<p>“It’s a wonder MIT, Stanford or Harvard are able to find qualified candidates in the US with the educational system describes on this thread.”</p>
<p>I posed this question in jest. We all know there are more than enough qualified US students for the top US colleges.</p>
<p>Differential equations IS calculus as well as multivariable calculus and calcBC which is integral calculus.</p>
<p>Yes, all students don’t need to take calculus in high school, but many do need to. There are probably more kids remedial in algebra in college than calculus.</p>
<p>This attempt to do away with calculus in high school is another attempt to DUMB DOWN the schools. NOT GOING TO HAPPEN.</p>
<p>Are people here actually positing the notion that engineers do not need to learn calculus? I don’t think everyone needs to know calculus, but certainly it is not too much to ask engineers and scientists to have some idea of where the tools they use come from. Granted, it is not something you need to use on a daily basis most places, but I have used it in my work at various times. Since my duties have changed I don’t use it much, but others in my office do. But that’s true for a lot of things that I don’t use but I think were worthwhile for me to learn.</p>
<p>I don’t think I’ve ever had to “complete the square” to solve a quadratic equation, but it helped me understand where the quadratic formula came from. How often does one use trigonometry or power series? Do we really need to know Euler’s formula? Cut that out too. What about linear algebra, I always hated Eigenvectors, but I don’t think we would be able to have the sophisticated data analysis tools we have today if at least somebody didn’t understand it. And who can tell who that somebody is unless the technically adept are exposed to it? </p>
<p>In my office a young kid just developed his own electric-pole loading model because he was dissatisfied with the commercial programs available. In his derivation he had to integrate the catenary curve. It turned out to be faster for him than searching for acceptable software. Someone developed a program for conductor heating, again because of dissatisfaction over the existing models. He ended up with a rather hairy partial differential equation which he had to solve using Runge Kutta, but I still consider the development of the equation to be calculus. </p>
<p>Is the idea that we have people learn physics without calculus? How do we teach Laplace transforms, or Fourier series, or Bessel functions, or just about anything in classical communications or control theory without calculus? Or is the plan to just tell everyone to forget about that, get some software and don’t worry about where your power spectral density or anything else for that matter comes from. Isn’t calculus involved in FEA. So again, we just learn to use the software and no point in any of the engineers understanding the math? I’m not sure that’s even possible.</p>
<p>Anyone who thinks there is no relation between statistics and calculus I would refer them to a book by Papoulis. Granted, for a lot of applications you don’t need to know the calc, and for others you can just plug into some software someone who actually understood the math made for you. But as someone who has been an expert witness on technical matters, at least someone on your team better understand the math behind the software or you are screwed.</p>
<p>Finally, to end a rather rambling post, almost anything at a college could be labeled an expensive boondoggle. I highly doubt universities are building billion dollar endowments by requiring some students to take a year of calculus.</p>
<p>Not to metion almost every kid in the world takes calculus in high school and clearly do well. If we can build a wall around the US and tune out the rest of the world, we can go on and on mulling over finer details. Given the implication of science and technology to the future economic well being, I don’t see how we can afford not to push our kids do well in math.</p>
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<p>I went to a charter middle school, where at the end of our 6th grade year they tested us for algebra readiness. Depending on how “ready” we were, some of us were put into algebra 1 in 7th grade. I remember being in the 8th grade algebra class with all the “big kids” and was fairly intimidated but loved the challenge. Then in 8th grade, about 10 of us were put into geometry (for high school credit). It was nice to be in a very small class with kids my own age and who are all advanced. It was also great to be entering HS with two math credits.</p>
<p>Then I went on to take algebra II in 9th grade, precalc in 10th, AP Calc AB in 11th and now I’m doing BC in 12th. I only really struggled with algebra II but I really attribute that to having a very rigorous teacher and taking the class in 9th grade when it wasn’t meant for 9th graders. I was still adjusting to HS while the rest of the class had already been there for 1-2 years.</p>
<p>Anyways I really think math is important and although I technically didn’t even have to take a math class my 11th and 12th grade years (for graduation purposes; obviously it wouldn’t be the greatest thing to show for colleges), I still felt like I needed to. I didn’t go straight to BC because I knew I had two years left so I could do both AB and BC. I’m glad I did because at my school AB and BC are in the same class, we learn the same stuff, take the same tests, etc. so I’m essentially taking the exact same class for the second time, and am therefore able to get away without ever doing my homework or stressing about the class (unlike last year, where I struggled to make a B in calc AB). </p>
<p>I think it’s really good for anybody to get ahead in math wherever they can, but only as long as they’re ready to… AP Calc looks intimidating when you’re not in it, but just like any other class, you learn along the way.</p>
<p>Leaving aside the arguments on calculus, my experience with a lot of different elementary and middle schools was that the “traditional” math sequence for 5th/6th/7th/8th grade and a really, really high degree of repetition. The Addison Wesley texts used by a lot of districts certainly showed that. For kids with much math talent at all, the excessive repetition and low pace of introducing truly new material was painful.</p>
<p>For kids who are performing well (not fabulously, but well) Algebra in 8th grade is a completely reasonable and feasible concept, and similar to what is done in many other countries around the world. Some kids are ready for that full algebra class in 7th grade, and a much, much smaller number in 6th grade. Just as we would hope not to hold back a strong reader in first grade (“No, you have to read The Magic Treehouse books, not Charlotte’s Web”) we should not hold back kids that have the capacity to move through math just to keep all the seventh graders together. </p>
<p>Perhaps we were especially lucky, but D’s 2000 student high school offered many paths through the math program – typically three, sometimes four courses at each level. Kids seemed to get into very highly ranked colleges whether they took Calc AB, Calc BC, IB Math (lower level), HL Math, or the Multi-variate calculus the university sent a professor over to the school to teach.</p>
<p>I’m also a big fan of at least Calc AB for any students who can complete the prereqs:
<p>Very interesting thread. Some pages back, alh asked my opinion. I didn’t post then, because I have only a limited angle on this topic. However, here are a few of my takes on things:</p>
<p>With regard to ucbalumnus’s original question: I see no reason for a school to insist that students take Calc AB before taking Calc BC, nor to enforce this by structuring the curriculum so that part of the Calc BC curriculum is not covered in the class labeled “Calc BC,” because that was covered in Calc AB. A student who is well-prepared with the pre-calculus material and has strong aptitude in mathematics can cover the entire first-year college calc curriculum in one year (with a reasonably good teacher).</p>
<p>I suspect that some of the students who are reluctant to take Calc BC are more concerned about not getting an A than about not being able to understand the material. Grade inflation, which has become extreme in some American high schools (more so than at the college level), plays into this, because a sub-A grade will adversely affect the student’s class rank. In the schools that are familiar to me, grades in math classes tend to depend more on test/examination performance than in any other subject. This means that a student will have to invest the time to “get” Calc BC. Meanwhile, students with high-level college aspirations face expectations for considerable accomplishment in their extra-curricular activities, and this reduces the time available to think about math. Also, because of college admissions practices, the return on EC accomplishment can often be greater than the return on taking the more challenging course. So in some sense, it’s a rational, “safe” choice to postpone or avoid Calc BC.</p>
<p>As far as the comparison of the top American math students with international students goes: A reasonably large number of American students who live near universities complete single and multi-variable calculus, differential equations, linear algebra (the matrix course), and some university mathematics beyond that, such as real analysis and/or abstract algebra. Their mathematical preparation goes three or four years beyond Calc BC. This has been true for at least 40 years. I didn’t take courses at that level pre-college, but had a high-school student in the senior-level college math course I took as a sophomore. I think this kind of coursework is comparable to the math preparation anywhere in the world. A few students take graduate-level math pre-college, but they are very unusual.</p>
<p>In quite a few countries, high-school mathematics includes vector calculus, differential equations, and linear algebra. US students who take Calc BC by sophomore year are on track with these students, and those who take Calc BC in junior year are close to that level.</p>
<p>On the question: Who’s pushing? I think this depends on the locale. Locally, the schools take a strong stance against acceleration in mathematics, for reasons that I don’t understand. In literature classes, a student who has developed a high level of understanding can still find the works interesting (even if the discussion is not), by engaging with them fully, as the author intended–particularly since the books in a reasonably good literature class were written for adults, by adults.</p>
<p>Not so, with the quadratic equation. Once you’ve derived the solution of it, and understood the possibility of complex roots, and explored the dependence of the solutions on a, b, and c, there is not much more interest available–until the student has taken abstract algebra, when it becomes interesting again. However, the student is very unlikely to discover results from abstract algebra for her/himself, and the number of high-school teachers who can introduce their students to Galois theory is rather limited.</p>
<p>This makes acceleration in mathematics sensible.</p>
<p>Back when I took Calc BC we did some differential equations, does that not happen anymore?</p>
<p>Calculus was a prerequisite for architecture grad school and we took a bunch of engineering courses there (one semester of statics without any math at all, then a statics course with math, wood, steel and concrete each as semester courses.) We never used calculus at all though there were a few times when I’d recognize that the calculus that lay behind some of the formulas.</p>
<p>The people at the Art of Problem Solving site make a strong argument for avoiding “the calculus trap,” of rushing ahead in mathematics to reach calculus, and then taking a course at a community college. They have a good point. If the US had a sufficient number of well-qualified mathematics teachers to implement the type of rich, pre-calculus curriculum they are advocating, this would be excellent. In practice, students have to choose from the course possibilities that are available to them; and in this setting, acceleration is often the best option, actually.</p>
<p>I think very highly of the AoPS people. But I think they ought to explore the “lay of the land” a bit more.</p>
<p>Hi, mathmom: Yes, some of diff eq is covered in Calc BC. I meant the full-blown, semester long course, that includes topics such as Laplace transformations to solve the equations, and matrix methods for coupled diff eqs.</p>
<p>I think the real trick is to match the level of math course to each individual student, as far as possible. I’d consider the course level a match if the topics, though initially unfamiliar to the student, seemed natural once the student worked on the course material for a while. </p>
<p>If a student has to resort to “this is how you do this type of problem,” there is a mis-match.</p>
<p>Several posters have noted the phenomenon of “hitting the wall” in mathematics. I think this is very real. If I could design a math curriculum from scratch, I would insert interesting “holding courses” that a student could take–different courses at each major blocking point–to develop additional mathematical maturity, so that the once-tough course would no longer be difficult.</p>
<p>Mathematics is a very weird subject in the sense that once one understands something, it is very hard to reconstruct why it ever caused problems.<br>
And yet . . .</p>
<p>Finally, on the issue of “calculus for everyone,” I think there is a real benefit for a student to take a course where the material is puzzling initially, but becomes clear after further effort. This would not have to be calculus.</p>
<p>I think that students should generally be encouraged to take calculus, if it is the next logical course for them, even if their current career plans do not involve the use of calculus. This would keep their options open. If the career plans change, then the student is less likely to need 5 or 6 years to complete undergraduate work.</p>
<p>By now, alh has almost certainly fallen asleep, and is regretting posing the question about my opinion!</p>
<p>“Starting off in Calc I in college – as most students do – is not going to hold her back from academic opportunities.”</p>
<p>^ I don’t agree. D came into her upper tier university having only taken regular Calculus (not AP) in high school. Having to start out lower than the majority of her peers on the calculus sequence in college, prevented her from taking her major’s intro level courses in freshman year. Now when she is looking for a summer internship for after soph. year, she has less relevant coursework under her belt than many other applicants who are taking the next level this year while she is only starting the 101 series.</p>
<p>True, the GFG. At many universities, the most challenging first-year courses in physics and chemistry are designed around the assumption that the student has already taken introductory calculus. Electricity and magnetism requires vector calculus as a co-requisite, at least, but in my opinion it works far better for a student who has already completed vector calculus before starting E&M.</p>
<p>I don’t know about you guys, but the math honors program in my school is a total joke! There are so many peopl who aren’t good at math and are in honors. There’s a sophomore in my Trig class and he’s completely lost already. </p>
<p>One time a friend of mine who’s in math honors asked his calculus teacher what he got a test they’d recently taking. The teacher told him he received a 48/50. When we went back to class he told our other friends that he wasn’t happy with his ‘95’. NINETY FIVE. I pointed his error out to him (in a nice way; I wasn’t b!tchy about it) and he insisted he got a 95. Everyone else (who btw aren’t in honors math) told him it was a 96; not 95. And finally he realized his error… We all laughed at the fact that he was the only calc student there, but really, in the back of my head I just thought “wow; theese are the kids who are in advanced math.” </p>
<p>I think the main problem with these honors math classes is the fact that it’s decided who’s in them in early middle school years. Not everyone who’s good at elementary/early middle school math will necessarily be good at algebra, geometry, trig or calculus.</p>
<p>Actually I’d even go out on a limb and say that most honors programs are extremely flawed. How do I know this? Because I’M in honors English and USH.</p>
<p>(I really suck at humanities)</p>
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<p>Is learning something useful only if we use it directly in our careers? I guess all those essays I wrote for my literature classes were useless because I don’t see myself writing anything more than e-mails or maybe math papers.</p>