Sunday, September 16, 2007

AP Calculus

Professor Posamentier, the dean of education at City College recently wrote an article in Newsday saying that AP calculus should not be taught in high school, that school's should emphasize algebra skills instead. He said kids are coming into the colleges with AP credit, yet their skills are severely lacking.

I know Professor Posamentier quite well. He was my main math education teacher as an under graduate, graduate and post graduate student. He is a brilliant teacher and I credit him with my skills as a teacher. I completely and whole-heartedly disagree with the premise of this article. Kids might come into college ill prepared but, it is not the fault of AP calculus. Instead I would blame the poor curriculum of Math A and Math B. I would blame the early emphasis of calculator use. I would blame the use of manipulatives, instead of memorization (multiplication tables) in elementary school.

I have been teaching AP calculus for over ten years. The students I have must qualify for the class by taking pre-calculus and doing well in that class. By taking away AP calculus from these kids we are dummying down the curriculum and once again teaching to the lowest level possible. We should not assume that all kids are not capable. The kids in my school are. Sometimes I do end up with students who "sneak in." They don't have the prerequisites. When I find out, I try to "hide" them in the class. The five on the AP exam is not that important. The skills they learn in my class are. AP calculus is a class that teaches kids how to think, how to apply their knowledge to problem solving. They are able to solve problems numerically, analytically, algebraically, and graphically. They must be able to support or confirm answers through written exercises. They must understand that technology is used to support results. Students are taught that mathematics provides the foundation that allows technology to solve problems. No other math class does this.

Calculus taught on the college level does not delve as deeply into the subject as it is on the high school level. There simply is not enough time. Rather than remove AP calculus from the high school curriculum, we should be doing more to insure more students get to take it.


Anonymous said...

I'm not with either of you on this. You are right that there are big problems with the algebra along the way. And this should be addressed. Having an actual Algebra Regents might, just might, help.

But schools that want to support calculus programs need to put extra emphasis on the particular algebraic skills that will be necessary. No one has banned trinomial factoring. Why does a=1 and never anything else in so many schools?

The rush to calculus bothers me. Math is not a foot race. Algebra, geometry, algebra II, precalc... most kids won't hit calculus in high school, unless stuff gets cut out.

So if an accelerated kid reaches calculus, and a regular kid who does well gets a highly enriched precalc, well, those are both very positive outcomes, and they will likely both meet success in college mathematics.

Pissedoffteacher said...

I agree that math is not a foot race. But, there are plenty of kids that are ready for calculus in high school. They should not be deprived of taking it because so many others are deficient in math.

For your information, my school teaches trinomial factoring with a >1. In our pre-calc classes we are doing the type of factoring you are discussing on your blog.

We have plenty of kids that will never make it to calculus also. We need to go back to teaching them good algebraic skills. We also need to go back to business math.

druin said...

As an AP Statistics teacher, I agree with almost everything you wrote. It does frustrate me to be the "stepchild" of the math department, but that's for a different forum. heheh

Anonymous said...

I guess what's needed most is reason and flexibility, and Posmentier doesn't show it.

How much do kids factor in your Math A course? I can tell you, even teachers at Science complain that the kids reach calculus without adequate factoring, algebraic fraction, and radical manipulation skills.

I didn't mean to somehow single you out or point at your school. And I am favor of keeping calc in high school. But not as the norm. The rush to maximize the number of kids who get in, that happens all over, and it makes me sad.

Pissedoffteacher said...

We do all the factoring, the problem is that it is not tested on the regents so kids do get passed through without a knowledge of it. The problem is not calculus, it is the syllabus leading up to calculus.

I know you weren't picking on me or my school.

LSquared32 said...

I'm a college teacher, and I get those students who can't do algebra, but most of them didn't get AP calc credit. The ones who can do AP calc are getting a fine calculus education in high school, and all of you high school calc teachers should be congratulated on doing a good job. The only places where they are missing something they would be getting in college are places with large math programs where they can do a lot of proof in beginning calc (math major only sections, etc). Math majors at those places should take calc again, but that's because it will be an entirely different class from the one they had in high school not because it will be a better version of the same class.

Pissedoffteacher said...

That is true. My daughter got a 5 when she took AP, then took an honors version of it in college (got an A). She ended up majoring in math and has a really good job using it. She had a great AP calcu teacher that gave her a wonderful background and helped instill her love of math.

le radical galoisien said...

I just took the exam last year in May.

I got a 5, but the curious thing that some principles suddenly "clicked" only when I was under duress and under the pressure.

The foundations to calculus must be started way before hand.

Firstly, the simple idea that the derivative of a parabola is a line -- could be easily anticipated years before the course is taken.

What really needs to happen is to drag out the calculus course over several years.

In Singapore this begins at secondary three -- at age 14-15.

For example, in a typical calculus course, many topics are covered within a short span of time, such that by the time you've begun to get the hang of a particular problem, so quickly move on to the next one. You never learn to mastery.

You don't have to wait till college to think about the idea of instantaneous change.

I remember my first thoughts began when I was about age 10: given two variable factors that always sum up to a constant, what is the pattern for change in the product? This was triggered by the simple thought that 5*7 was not the same as 6*6, but it was quite close, but much farther than say, 11*1 or 11.9 * 0.1

These are topics that students should begin exploring from a young age.

And factoring cubes is important, but not as critical to the need to expose students earlier.

What frustrates me is that some of the topics were really thought-provoking and interesting but the need to cover everything on the AP exam led to what I felt was a highly rigourous but a highly frenzied course.

In Singapore I used to do the Math Olympiad and study number theory, etc.

I suddenly feel very inadequate, because the colleges I am hoping of applying to, my 5's are mundane and not exceptional, while they look for success in international competitions like the IMO, etc.

Like suddenly I find that I am competing with students who take 16 different AP courses (with all 5's) and have won 10 different science fairs and had separate classes for number theory in their junior year, easily solving problems like this.

The other thing is that I think isn't practiced enough is graphing tedious functions to the point that you could look at any algebraic and transcendental function and graph i mentally.

In Singapore we started this process halfway, where we would spend months and tons of graph paper (and using bendy rulers!) learning how to graph basic polynomial functions accurately (this was in sec 2).

Like not sketch. Graph precisely, by hand.

IIRC, graphing calculators are unheard of until sec 4 or junior college.

It occurs to me that the students I am competing with could probably look at functions with arbitrary products of trig functions and graph their second derivatives in their heads.

What I would consider useful preparation (you know, before exposing us to revolving random integrals around an axis) is learning how to apply most f(g(x)) transformations mentally.

Like given x^3+1 and sqrt(x), you can immediately envision in your head how the graph looks like when they are combined as sqrt(x^3+1), without the use of your graphing calculator or even without the need of paper.

It strikes me that with proper training (like if I had been trained properly) I wouldn't have to rely on graphing calculators to graph most algebraic/trigonometric-based functions.

I'm pursuing a further year of calculus at a local university in my senior year after having taken AP in my junior year.

What I would really feel would hav been useful would have to be given more time to grasp key concepts like many of the identities that were given out: the product rule, the chain rule and so forth.

Like we know when to apply them and do problems involving them and do them perfectly.

But we couldn't see or imagine what was happening. We only did it because we memorised the rules and we were effectively told to.

At most we were given preliminary proofs at the start of each new unit involving new identities or formulas, without having time to digest the formula.

In lower-level algebra, it seems that many weeks if not months would be spent on learning how to graph a quadratic properly and how to apply the quadratic formula.

Whereas with AP Calc, when it comes to say, using an integral to get an inverse trig function, you are given a bunch of formulas/identities to memorise. Sure you had some idea how they worked (but not vividly). The idea was to do well on the AP exam and that was it.

Who cared about the various ramifications and exciting possibilities of certain principles being introduced if the ramifications had no practical application on the AP exam? "My first priority is the AP exam," my teacher said. "You can look at the proof yourself in the textbook if you want to."

I don't blame her -- the schedule of the course made her pressed for time.

The other thing is that the stuff I learned in Precalculus really didn't seem to be really useful on the AP exam. Like all those trig identities and learning how to write equations for a tilted ellipse on its side? Totally unused in the AP exam, except for the occasional case of 2 sin x cos x = 2 sin x.

And was it a big surprise that come the math meets the students taking AP calculus (one year more advanced!) were at a DISADVANTAGE compared to the Precal kids because for them the material was fresh in their minds while the AP syllabus had practically never revisited any of the stuff taught in Precal?

My biggest issue is being taught really cool and neat things but then because the focus is doing well on the exam, you end up forgetting those neat and exciting things six months after the exam.

The questions were really challenging on that exam. You really had to think. I enjoyed doing them (though I probably didn't notice that I was enjoying it being all stressed out). It was actually one of the first exams I had actually ever studied for (beyond the PSLE in Singapore).

But I am frustrated that come my college class, I struggle so hard to remember all the principles I applied on the exam. A lot of things are review and in fact my fellow classmates haven't seen some of the types of the problems that we covered on the exam. Should be an advantage for me right?

But even though a lot of the questions are deja vu, I have to do a lot of rediscovering.

It is rather frustrating.

Pissedoffteacher said...

In my classes I emphasize thinking and applying principals. I treat the math like a detective solving a crime would--we look for clues and put them together to arrive at a solution. The AP exam is like the icing on the cake--it is just an extra treat. I want my students to do well but it is more important to me that they learn and understand. As you wrote, calculus is a topic that needs to be studied for many years before it can be mastered.

The emphasis in AP calculus now is understanding and using, not doing tedious calculations and algebraic manipulations. While those are interesting and useful, not everything can be taught in every course and they are just not a part of this one. That does not take away from the value of AP, it just makes it different.

mathmom said...

I agree with you that getting rid of AP calculus will dumb down the high school curriculum.

However, I disagree with you that it's important to try to make sure more students get to take it. I think the relative difficulty of "qualifying" for AP calc is what helps ensure the caliber of students in that class, and allows it to be a high-quality class for highly able and highly motivated learners. If "everyone" took it, you would have to "dummy down" the calculus class, IMO.

I also disagree with your statement that:

Calculus taught on the college level does not delve as deeply into the subject as it is on the high school level. There simply is not enough time.

In my experience as a math major, (single variable) calculus was a 2-semester course. Everyone was expected to take it, even though we'd all had calculus in high school. (Because I was not in the US, I'm not talking about AP calc specifically, and besides, my experience is a little out of date at this point. But from what I've seen of AP calc since we moved here, I believe my arguments still apply.)

The college calculus course taught almost exactly the same topics as high school calculus, but they went way deeper in the problem sets. It was my first experience where I could not just sit down and whip off my homework problems. I had to think about them. I had to work on them. I had to try different approaches. What would have been considered "contest math" in high school was now "homework".

I didn't realize that I had learned anything in calculus that year until I came across an old high school math contest as I was packing up to head home that summer. I was shocked at how easy & straightforward the problems seemed! That was when I realized that what I had learned that year in calculus was not more calculus, but improved problem-solving skills.

I think that AP calculus is a fantastic opportunity for highly-able and highly-motivated math students to begin to learn all the things you've so eloquently argued that it teaches. But rather than being "better than" a college calculus course, in some cases it is what prepares students for success in an even more rigorous college course.

Pissedoffteacher said...

I didn't mean that it is better, but it is different. And, since there is more class time, we do delve deeply into many problems that there would just not be time to do in college (except for hw).

In AP calculus we approach things differently than college professors. I'm not implying that it is better or worse, just a different exposure. Calculus is the type of class that needs to be taken multiple times to grasp a full understanding.

I don't think we should dummy the course down at all. I do think that in my school some very good kids are excluded for some not so great reasons. More must be done on the lower levels to prepare kids for the course.

I agree with everything you wrote. Thanks for sharing.

Anonymous said...

First off, I'm a physicist who adds and subtracts with his fingers...

I wholeheartedly disagree with you that "memorization" is a form of math. In fact, I think multiplication tables, algebra tricks, and geometry isn't math at all. Sure, whoever came up with the trick was a brilliant mathematician, but realize when I see a linear homogeneous ODE, I have memorized the answer...I'm not doing any math.

The other day I had to look up a Fourier Series. So what? Do you think I can remember how to "complete the square"...nope, I'd referemce it out of the CRC amd then be fine. Do you know how I manipulate trig identities?? Mathematica!

Early emphasis of a calculator is crucial (esp a graphical calculator). 5*5 is not math. Trinomial factoring is not math. Figuring out the area of a sphere on one's own by integrating r d(theta)d(phi) is math.

However, I will agree that getting rid of AP Calc is a huge mistake. i was capable around 13-14. I had to wait until I was 17, and my father had already taught me it so I was insanely bored in HS.

But don't let the college profs get to you. It has been my experience that math majors can't solve a real world problem (i.e. they can only solve 'factor x from y'), and engineers really never deviate from static/closed systems.

If you can supply a well thought-out examination, it doesn't matter if the kid's got a TI-99999 on his side. Sounds to me like Professor Pontification is just pointing his finger when maybe he's not such a great teacher.

And anyway, one should never take an education profressor's advice on anything math/science. They just never got there mentally.

If you wanted to stress something useful in HS, I'd say shorten/quicken the statistics and give 'em a heavy dose of complex number theory after Calculus. If their algebra skills were weak before, that'll hone 'em as well as make Euler a hated word.

Pissedoffteacher said...

Maybe you add and subtract with your fingers, but you have the skills to do otherwise, if you so choose.

Without knowing basics, kids can't go on. Maybe it is not math, but it is crucial. Kids have no idea that 8/2 is not the same as 2/8 and thus put down wrong answers all the time.

I whole-heartedly agree that when you need some things, you look them up. But, you have to have an understanding of how they work before you do. How else can you know if your answer makes sense. You can't get to the well thought out questions, until you can master the easier ones.

My husband and son are engineers. They think engineers are the only ones who know anything.

mathmom said...

My husband and son are engineers. They think engineers are the only ones who know anything.

I have noticed that about engineers. ;-)

I disagree strongly with the anonymous physicist above regarding early emphasis of calculator usage. But you already know that. :-D

Adrian said...

Early emphasis of a calculator is crucial (esp a graphical calculator). 5*5 is not math. Trinomial factoring is not math. Figuring out the area of a sphere on one's own by integrating r d(theta)d(phi) is math.

How is that math? That is no different than factoring a polynomial -- all of it is just letter arithmetic. You are right: arithmetic is not math. Arithmetic includes memorizing times tables but also mechanically applying some memorized theorem be it in the context of solving a numerical problem, an algebra problem, or even a calulus problem.

Figuring out the area of a sphere is more like a theoretical physics problem. For the true distinction between math and physics, I'll take yet another example. The physics is using the pythagorean therem to determine the distance between two objects in space. The math is proving that the pythagorean theorem is actually true in the first place. And the problem with the fact that we don't teach that stuff anymore and increasingly just concern ourselves with applications all the time is that no one even seems to know what it takes to formulate a coherent logical argument.

At any rate, I shouldn't try to justify why we should teach students how to prove theorems in this comment -- I just wanted to dispute that upping the ante on abstraction or merely making the student think suddenly makes it "math". In fact, your example is a pretty clear cut example of applications of math -- not the math, itself. If the student was taking the limit of finite sums to arrive at that area, then it might be a different story. But, otherwise, even if it is up a level or even up several to solving some hairy nonlinear PDE, say, a simple application of a mathematical theory is just as much letter arithmetic as learning to "FOIL" is. It doesn't change the fact that it is mere calculation just because it is a word problem (i.e. "find the volume of this sphere" as opposed to just presenting the student with an integral to carry out). What makes the difference is the need for a formal, coherent, almost philosophical, really, argument that justifies a conclusion.

That's the difference between the mathematicians and the physicists when it comes to the math. The physicists just use math and are willing to approach math just as empirically as they approach the physics. Mathematicians make the math and take great care to approach their subject properly in an a priori fashion.

Pissedoffteacher said...

High school math is very different than the things you talk about. Most of my students never get that far. And, I still say that it is impossible to succeed at the level you are talking about unless you master the earlier ones.

Adrian said...

Well it is very different than nonlinear pdes, that's for sure. You would be hard pressed to teach your typical high school student how to prove that the real numbers are uncountable. But, what I am talking about is just proving the Pythagorean Theorem. An axiomatic approach to basic geometry and algebra. It's just a question of what we teach not how to teach it. Instead of spending time covering a new theorem and application we could go back and learn not just the statement of the theorems we already think we know, but also the justification for those assertions and really know what we think we know. It is specifically about not going up to the next level until we have mastered the one we're at. Moving on just because most students can state and apply a theorem based on some loose heuristic rationale for why someone might have thought it was actually true is the opposite of mastering the basics. That is the mad rush forward to cover as much ground as possible.

And that is done for the sake of physics and engineering not math. Now, maybe that's a good trade off, but let's not deny the trade off that is taking place. We are specifically getting rid of the proofs in favor of breadth -- of being able to "talk a good game" about a much wider set of topics. Personally, I think it's cutting corners, and it works out because most people don't even know that a corner is being cut and there are a lot more people interested in getting on to, say, calculus based physics than dwelling on such tedious details as axiomatics and being able to prove theorems. We simply just don't think proving theorems is that important -- it's really just that simple. Making new scientific discoveries are far more important to us. Personally, I think it's a false economy, even if we are just interested in science, to make such a trade off especially at that early in someone's academic career.

Pissedoffteacher said...

It is obvious you are not familiar with the students we teach in NYC. What should be is not realistic and will never happen with the kind of students we have and the curriclums we are expected to teach.

Unknown said...

You were wrong as soon as you said that manipulatives should be use less and they should just memorize things. Memorization does very little to promote understanding. Manipulatives help students make the real world connections which is the point of Mathematics. If all you are doing is asking your students to memorize things, you must not be teaching at a very high level; reference blooms levels of taxonomy (that would be 2nd level at best out of 6, 6 being the highest). The real truth is that Mathematics is hard and if you aren't willing to practice enough discipline within your work ethic to grasp the concepts then that is your own individual problem. Some people work harder than others. You get what you put in essentially. Don't blame the high school teacher's, I am one, most of us are still trying to fix basic algebra 1 issues stemming from middle school. Now that doesn't mean they had bad middle school teacher's necessarily, to be honest they probably weren't even interested enough to begin with so they tuned out. This is why K-12 teacher's plan; engaging instructional activities, interesting topics, etc. We grade school teacher's are salesman. It's actually pretty irritating to me that some "professor" feels like he can bad mouth high school teachers when he doesn't have to cater/accommodate to half of the ridiculous criteria mandate by the state and federal government. This guy doesn't have to write lesson plans or deal with parents or deal with behavior problems or answer to a state testing board nor does he probably ever get observed. This guy can "wing" a 50 minute college class and look like a superstar when he hasn't done the first bit of research on his student dynamics. In college they don't have to make it interesting or engaging they simply show up and demonstrate a couple concepts and call it a day and you either get it or you don't. Oh yea and he makes probably $80,000 a year and he's MAD! This guy needs to slow his role honestly. Finger pointing is easy. What are you really doing about it.

-High Math teacher
-B.S. Mathematics
-B.S. Electrical Engineering
-B.S. Physics

Anonymous said...

Jonathan, I hate to be insulting, however, you write incredibly poorly. I fail to even understand what your point is. I have taught all levels of mathematics in four differenct schools and all of them factored using coefficients for a not equal to 1. It is ridiculous to make a statement that AP Calculus should not be taught. Weak algebra skills are the result of using too much technology and "other" factors that unfortunately cannot be solved. AP Calculus teaches lateral thinking unlike any mathematics course on the highschool level. Students that are truly capable mathematics or science majors in highschool will benefit from the AP Calculus curriculum more than additional algebra study. It is the weaker kids that truly need to be addressed. These kids have absolutely no skills at all and are graduating highschool.

Anon said...

honestly, ap calc bc needs to be scraped. what kind of calc course doesn't even teach delta-epsilon proofs. For an example on how calc should be taught, take a look at Spivak's book on calculus. maybe they can keep calc ab with some modifications, but they should really focus on teaching proof-based thinking for calc bc, however that is something which even most high school teachers are unfortunately incapable of doing.