<p>Interesting. The only math required for my S major - Biochemistry- is calculus 1 & 2. Prereq is College Algebra & Trig for everyone - which he picked up in HS dual credit. I also wondered why calculus is what they required.</p>
<p>It depends on college math student is taking. D did not need more calc., she took Stats at college, much more useful for her. Stats is math but it is not calc., so not all college math is Calculus-Focused. But engineering majors need lots of math., including calc.</p>
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<p>Name one. </p>
<p>One of the features of the earlier thread was adult after adult (including scientists, doctors, and an architect) saying that they never, ever came across calculus applications in real life. No one but a few engineers ever used it.</p>
<p>One person pointed me to a supposedly real-life-based calculus curriculum online, and the introduction I read was laughable: an absurd hypothetical for which the use of calculus provided no meaningful advantage over basic arithmetic. With a fair amount of effort, I was able to think of a few problems that would interest ME that one might use calculus to address, but they are not the kinds of problems that would grab the attention of the world’s 12th graders.</p>
<p>Here’s how I’ve heard my H, a math prof at USC, describe it, by way of analogy. </p>
<p>Take a college football player. They do a lot of training specific to football. They also do a lot of exercise doing things like push ups, etc. Now, has anyone seen them doing a push up during a game? Probably not. Does the push ups help strengthen the body so they can do football specific skills during a game? You betcha.</p>
<p>So, subsitute calculus for football, and the mind being strengthened in lieu of the body.</p>
<p>This goes along with an early point, mentioned by several posters, that calculus is one of the core areas of math that is hopefully covered in HS. Realistically, we know that except for students thinking of going into college to study in a STEM area, it’s probably not.</p>
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Since he probably has to take Physics, he will be adequately prepared to take either the algebra-based OR the calculus-based physics courses. Calculus based physics is much better :). Algebra based physics is watered down for those who don’t have adequate calculus backgrounds.</p>
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If the measure of the value of teaching something is that the majority of the population needs to use it in their daily life, then you are correct.</p>
<p>Of course nobody really uses calculus in their daily life. But I can’t remember the last time I needed to use the sine of an angle in my daily life, or half of what I learned in Algebra either. Or know who Cotton Mather was either for that matter. So the question should be expanded to ask whether anyone really needs any specific math beyond arithmetic, or the specifics of US history, since they don’t often really use it in their day-to-day life.</p>
<p>However,although most people don’t really even use calculus in their jobs (because a lot of the work is now done by computers), at some point * somebody * had to know calculus in order to model almost every system in the world. And since we don’t know who that somebody will be, it is valuable to teach if we are going to teach any mathematics at all.</p>
<p>Doctors may not need to know calculus, but most modern monitoring and control systems in hospitals are based on analog and digital control - often something called PID proportional, integral derivative control. Somebody probably had to know LaPlace transforms and something about stability at some point to design them.</p>
<p>I’m no expert on it, but I imagine calculus is needed to model chemical reactions, which may be the basis of some blood tests. After we throw some reagent into a cup of goop we need to know when we will get to the maximum amount of product. I assume this required someone to know calculus at some time, although obviously they don’t need to work out the equation every time.</p>
<p>I believe some types of economists use Calculus to predict things like price elasticity. At least they have computer programs that do this for them, which somebody had to model and then program. Similarly, I bet a lot of investment modeling/trading software has calculus built into it somewhere.</p>
<p>I don’t think I’ve ever come across an eignevector in my daily life either. But I’m sure glad somebody taught them to Larry Page every time I Google something.</p>
<p>I’m no expert either … but I wonder what might replace pre-calc as an indicator of math ability. Most people don’t use Chemistry or Biology (or Physics or Literature or History or Foreign Language, etc.) in their daily lives. But colleges surely use results from those HS classes to evaluate applicants. No engineering school is going to accept Statistics as proof of potential … just as no Medical school is going to look kindly on an applicant who earned straight C’s in college science courses. (Just curious … when was the last time your primary care physician used Organic Chemistry? Does that mean prospective physicians shouldn’t take O.Chem?)</p>
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<p>I guess I must not have been really clear, and I apologize for that, but I agree one-hundred percent that Calculus is important, for the reasons listed above, and everything in bovertine’s post also. But like you said, Calculus is one of the few branches of mathematics that has made it into the high school curriculum, and my impression is that Algebra 2 and Pre-Calculus curriculum are based off of the idea that the student will be taking Calculus later on. This is what I don’t understand. Why is Calculus so much more important than let’s say, Algebra? The vast majority of my friends will not have taken a math class that is not preparation for Calculus, is Calculus, or based on Calculus since their high school Geometry Class. This seems extremely odd for a subject that is usually taken every year until at least the first year of college. And I don’t see how being exposed to only one part of math is helpful at all.</p>
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<p>So I guess this shows my bias, but the exact opposite was true for me. In high school, calculus was so overemphasized as what higher-level math consisted of, that once I actually took my AP Calculus class, I was really disappointed. I didn’t really have a strong desire to take higher-level math classes until I learned a little bit of group theory over the summer.</p>
<p>bovertine, I have pretty much written myself out on this topic, but I want to make clear that I don’t think the measure of value of teaching something is whether you use it in your daily life. On the other hand, if people are going to defend a universal calculus requirement on that basis, they ought at least try to be specific.</p>
<p>No one is suggesting, by the way, that economics students, or biomedical engineering students, or certainly math students shouldn’t learn calculus. Some people may suggest, but I’m not one of them, that no one should have to learn more math than, say, Algebra II, to be considered well-educated. But right now, the near-universal model, at least for smart, academically ambitious students, is that everyone has to take a year of calculus, either in high school or as quickly as possible in college. And I find that curious, both because personally I can’t remember anything about the calculus I learned, mainly because I never encounter it in real life (despite using math all the time), and because math-sophisticated people (math professors, grad students) seem to regard it as considerably less important than other things people might study.</p>
<p>I’m not suggesting, either, that calculus doesn’t have valuable applications in the world, of the sort you describe. (My rough understanding, however, is that in areas where the world is not actually continuous, which are many, discrete math + computing power can be more accurate, and that many problems that used to be solved with calculus are now being solved that way.) I, too, am glad SOMEONE understands it.</p>
<p>Everyone knows that calculus is required, and therefore that it must be terribly important. But calculus courses fail to convince many students that calculus is terribly important, and lots of knowledgeable people also think it’s not that big a deal. And there’s no question the ubiquitous calculus requirement crowds out, say, a ubiquitous statistics requirement – something that would seem a lot more valuable from the standpoint of citizenship – as well as other possible areas within math.</p>
<p>I took calculus, not statistics in high school, like everyone else who wanted to go to a good college; almost every day I wish I knew more about statistics, and I never wish I could remember my calculus. I even understand that calculus can be useful for statistics, except that (a) you couldn’t have told that from my calculus course, and (b) no introductory statistics courses, and surprisingly few advanced ones, seem to require calculus. (That’s not true at MIT, and I’m sure some other places, too. But my son, for example, has taken graduate statistics courses in his social science field that are supposedly covering the latest super-hot methods, and he reports that he’s still not using any of the calculus he learned twice.)</p>
<p>The push-up analogy is a great one, because it cuts in all sorts of different directions. Yes, no one does push-ups in real life. (I can, however, think of a common activity in which most of us participate very enthusiastically in which push-up skills are often useful.) But it’s easy as pie to explain what muscles you are trying to develop with push-ups, and what aerobic functions are being served, and why those things are important. And people understand quite easily that push-ups make them look better and feel better. So the question “Why are we doing these push-ups” doesn’t even come up that much. At the same time, push-ups are not the latest, greatest, hippest exercise around. Exercise science and (yes) fashion have moved beyond the push-up, and I venture to guess that people don’t do as many of them as they used to, and that sophisticated exercise programs are no longer push-up-centric. It’s not that we don’t care about the goals of push-ups, but we have more efficient ways of achieving them.</p>
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<p>Obviously pre-calc is based on the idea that one will take calc later on (hence the name). My impression of Algebra II is that it is designed more to be practical for everyone - to teach everyone the basics of things they might find useful later on, such as the concept of an exponential or a logarithm, or techniques for interpreting graphs. I can’t remember everything else that was in Algebra II.</p>
<p>Maybe you could be more specific. Exactly how would you redesign the math curriculum, and specifically courses such as elementary algebra under your vision? Would you have paralle tracks leading to either Abstract Algebra or Calculus? Or would you just include additioonal topics? I guess I’m not getting what you would leave out of today’s Algebra II, and what you would add. Since there is limited time. And like I wrote before, I’m not really a math guy. And to tell you the truth, I guess I never thought about it.</p>
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To a certain extent that may be true. I would say more accurately, that the world cannot be modeled by simple linear equations, and the necessary complicated non-linear differential equations need to be solved by numerical methods. But that still requires a knowledge of calculus in many cases.</p>
<p>Discrete modeling, such as something like digital filtering, requires the use of difference equations. My belief is that to the extent I understood these, the fact that they were analogous to continuous differential equations helped me out somewhat.</p>
<p>I guess I wouldn’t take a huge level of exception to the other stuff you wrote. I would never try to justify the teaching of calculus based on its general usefulness in everyday life.</p>
<p>I’m the architect from the previous thread, who never uses calculus, but only wish I’d written as eloquently as b@r!um on that thread! That said, I do think it’s too bad that some of the other interesting byways of math never get taught in the high school curriculum. My son learned about matrices and modular arithmetic for example in a cryptography class in middle school. There are all sorts of interesting topics in topology and number theory that are easy to understand. One of my son’s calculus teacher was lamenting that she would like to teach a course that would cover these subjects, but has been unable to persuade anyone to let her offer it. Some of this fun math is here: [This</a> Is Mega-Mathematics!](<a href=“http://www.ccs3.lanl.gov/mega-math/index.html]This”>http://www.ccs3.lanl.gov/mega-math/index.html)</p>
<p>The OP’s original question is why calculus in high school? I believe that calculus it critical for some fields, but not all. If you want to be the best candidate for engineering school, AP calculus is on your must list of classes, or so it seems today. If your not going to use it in your choosen field, skip it and take something more relevant in high school. You can always take it later in college, if required.</p>
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<p>Algebra courses in high school are stand-alone subjects that are even more critical and fundamental than calculus. This is my experience, at least. I don’t think they exist in order to prepare one for calculus - they are important for every student, whether or not that student goes on to take calculus.</p>
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<p>What does it mean for it to be over-emphasized? And what made you disappointed? Was it too easy? Not interesting enough? I mean these issues might have to do with the curriculum, the teaching in the class, or other specifics, rather than calculus as a branch of math. </p>
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<p>Some of these things are taught in other classes. For example, matrices are covered in algebra courses (again, in my experience.) My high school’s higher math sequence worked in the following way: Algebra 2 in 9th grade, Trigonometry/Discrete Mathematics in 10th grade, Pre-Calculus in 11th grade, and one of the AP Calculus courses in 12th.</p>
<p>The sophomore year class had a semester of trigonometry, and a semester that covered various topics in discrete mathematics (combinatorics, probability, some graph theory.) So although this is just one specific example - there do exist other areas that get coverage in a public high school math curriculum that ends with calculus.</p>
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<p>A lot of your argument seems to focus around this; ie, that a majority of people who should be considered “experts” actually don’t care about calculus or don’t think it’s important to learn. Is there evidence of this beyond the few anecdotes presented in this thread?</p>
<p>I would be absolutely shocked if the majority of pure mathematicians genuinely believe that students taking calculus should be taking statistics instead. Perhaps math graduate students and math professors who are focused on certain fields that do not directly relate to calculus (ie, those who aren’t in analysis) would prefer their areas to get more attention earlier on, but that’s just their passion for their field speaking.</p>
<p>Similarly, perhaps the math people would prefer these courses get taught in a more rigorous way. So an introductory analysis class rather than calculus. This isn’t really appropriate for the majority of students, but math professors often only interact with the very most talented students, so they lose sight of this fact.</p>
<p>Still, can you give us some support for why you think this is true in general?</p>
<p><a href=“b”>quote</a> no introductory statistics courses, and surprisingly few advanced ones, seem to require calculus.
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<p>Any mathematical statistics class will require calculus, because you can’t do the math without calculus. You just can’t; knowing calculus is fundamental for mathematical statistics. I happen to have taken the first Mathematical Statistics class at Stanford this summer (just for fun, because I like math) and we spent plenty of time integrating.</p>
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<p>I agree, but I think the counter-argument is that there exist statistics courses that do not use calculus at all - all the fundamental aspects are abstracted away.</p>
<p>I think a close analogy might be physics - even fundamental mechanics and E&M courses require calculus, and lots of it, but it’s possible to create a very stripped down version and avoid those topics altogether.</p>
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<p>I’ve only taken intro statistics courses, but my impression is that most statistical techniques don’t require much math at all to actually use. OTOH, they’re all based on probability theory, which (in continuous cases) is entirely calculus. If you want to find the probability associated with a z-score, you don’t need to know anything about the Gaussian Integral. You don’t need to calculus to use the Central Limit Theorem or construct confidence intervals either.</p>
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<p>Yeah, the first actuarial exam is just one integral after another.</p>
<p>CC’ers are a funny bunch–very thoughtful people who can over analyze the h*ll out of something. I must admit that all the viewpoints are quite interesting. </p>
<p>Unless a student is in a special gifted program that recognizes their mathematical brilliance, high school’s did not put this much energy into “why” students should take calculus in high school. It was the next logical step after pre-calculus. Parents found out that some students at some high schools seemed to have an advantage over their child because they took AP Calculus. And we all know what happened next. So now every high school has AP Calculus. </p>
<p>How do we one-up it so our kids will get that edge? How about offering the whole first year of calculus instead of just the first semester? Now everyone is taking Calc BC. But what about the real superstars? Let’s do second year calculus (analytic geometry, linear algebra, differential equations) at the local community college (or if we can at a 4 year college).</p>
<p>What percent of students that take AP Calculus successfully go on to the succeeding college calculus course? Engineers? Math and Physics majors? Well, yes! Statistics majors? Yes–lower division statistics does not require calculus, after that calculus is required. Statistics courses offered for grad students (in non math majors) are “usually” nothing more than beefed up lower division statistics courses.</p>
<p>It’s all about what looks the best on a high school transcript–not what is most useful.</p>
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It would certainly not be my experience that it is true in general, or even for a small percentage of mathematicians, other than to the extent that many of them do not like teaching calculus lite to recalcitrant non-science/math types.</p>