Remarks on AP courses in high school

[Astronuc]

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

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

I think this would blow most high school math teachers away.

 

[mathwonk]

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

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

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

 

[mathwonk]

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

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

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

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

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

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

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

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

 

[mathwonk]

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

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

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

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

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

 

[courtrigrad]

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

 

[mgiddy911]

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

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

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

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

 

[Hurkyl]

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

Only to some extent.

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

 

[jbusc]

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

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

 

[Pseudo Statistic]

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

 

[0rthodontist]

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

 

[mgiddy911]

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

 

[jbusc]

I think, it's for two reasons:

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

 

[0rthodontist]

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

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

 

[Lucretius]

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

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

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

 

[Pseudo Statistic]

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

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

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

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

 

[Lucretius]

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

You got Form B of the free-response questions, from your description. I didn't have any sort of table for section 1. Was this the test you had? Here is the test I took. I didn't think the multiple choice was very hard; the practice multiple choice exams I had taken in class were much more difficult than this years'.

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

 

[jbusc]

You got Form B of the free-response questions, from your description. I didn't have any sort of table for section 1. Was this the test you had? Here is the test I took. I didn't think the multiple choice was very hard; the practice multiple choice exams I had taken in class were much more difficult than this years'.

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

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

 

[Pseudo Statistic]

You got Form B of the free-response questions, from your description. I didn't have any sort of table for section 1. Was this the test you had? Here is the test I took. I didn't think the multiple choice was very hard; the practice multiple choice exams I had taken in class were much more difficult than this years'.

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

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

 

[Lucretius]

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

 

[Pseudo Statistic]

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

 

[jbusc]

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

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

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