<p>I've noticed that in a lot of colleges, the curriculum of Calculus I and II is different from AP Calculus BC. The actual calculus seems much more challenging than the AP Calc one (which isn't saying much, but you know what I mean.) For example, they still teach trig sub, and certain series manipulation hardly seen in AP Calc. ANother thing usually left out as well in AP Calc is the delta-epsilon proofs. </p>
<p>Now, clearly, I'm not expecting AP Calc to go on in-depth for delta-epsilon. In fact, teaching a small amount of content is not too bad, if managed right. After all, if you introduced difficult question that actually require some thought besides simple manipulation, then you'd be clear that the students know AP Calc. </p>
<p>It's not enough that the questions are ridiculously easy (such that one could only really mess up if one screwed up in algebra, or any that nasty stuff that happens in computations), but the curve itself. I mean, you could get a failing grade and still pass with a decent score! Of course, college classes are curved and whatnot, but this is ridiculous. What's the deal?</p>
<p>All the questions that try to be "applied" and whatnot, they are nothing but "plug it in your TI". The open-calc section was literally just "hopefully you brought a calculator that has a Solve function." Why aren't there less computational problems and more "show this" or "show that"?</p>
<p>Is it that the content is too hard for the students? Or is it the teacher? After all, teachers exclaim their difficulty is using text like Apostol, which understandably may be too dry for ordinary students, but perhaps a softer text like Spivak or the likes as an introduction. But if you check the official ap calc listserv, you see a lot of teachers asking silly questions that any freshman who actually learned Calc I-II in college or equivalent could answer. </p>
<p>So my questions is, what's going on? Why is AP Calculus the way it is? Is it just easy, or is it that I know too many people who know their Calculus and thus makes my opinion biased?</p>
<p>I propose we randomly select people who scored 5s on the AP Calculus BC exam and have them do the same “inane” calculus test.
Example questions will include: “What is a definite integral?” “What does d/dx mean?”
Sadly, many of my fellow classmates in Calculus III who had taken calculus at the university do not know the answers to these questions. So I guess AP Calculus was the lesser evil.</p>
<p>AP Calc BC is easy to some extent, it’s not as easy as you’re making it seem. First of all the class is not a joke. For some reason i get the feeling that you think this is an Algebra 2 class. You need some analytical thinking and when you get into the more complicated stuff (diff eq’s and series) you really need a solid foundation with a perceptive mind to understand things. You can’t just “plug and chug” and expect some magic answer; many of the calculus problems assigned for homework and on tests have to deal with you using the skills you’ve learned and applying them to problems you’ve never seen before. </p>
<p>For the AP exam, i would agree with you mildly. Keep in mind that despite this “amazing” curve, people still score 1’s, 2’s, 3’s, and 4’s, so getting a 5 isn’t entirely easy even if you aced the class because preparing for the exam is a lot different than preparing for specific tests. All it takes is a few careless mistakes, a FRQ or 2 you don’t know how to answer, and you’re grade is totally messed up. You can’t really base the percentage of 5’s as an indication that those kids are definitely prepared for college calculus, because as we know, many of the AP classes can be self-studied with a prep book in 2-3 months max. The AP exams may be crap, but the class is definitely not. </p>
<p>Calculus differentiates itself from other math classes just because it involves so much thinking into problems you KNOW how to solve, but just can’t seem to “get the gears moving” right away. Whether i would have taken Calc I + II at the college i believe i would’ve gotten the same education. An AP classes instruction can be subjective, as with any class, so take what you’ve heard with a grain of salt.</p>
<p>Could’ve fooled me! After all, considering every problem can be solved as follow:</p>
<ol>
<li>Set up equation.</li>
<li>Simplify equation if necessary.</li>
<li>One shot calculus (whether it’d be reciting an identity, basic integration, basic derivative, etc.)</li>
<li>Algebraic manipulation / simplify. </li>
</ol>
<p>If you take step 3, you get Algebra 2. There’s really not that much difference. </p>
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<p>Untrue. I took that exam last year ( or I guess, this year, since it was in May), and I know for a fact I made careless mistakes on ALL the FRQs (barring the Diff. Eq one). Hell, I even got the question about counting the little boxes wrong, : ) . I still got a 5. </p>
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<p>Exactly my point. The fact is, a lot of this kids are NOT prepared for college calculus. I only shudder to think of those who get below that a three and actually tried. </p>
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<p>True, but this is not the case with AP Calc. I mean, they made it so instead of trig sub, you merely do some algebraic manipulation, and turn it into the integral of arctan. Yes, a nice shortcut, but how does that teach anything other than showing the fact you have algebra skills and that you memorized that formula? </p>
<p>Also, just one thing, I suppose the class itself may be good (some schools teach the subject well, I’d imagine) but the test and the limited content is pathetic.</p>
<p>Our class learned trig integrals, delta epsilon, and “shells” (a way of calculating the volume of rotated solids). These topics were not on the AP test.</p>
<p>Speaking briefly to the teacher end of things, I truly believe that you need to be able to competently perform at least two years ahead of the highest class that you plan to teach. Considering that:</p>
<ul>
<li>calculus is one of the highest levels offered at most high schools,</li>
<li>most math teachers don’t have the opportunity to practice the higher levels after graduation from college unless seeking an advanced degree in mathematics,</li>
<li>the track for where mathematics goes splits pretty significantly after Calculus into many different branches, and lastly</li>
<li>most schools assign the more advanced classes based on seniority, meaning that the teachers first teaching calculus haven’t taken the class in 15-30 years, and haven’t used it in about as long</li>
</ul>
<p>it’s no surprise that it’s difficult finding the most competent teachers of calculus out there.</p>
<p>There are many teachers in AP Calculus who do a superlative job, but between the overall growth of the program over the last 10-20 years and the other factors mentioned above, I think it’s extremely difficult to get the number of highly qualified teachers to teach the course.</p>
<p>Consider that about fifteen years ago, I myself took calculus in high school, in a class of less than ten people, and took the AP Exam in a gym filled with ninety kids from ten of the local area high schools. I now teach at one of the other high schools in the area, and our school has over 100 students taking it from just our school alone…</p>
<p>That’d be cylindrical shells. And I’m glad you learned those. You /may/ have had a better education that most! But, as said before, what I’m complaining about is not the class itself but rather the AP exam and its content and how many teachers just stick to it.</p>
<p>And even, many of the classes that teach a lot of content also give out easy questions. Doing delta-epsilon proofs for lim x–>3 of x + 2 is not exactly rewarding. More tricky problems should be employed: For example, the integral from 0 to pi of sec^2 x dx always gets people. It helps them actually use the analytical skills they are supposed to be learning from calculus and apply it, and not just “Oh, definite integral, let’s apply fundamental.”</p>
<p>
The “higher levels” like Real Analysis and the likes (as far as calc goes)? They should have those that down-pat by the time they are teaching AP Calc, considering that it is (or it should be) a requirement for most colleges if you want to get a math major. </p>
<p>I just don’t understand why it’s being so watered-down and why most teachers are afraid of Calculus (most of the teachers at my high school didn’t want to touch it).</p>
<p>I think TheMathProf gave a good reason why some (most) teachers are afraid of Calculus.</p>
<p>TheMathProf, my friend who teaches at the local CC talks with the math profs there. They say Calculus has been watered down a lot over the years. Your thoughts?</p>
<p>I also think that the depth of Calculus must vary a lot from school to school, and the AP exam must not be designed to replace Calculus at the more rigorous schools. That’s why I’m not enthused about placing kids into AP Calculus generally. They might be better off branching out into other math, then taking rigorous Calculus in college.</p>
<p>L’Hopital, my middle 2 years of HS math were independent study. We were never taught; we just sat with the book, did problems, took tests and learned. It was Bliss. If you want more calc, get some good used books and start doing problems (or get the Teaching Company DVD if you’d rather listen).</p>
<p>“Easy” is subjective, but I feel that AP Calc is a rather misguided effort. A mathematics course not based on rigor and proof accomplishes very little. Most schools teach math with a focus on quantity over quality - a student who can get through Calc BC is seen as successful simply based on the amount of material “covered” rather than the actual skills developed.</p>
<p>Calculus is watered down because the AP program has tried a great deal to expand and have more students take the class each year. If you see exam’s from the 60’s/70’s, it was a lot more computational stuff and actually knowing calculus. Now it’s all “analytical”, which is just another name for “TI time.” </p>
<p>I think AP Calculus is meant to serve as an introduction to the basic concepts of calculus, but it shouldn’t be the basis for credit to higher math courses. Personally, I realized that AP cal was a joke, so I decided to learn real calculus. I studied out of a book called Calculus by Micheal Spivak, and I feel that I know calculus much deeper than AP ever could.</p>
<p>TreeTopLeaf: I’m not worried too much about my own learning, (I’ve read about 3/4 of Spivak), as I am to the people who THINK they learn Calculus, or who struggle in it and claim it is because “it is very difficult” when it’s nothing compared to the real thing. Those who struggle will struggle regardless rigor, but without adding rigor, nobody benefits, except maybe College Board who can boast such high passing scores.</p>
<p>An0maly: Spivak is an amazing book. Granted, there are more “rigorous” books, but I don’t think anyone has that style that Spivak has. That style that makes you feel like he’s right there, just chatting with you informally about the subject. You should look up “Pig, Yellow” on the index. ^^</p>
<p>I find it insulting that they say “analytical” when it’s nothing but. Like, I wouldn’t be so disgusted if the test wasn’t so ridiculously easy. I mean, I think they could even cut some of the material out, like the dumb logistic equations, and just make the problems significantly harder. I’m not saying “Hey, find the integral from 0 to 1 of [ ln (1 + x)]/x” and expect to them to a series expansion, and happen to know the sum of alternating reciprocals of the squares. Just something that will get them thinking.</p>
<p>Something like that int. sec^2 x dx from 0 to pi, to get them thinking about the possibility of things diverging. Or something like, find y’ if y = 1/(x + 1/(x + 1/(x + … ) to get them thinking about patterns. Just something that is just find " find f ’ and set it equal to zero."</p>
<p>My own personal hypothesis about the “watering down of calculus” lies in two major arguments.</p>
<p>The first of these is that much of what the AP Exam focuses on is the distinction between applied calculus and theoretical calculus. Many of the topics that the poster L’Hopital mentions would fall within the realm of theoretical calculus; it’s mathematically quite fascinating, but when in the world would we ever use it? Finding practical applications of some of these more theoretical applications is relatively small. And while certain topics certainly get covered in AP Calculus despite still being largely theoretical, the emphasis that I’ve noticed on the exam has been a shift towards the nebulous idea of “conceptual understanding”. (On the plus side, I have seen an increasing shift towards needing some sort of algebraic proficiency over the last couple of years with the Free Response; this is a trend that I hope continues, but that too many of my students hope reverses.) Some people would argue that an “applied calculus” is watered down, and perhaps it is, but what are typical high school students capable of in terms of deeper conceptual and analytical thinking? By an interesting note, AP Stats is and always has been an “applied statistics” class; it’s doubtful that we anytime soon will see an AP Stats class that talks about the development of normal distribution curves and how z-score tables are really just generated by integrating. :)</p>
<p>The second concern is frankly more systemic. We ultimately designate a series of requirements that are designed to separate the cream from the crop in evaluating candidates (whether it be for college admission or business), we decide that everybody should have the opportunity to meet those requirements so that they can reach “The American Dream”, and then in order to let everybody meet those requirements, they become so watered down and/or so many applicants achieve them that new requirements are needed in order to now distinguish the cream from the crop.</p>
<p>It’s gotten to the point where a high school diploma has gone from being something that is a source of pride to an expectation, when frankly, we all know some students who frankly don’t have the skills or the minimal work ethic necessary to deserve one. Somehow, we decided that a high school diploma was something that so many people couldn’t live without, and yet, without some kind of standard involved, what does the diploma even mean?</p>
<p>And L’Hopital, I think that when you’re separated from these classes by a large number of years, you’ll find that math that you had down pat all those years ago doesn’t come as easily as it does today. Shoot, take a look at how many of the average parents can’t help their high school students with basic algebra… :)</p>
<p>Then what’s the point of teaching AP English Literature? What kind of application does that entail? At least applied math is derived from theoretical math. When in the world would you ever use extensive knowledge of The Great Gatsby. Or the point of AP [Insert Humanities Class here]? Don’t get me wrong, the knowledge is not trivial, but if you think about it, its range of applications are smaller than that math. </p>
<p>Actually, let’s go to an even stronger statement. When in the world would you ever use calculus, applied or not? We both know life doesn’t work out with nice pretty integratable functions, nice little differential equation with analytic solutions. And the few cases where we ignore certain variables to produce it, who would apply this stuff in life? Why, someone who actually cares math. </p>
<p>So, why not just erase AP Calculus, and introduce AP Numerical Methods for Calculus? At least this way people see REAL Applied Math, and not “If a car goes at a velocity of v(t) = 5t^2 + 6, what’s the acceleration?” </p>
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<p>Well, considering I’m planning on a math major, I truly hope that I don’t lose my mathematical knowledge. I plan to be submerged in the “mathness” for as long as I can, as any math major should. And if not, then why get the math major anyways? You see history buffs know their stuff, literature majors know their stuff, why is it “acceptable” for someone who has a degree in mathematics to not know anything beyond pre-calculus, or to have such a minute knowledge of calculus? Granted, a lit. major does not know everything of literature, but he’s very knowledge of authors and titles, and of the many he has read, he can tell you tons of it. However, what can an AP Calc Math teacher tell you? It’s not like he has some knowledge of [insert advance math here], so what gives?</p>
<p>Well, personally, I would use my copy of The Great Gatsby to build a bonfire. It might not be very bright or last very long, but I would at least feel more satisfied that I got something useful from it. :)</p>
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<p>Isn’t that why we have TI Calculus now? I think the more interesting question is the reverse: if the majority of situations with actual calculus applications outside of theoretical mathematics consist of largely non-integrable functions and differential equations without tidy solutions, then what value exists in being able to calculate antiderivatives or to solve differential equations with nice tidy solutions?</p>
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<p>I think you actually have partially answered your own question without realizing it. English buffs and history buffs can certainly specialize in particular areas of interest without having to know everything about the entirety of literature or history. Nothing prevents me from being a Shakespeare buff without a knowledge of Chaucer, or an appreciation for poetry without knowing e.e. cummings. Similarly, one can be deeply familiar with the causes of the American history without delving into Europoean history. Further, there’s no predetermined sequence that history or literature must be studied in. Some texts/topics generally come earlier than others (for instance, most would study Romeo & Juliet before Macbeth), but this has nothing to do with the ability to understand the later ones as much as it does the emotional/mental maturity of the classes in question.</p>
<p>Math, on the other hand, by its very nature is sequential, at least in the early going through much of the early college track, and accordingly, much of the math knowledge that most math teachers have is incredibly common to all math teachers. The concepts of what math teachers should have beyond that track is extremely diverse, not necessarily dependent on calculus, and frankly not something that many teachers could apply to their classrooms.</p>
<p>If a history teacher continued to study the Civil War in more depth, wouldn’t that have direct application to the high school classroom when teaching that topic? Wouldn’t that high school teacher have a direct opportunity to show some of the more unique aspects that they had discovered to enhance student learning?</p>
<p>On the flip side of that, if I learned how to solve triple integrals, my primary applications are going to be in either teaching a course that involves triple integrals or in pursuing other math courses that require their proper usage. There would be no way to reasonably introduce such concepts to math students prior to the proper time in the sequence.</p>
<p>So this later learning for high school math teachers, in general, isn’t something they have an opportunity to regularly employ in practice. That’s not really a good reason for mathematicians to not at least have a passing familiarity and an ability to reacquaint themselves with the mathematics when need be, but it’s a fairly realistic one.</p>
<p>Another side point is that most English and history buffs don’t have much that they can do with their degree, besides… well, teaching. On the other hand, most people with a more advanced understanding of mathematics could easily make six figures, so there has to be a significant draw to the field of teaching. I’d argue that some of the best possible people for the math classroom are lost due to the inability of the teaching profession to match the kind of dollars that can be made elsewhere.</p>
<p>I agree that AP Calculus BC is ridiculously easy. I took a one-semester online calc 1 course from a community college this summer. It took about two months to learn all of the material, plus a few weeks on the chapter “techniques of integration” for the hell of it. I just looked at the 2009 BC exam - could get the FRQ done in 20 mins. The material is almost trivial. I studied like a dog for Chemistry last year (and spent a year in the class!), and even after that the test was much harder. Same with AP Euro two years ago.</p>
<p>In response to the comments about theoretical vs. applied calculus:</p>
<p>I would argue that theoretical, proof-based calculus is actually more useful for most people than plug-and-chug applied calculus. Let’s face it, most people will never need to compute an integral. Even if they do, they will almost certainly have access to a computer algebra system or graphing calculator that can do it for them. Thus, the vast majority of AP Calculus students will not really need to memorize integration strategies for later life. However, the critical thinking and logical reasoning skills developed in proof are useful in areas outside of mathematics (I personally found that my debating and essay writing styles became stronger after I began doing proofs regularly).</p>
<p>To carry this reasoning even further, it is probably more beneficial for most people to never learn calculus in high school and instead focus on rigorous, proof-based HS-level algebra and geometry.</p>
I’m glad we agree on something. Or rather, I would agree on it if I owned a copy of that terrible book. Haha. Then again, after Catcher in the Rye, all other school books seemed terrible.</p>
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<p>Come now, Prof. We must learn to crawl before we can walk. As a mathematician (or at least someone knowledgeable in math) you know that the non-linear can be approximated by the linear in some cases. It all depends on the level of accuracy and the likes. Not to mention, I want to solve the Navier-Stokes equation. If I can’t solve the basics, I won’t be able to solve that. : ). But in all seriousness, the complicated is based on the basic, hence the name “basic”. You look at a proof and wonder, “how the hell did he come up with that?” Just experience working with the matter at hand. What knowledge can you pull out if your first notion of computing in an integral is a matlab code?</p>
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<p>The problem is that this knowledge is usually equivalent to that of a junior in college in a math field (not even a math major, just someone who works with math), if not less. While a history buff who specializes in Civil War may not be as knowledge to other contents, I’m not seeing what math professors are “specializing” in. It’s definitely nothing with rigor, since most of them are afraid of proofs. Their concept of “proofs” is the following: “Well, see, it works for this example” or “Well, I mean, you can tell this works like this if you just think about it.” Try writing that on a Real Analysis test. </p>
<p>So let’s see if I get you:</p>
<p>Either
A) Math professors specialize in something that they are not able to produce in the classroom,</p>
<p>or</p>
<p>B) They don’t specialize because it can’t be brought in the classroom.</p>
<p>Let’s focus on A)</p>
<p>So, what do the math professors specialize in? It’s obviously not Calculus, as one can see. It’s more than likely something that’s not proof-based, considering their lack of fluency in the language of proofs. Wait! Hold on. What math is left? Non proof-based math. Now, this is not necessarily bad. We need numerical analysts and the likes. But these people are not numerical analysts! If you had SOME kind of advanced math knowledge, it means you at least have familiarity with proof-language. </p>
<p>As for B)</p>
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<p>The question is, if you didn’t learn to solve triple integrals, how the hell are you a math major? The question that still here is that to be a math teacher, as I understand it, requires a math major. If you can’t do triple integrals, how did you graduate from college to begin with? Now, of course, if you don’t apply the knowledge, you’ll probably forget it. Granted, I don’t see why one wouldn’t keep reading books to expand one’s knowledge, especially if one loves math, but we’ll ignore this. </p>
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<p>Related with the last paragraph, I’ll concede this fact to you. Mind you, that’s a terrible excuse and, realistic or not, it should not be acceptable. Granted, they are not fully-responsible for this.</p>
<p>Let’s tie back up to the main topic. If the AP Exam actually required a good depth of calculus knowledge, then this intellectual laziness/incompetence of certain calc teachers would not be acceptable. After all, should the exam at least offer difficult questions, then this would force the teacher to get a better grasp of the subject just so the students could also grasp it. </p>
<p>I don’t think you ever gave your opinion of the AP exam. Would you like to share?</p>
<p>I would have thought that students couldn’t learn how to factor polynomials without first learning how to do basic multiplication and division either, but as it turns out, I’ve actually seen an increasing number of students learn how to do this. Could someone learn what they need to know of the theory of calculus without actually learning how to do some basic integration? I suppose it’s possible, but I wouldn’t want to attempt to teach that course. I know some folks who probably would, though.</p>
<p>I actually disagree with your comment about math teachers being afraid of proofs. Your examples that you cite are cases of inductive reasoning and poor reasoning, respectively, and perhaps your teachers are indeed guilty of arguing that these formulate any basis for a proof.</p>
<p>But I do think that teachers have gotten inherently lazy in the presentation of mathematics, and I would argue that this relates back to the concept of providing a high school educational program that every student is capable of handling. The appreciation for the proof is a dying art form, in part because we (generally, not specifically) are no longer interested in why things work but that things work.</p>
<p>I think another failing of the modern-day mathematics program is that we’re more interested in assessing pieces that occur on standardized testing. That will never include proofs, unfortunately, and until our over-emphasis on testing fades, I believe we’ll see teachers continually teach to the test, so to speak. There’s actually no reason why AP Calc teachers couldn’t – and dare I say, shouldn’t – teach delta-epsilon proofs in their classroom, but many don’t, knowing it will never be assessed on the AP Exam itself.</p>
<p>As for the commentary regarding math majors, believe it or not, Calc III and Diff EQ were not required courses in my math degree at a relatively competitive college. I took Calc III since my math education degree required it, but regrettably didn’t have time to take Diff EQ without investing an initial semester at ridiculously high tuition dollars or seeking a waiver to overload my class schedule. I think you would be surprised at how diverse the requirements for a math major are.</p>
<p>I also think you’d be surprised on the demands on your time that come with teaching in general. I work an average of 70 hours per week on the various tasks related to my position, and have such a sick sense of fun that I’ll post on a forum since I don’t have enough students of my own to work with. Other people choose to have families. C’est la vie. :)</p>
<p>I actually am largely a fan of the AP Exam in its current format. There are topics excluded that I believe should be included in the curriculum (I’d personally reintroduce Newton’s Method and integration by parts back into the AB curriculum, as just a few examples), and I believe that some shoring up needs to occur to ensure some consistency between what the BC classes mean from school to school. Having students who start after Labor Day in a stand-alone BC class competing against students who start mid-August with AB one year and BC the following year can’t produce an equitable system for representing student ability on that test.</p>
<p>I also would generally like to see the algebraic rigor on the AP exam increased, not so much because it relates to the ability of one to do calculus, but because facility with algebraic manipulations is an essential factor in determining a student’s likelihood of being successful with mathematics at the college level.</p>
<p>But I would be careful not to introduce questions that require students to do too much more thinking outside the box than the current exam already does. While there are certainly students who can quickly and easily identify how to do every problem on the AP Exam, we already knew these students were ready for collegiate mathematics and then some. And I truly believe that the vast majority of these students have the ability to do well in the following college course at any college in the nation. The ability to do these kinds of problems would only help to distinguish the super-elite from the elite, and that’s frankly not the goal of the AP Exam, nor should it be.</p>
<p>The goal of the AP Exam should be to determine which students have demonstrated competency with basic calculus. And while we can certainly debate whether or not the current cut scores do this (I would argue that the cut score for a 3, considered passing by some, should be increased), I think the overall exam does an adequate job in most respects of determining who has sufficient preparation and who does not.</p>
I think the more interesting question is the reverse: if the majority of situations with actual calculus applications outside of theoretical mathematics consist of largely non-integrable functions and differential equations without tidy solutions, then what value exists in being able to calculate antiderivatives or to solve differential equations with nice tidy solutions?</p>