class: as calculator buttons, and nothing else!

 

You mean there's a different way??? :lol: EVERY single time I've been in for a class doing any sort of trig ratios I always ask them to tell me what they are in plain English. I've never had a student give me the correct answer (who hadn't had me before). I do the same for square (or cube) roots, division by zero, and a bunch of other basics. Our schools teach memorization, not math. Then they wonder why the scores are so poor.

 

Actually it is my conjecture that a math course in which creative thinking is encouraged would be better preparation for a medical student who intends to make diagnoses from limited information, than one which is all computation. Again I am not talking about impressing an admissions committee looking for all A's, but impressing a medical professor (think "House") who is looking for depth of understanding and reasoning ability. Hard math courses also help teach students how to learn. (My wife is a math major and a physician - I think her favorite math course was abstract algebra, but she also took logic and proof, complex variables, and advanced calculus. After finding out public school teaching was not her thing, she went to night school (with two children), took the pre med courses and nailed the MCAT's. Her scores were quite high and she was admitted to the only med school she applied to, Emory.)

 

True, but there are many ways to learn/use creative thinking from math to art. Every student should delve in where their passion lies. My guy is currently into neuroscience, but is also loving his little bit of independent study of organic chem. I'm not the least bit worried about his ability to have or carry over creative thinking. He's proven that trait already and not just to me, but to every teacher-type who has come in contact with him from elementary school up to college profs. It's ok with me if he doesn't want to be a math major and spend his limited extra time on that. I want him to do what he enjoys. ;)

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:banghead: This brings to mind the time that I did a two week sub in a Precalc class for an instructor who had a family emergency. She had given the students a seven page packet of trig junk and told them to memorize it. When I reveled that all one really needs to "memorize" are a few relationships that could be penned on an index card and the rest is easily derived, I had people in the class cheering. The class was in a corporate setting and populated by "non-traditional" students. They had no interest in memorizing and spitting out material--they were there to learn.

 

And that is really my problem with the algorithmic approach to teaching mathematics. If one memorizes methods without asking "what does it mean?", does one retain much let alone understand what is going on? I recognize the fact that engineers do not necessarily need to learn the proofs of mathematical theorems, but I will posit that a good engineer will do more than plug and chug. Unfortunately there are a number of students (and possibly teachers) who feel that this is what mathematics is, just a series of problems that are cranked out to get "the answer".

 

I always stunk at memorization, so part of the appeal of math to me is that I could figure out most everything on the fly from first principles. It really is SO much easier than memorizing algorithms. But I don't have to tell you that - I'd be preaching to the choir!

 

I think that there's a real void and need for a new kind of calculus class in the middle of the two approaches (the crank-it-out approach and the future mathematician approach) for our future engineers and scientists, etc. Just thinking, but it might be fun to develop one.:)

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I think that there's a real void and need for a new kind of calculus class in the middle of the two approaches (the crank-it-out approach and the future mathematician approach) for out future engineers and scientists, etc. Just thinking, but it might be fun to develop one.:)

 

In full agreement.

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I'm really enjoying this discussion about math instruction and AP calc.

 

Just to insert a very practical thought into the discussion, I would not generalize too far from BC calc to the rest of APs. There are range of APs that may make sense for different reasons. Just to take a wildly different example - AP World or AP US. The exams don't impress me as a learning tool and I would not build preparing for the exam as the central way to learn World history.

 

But, just as a practical matter, many WTM type homeschoolers are going to be much, much beyond intro history courses typically offered at state universities. History instruction has probably been much more central to their education for years and they will may be totally out of step with intro courses. So, if the kid is headed to the state u (as are) it may make quite a lot of sense for them to knock off the AP. Instead of wasting a year in a huge and poorly taught section survey of world history for people who have never studied it before, many homeschoolers would prefer to go directly to a more engaging upper level course on an area they've wanted to study more in depth. For a student headed to a top liberal arts college this advice won't apply, but I would suggest students planning on the state u look at APs seriously as an option to help shape a better experience at the state u. APs on the record can help get the kid into smaller, and better, classes much faster.

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I always stunk at memorization, so part of the appeal of math to me is that I could figure out most everything on the fly from first principles. It really is SO much easier than memorizing algorithms. But I don't have to tell you that - I'd be preaching to the choir!

 

I think that there's a real void and need for a new kind of calculus class in the middle of the two approaches (the crank-it-out approach and the future mathematician approach) for our future engineers and scientists, etc. Just thinking, but it might be fun to develop one.:)

 

I fondly remember at least one Physics class where we were only allowed to use F=ma as a given. Everything else had to be derived. Once we had derived other things, then we could use them. But, that's my passion - theory. Hubby's an engineer and prefers just to use formulas. He knows what they are for (understands the concepts), just feels no passion for the proofs. He's a superb engineer and has been able to figure out solutions to problems that have baffled others on the job. He has no problem at all with creativity.

 

The problem lies where the students have no idea of the concepts to be able to do either one and are just picking formulas at random because they seem like they could work and possibly can't do the math even if they got the correct formula.

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For a student headed to a top liberal arts college this advice won't apply, but I would suggest students planning on the state u look at APs seriously as an option to help shape a better experience at the state u. APs on the record can help get the kid into smaller, and better, classes much faster.

 

This I totally agree with!

 

The bottom line remains - different reasons for different paths. Each can be "right" for a student.

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