What's So Great About Calculus?

Parents Forum
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Of course it would be better to know both, but does it all have to be a prerequisite, one thing for the next?

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<p>It depends. What a shame that you had such an awful experience with trig! And rather odd that it was a separate course - my high school had it as a unit in precalc. I'm not sure that I ever used more than the first week or so of trig again in any of my subsequent math or science classes.</p>

<p>Perhaps unfortunately, much higher math, including geometry beyond the high school level, DOES require basic calculus as a prerequisite. I'm not convinced that either linear algebra or theoretical computer science do, though. Think of it more as - hmm, searching for a good humanities analogy - you need to know the basics of painting techniques (at least the theory) before you can study Italian Renaissance painting. Or any other sort of painting. But you wouldn't need them to study Greco-Roman sculpture; you'd need a different set of basic tools for that.</p>

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<p>I don't think any high school would require beyond single variable calculus.</p>

<p>However, a few high schools treat AP calculus AB and BC as two different courses, and AB is consider to be prerequisite of BC. This is really stretching one year of calculus into two years, and it is not truly two year of calculus.</p>

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<p>St. John's believes in studying ideas at their source. Calculus is one of the great ideas and either Leibnitz or Newton (or both) is the source. They read Euclid too. It's not about textbooks, it's about historical documents. I'm sure they also learn about why Newton needed to invent/discover calculus in the first place. It's actually the aspect I like best about the St. Johns curriculum.</p>

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<p>The web page <a href="http://people.hofstra.edu/Stefan_Waner/cprob/cprobintro.html%5B/url%5D"&gt;http://people.hofstra.edu/Stefan_Waner/cprob/cprobintro.html&lt;/a&gt;&lt;/p>

<p>is about</p>

<p>Calculus Applied to Probability and Statistics for Liberal Arts and Business Majors.</p>

<p>An example is given in the introduction:</p>

<p>You are a financial planning consultant at a neighborhood bank. A 22-year-old client asks you the following question: "I would like to set up my own insurance policy by opening a trust account into which I can make monthly payments starting now, so that upon my death or my ninety-fifth birthday - whichever comes sooner - the trust can be expected to be worth $500,000. How much should I invest each month?" Assuming a 5% rate of return on investments, how should you respond?</p>

<p>To answer the question on the previous page, we must know something about the probability of the client's dying at various ages. There are so many possible ages to consider (particularly since we should consider the possibilities month by month) that it would be easier to treat his age at death as a continuous variable, one that can take on any real value (between 22 and 95 in this case). The mathematics needed to do probability and statistics with continuous variables is calculus.</p>

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<p>Yes, the St. John's approach is something akin to "phylogeny repeats ontogeny". You understand a system better if you understand how and why that system developed. So you start with Newton. Just as American Government classes still often start with Locke and Rousseau, and read the Federalist papers.</p>

<p>I took shots at St. John's before, but I can appreciate that aspect of its program, and personally I probably would have benefitted from it, at least insofar as calculus is concerned.</p>

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<p>I think many colleges are changing their curriculum to better meet the mathematical "needs" of all their students. S's college offers very different calc classes to science/engineering students than are offered to other students. I wish now that I had a better math and science background than I do (I took only science/math courses for NON-science/math majors; and have never taken a single calculus course). I never dreamed that I would end up in a field where that background would benefit me. Or that I would ever discover that I might have actually liked studying something like calculus if I had given it a chance(I should have listened to my GC who encouraged me to go into math based on aptitude tests but our hs math program was pretty crappy...) I am a trial paralegal but work on patent litigation and have frequent contacts with technical experts who, on large cases especially will often provide the trial team with hours-long tutorials on whatever science or technology is involved in each case. We'd be lost without that.</p>

<p>Here's a really nice article, "Creating the Quilt of Quantitative Literacy" (I LOVE that title!!) that stresses the importance of "quantitative literacy" in our society (higher-level math beyond high school level):</p>

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Much of this (mathematical) richness occurs when patterns amenable to quantitative or logical analysis arise in other subjects - in history or agriculture, in carpentry or economics. For students to develop mathematical habits of mind, they need to see and do mathematics everywhere, not just in math class. As writing is now accepted as part of entire curriculum, so should math.</p>

<p>The benefits of quantitative literacy are broader than being able to balance your checkbook or knowing which size of detergent is the best buy. Quantitative literacy is a way of thinking and reasoning that cuts across all disciplines. It is the historian analyzing a document for authenticity, or the attorney carefully structuring an argument, or the social worker calculating the mileage he or she traveled to see a client, or the college administrator evaluating the cost/benefit of canceling a class. As information becomes more readily available the need to understand and evaluate that information becomes greater. Quantitative literacy is needed in everyday life as well as the workplace. Should a menopausal woman use hormone replacement therapy? What are the risks of investing in the stock market? Can I understand my financial consultant when she or he explains how bonds work? What does a "yes" vote on a ballat initiative really mean?</p>

<p>...Quantitatively literate citizens need to know more than formulas and equations. They need a predisposition to look at the world through mathematical eyes, to see the benefits (and risks) of thinking quantitatively about commonplace issues, and to approach complex problems with confidence in the value of careful reasoning. Quantitative literacy empowers people by giving them tools to think for themselves, to ask intelligent questions of experts, and to confront authority confidently.

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<p><a href="http://www.evergreen.edu/washcenter/newsletters/Fall2003Newsletter/Pg9-10.pdf%5B/url%5D"&gt;http://www.evergreen.edu/washcenter/newsletters/Fall2003Newsletter/Pg9-10.pdf&lt;/a&gt;&lt;/p>

<p>A few more real-life examples illustrating how calculus can be used in every day life:</p>

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For example, consider the mathematical relationship between the diameter of a pizza and its area. You know from geometry that the area of a perfectly round pizza is related to its diameter by the equation </p>

<p>A = 1/4 * PI * d^2</p>

<p>You also know that the diameter can be changed continuously. Thus, you don't have to make just 9" pizzas or 12" pizzas. You could decide to make one that is 10.12" or one that is 10.13", or one whose diameter is halfway between these two sizes. A pizza maker could use calculus to figure out how the area of a pizza changes when the diameter changes, a little more easily than a person who only knows geometry. </p>

<p>But it is not only pizza makers who could benefit by studying calculus. Someone working for the Federal Reserve might want to figure out how much metal would be saved if the size of a coin is reduced. A biologist might want to study how the growth rate of a bacterial colony in a circular petri dish changes over time. An astronomer might be curious about the accretion of material in Saturn's famous rings.

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<p>That is interesting and got me thinking. </p>

<p>I certainly didn't feel that there were scores of Humanities that I had the interest, aptitude, or preparation for when I was in college. At my school, I would have been as far over-my-head taking most of the intro humanities courses as my history-major roommate would have been in Digital Signal Processing.</p>

<p>mathmom-
My St. Johns/Newton's Principia comment was an engineer's attempt at humor (though in retrospect, I should have cited Euclid's *Elements[/] as a 3000-year-old text!), but this underscores how deep the difference is between different types of people. The future Aerospace Engineer isn't going to go to St. Johns. The future Classics Scholar isn't going to go to Georgia Tech. And neither of them can understand why the other is doing what he/she is. And as Dr. Phil would say, "It's OK to be different!"</p>

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<p>Bomgeedad:</p>

<p>No amount of calculus would convert your example into something not absurd. There is/can be no such thing as an "insurance policy" with a risk pool of one. What's more, even if I translate it into a non-ridiculous question -- like "I want a 90% chance of having $500,000 in the account when I die" -- the additional accuracy obtained by having mortality as a continuous variable rather than a series of discreet binary possibilities (on, say, a monthly basis) is pretty unimpressive for a project like this.</p>

<p>(It is actually kind of shocking that a published textbook opens with such a silly problem. I don't know whether the problem is contempt for humanities and social sciences majors, or lack of connection to the real world among the authors of math textbooks.)</p>

<p>A better example is understanding the pricing differences between a European option (exercisable only at a date and time certain) and an American option (continuously exercisable until expiration). The American option is clearly more valuable, but by how much?</p>

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<p>I was a math major and loved calculus (in fact I think it's one of the most amazing things ever invented by the human mind), so my knee jerk response was to defend it to the last ditch, but I tend to agree with the basic thrust of the OP's argument. Most people will not in fact "need" calculus -- or at least the specific set of tools generally taught nowadays under that name -- in any meaningful sense. I would rather that people be required to know (and I mean really know) some basic statistics, so that they can read newspaper articles and have a sense when they are being BSed. (I think this ought to be a requirement for newspaper reporters, who can be an incredibly gullible lot.) I also think that the move away from the axiom-and-proof approach to geometry is a major mistake. My kids seem to have learned a lot of results, but very little about how to reason with them. It's the latter, not the former, that originally made the study of geometry one of a cornerstones of a true liberal education. So I'd bring back the traditional approach to geometry as well.</p>

<p>I would also note that judging from the comments of a number of posters on this thread, there's an incredible amount of bad math teaching out there right now.</p>

<p>PS -</p>

<p>I think about the relative areas of pizzas whenever I order but I never use calculus to do it.</p>

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<p>Fascinating thread…who’d think Calculus would arouse such passions! My HS senior D – emphatically NOT a math person – is taking AP Calc and AP Stats this year. AP Stats is “okay but nothing special.” But she is absolutely loving her Calc class, she finds it “relaxing” (!!!) and has a charismatic teacher, which I’m sure helps. She is really (by her own definition) a mediocre math student, but is intellectually curious about all sorts of things, and likes exploring topics that are outside her primary areas of interest, for their own sake. She seems to be approaching Calculus like a new language, and loves being able to converse, even in limited ways, in that mysterious math-y language. (I remember taking Calc in high school also, and enjoying it, but can’t remember a lick of whatever I learned back then. In fact, I came across some Calc notebooks a while ago – in my high school handwriting – and thought I was having an out-of-body experience…did I write that? Did I really once UNDERSTAND all that?)</p>

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<p>{Re: my previous post. It's not a published textbook, just a professor's web page. He really ought to revise it. I understand that point of the introduction is to illustrate something you need calculus to do, and it doesn't have to be realistic. But it's presenting itself as a real-world application, and it isn't at all. In fact, it isn't a problem anyone would use calculus to solve. And it's completely a trick question: The only way to satisfy the client would be to have him buy an insurance policy. A better way to phrase the question would be to ask how much of a risk premium the client ought to pay to purchase an insurance policy vs. self-insuring in the manner he proposes. That is actually an interesting, calculus-y question, with real-world applications. And it has profoundly affected the structure of insurance products over the years.}</p>

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<p>Very timely thread for our house....
My daughter, also emphatically NOT a math person, but with a decidedly different outlook on Calculus than OrchestraMom's D. Although she also has a teacher who opines about the "art of Calculus" and feels so strongly about the subject and her students, my D just is miserable. Her pitiful meltdown Wednesday evening before yesterday's test made me sad. She is likely going to drop down to an Honors level pre-Calc and frankly, I'm thrilled.</p>

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<p>I think what people may be struggling with is how calculus is taught, along with other college level math classes. If they are taught as a series of proofs then it can be quite difficult for some people. Though it is the proofs that get people to think through the logic and the solution.</p>

<p>I have heard from many of my engineering buddies that often calculus for engineers is different. It is a bit more concrete in its application, give them an equation for a particular problem so they can see it in use. This also highlights a major difference in the way people learn, some prefer the theoretical where others prefer the concrete.</p>

<p>Regardless, I find math and in particular calculus a major component of a liberal arts education, though I do agree with CGM that 2 years of calculus is overboard.</p>

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<p>I just discovered this thread and haven't read all the posts yet.</p>

<p>Regarding one earlier question on why pre-meds have to take calculus. Here is my take. </p>

<p>Advances in medicines in the 19 and early 20th centuries were made chiefly by physicians who utilized contempory discoveries in chemistry in their research to identify metabolic pathways and physiology. As such an understanding of cutting edge science at that time (calculus being one) was essential. With time, calculus was incorporated as a pre-requisite to be a physician. From the late 60's to the present, basic medical science began to takeover the basic research side of medicine, and the need for physician to be able to do research is increasingly de-emphasized. Thus, whereas 50 years ago, almost all medical students had to do "biochemsitry" or "phisological science" laboratory courses, this requirement is no longer in today's curriculum. Thus, calculus is a leftover requirement. </p>

<p>As I said in the other thread, calculus is nothing but the tranformation of non-linear functions to linear ones. In medical teaching nowadays, the translation has been made so that an understanding of calculus is not essential. That is true for most things. For example, we really don't have to understand the math behind carbon isotope dating, we just accept the number.</p>

<p>Calculus in college courses for many of us is very much like taking one or two semesters of a foreign language. We learn the rudimentary but we tend to forget it with time without the opportunity to use it.</p>

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<p>I often wonder about the tyranny of math and have for years. Its basically one of those "gatekeeper" subjects, more than any other in HS, that divides up the "smart" kids from the rest. "Smart" kids do calculus in HS, while the rest of the bumpkins stop at trig, if they're lucky, go off to second-rate state universities, and prepare for lifetime careers as low-paid civil servants. I think what is so annoying to us social science and humanities types is NOT that calculus and math are unimportant, but the layers of perceived prestige and hosannas heaped upon kids who do well at it, at the expense of those who do not. That's an issue close to my heart as I am wrestling with school now about some of these issues with my own kid.</p>

<p>To study calculus in college as training for the hard sciences or even the social sciences makes obvious sense. HS is a different matter, and the problem, I suspect, goes to the weird idea college admissions people have that the 'best" kids are good at everything. (Other wise why all those prospective English majors sweating AP calculus?) In fact, HS is weird because it is the ONLY place in life where you are expected to be good at everything. At work, you have a job. In college you have a major, usually with distribution requirements in disciplines other than yr major. Which, I may add, is a good thing. But not do calc in HS and you're a "loser." (And yes, I know some kids don't do that and get into top colleges. I suspect that they are the exceptions. </p>

<p>Calc is simply another way to think. The biggest problem is not whether or even how it should be taught. The question is why this way of looking at the world seems now to be so overvalued relative to everything else. It shouldn't be.</p>

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<p>I tend to like the proof/logic approach, at least in theory. That seems like a valid component of a liberal education, at least if the teacher can communicate what's going on and why it matters. The applications approach is not terribly interesting unless you are interested in the applications, and a basic course is only going to deal with very simple, unrealistic applications. </p>

<p>Now, from 35 years ago, I remember liking the applications more, even though I had no interest in them. They seemed at least possibly useful, if a little dumb. But that's because the proofs just seemed like a boring game not connected to anything.</p>

<p>But I'm still struggling to understand statements like "Calc is simply another way to think." What's different about it? The people who love it all say things like that, but they are not communicating how it differs from any other type of math-thinking. I think in algebra all the time. What would I be doing differently if I were thinking in calculus?</p>

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<p>Our school required a year of fine arts in junior high and a year in senior high. Why? If nothing else, it taught the kids to appreciate the arts. Maybe they learned a little about music-reading, music theory, or what goes into creating a sculped work of art. My kids will never NEED to use what they learned in their years of fine arts, but it enhanced their overall education, made them think a different way, use a different part of their brain, exposed them to something that challenged them. Now when they hear someone play a concerto, they truly understand the skill, beauty, virtuosity involved. </p>

<p>If nothing else, maybe taking a semester or year of calculus helps people appreciate the engineers, mathematicians, and scientists who build our bridges and dams and develop technologies to make our lives easier, safer, and more civilized. If nothing else, it makes people realize that they are worthy of respect, even though they may not be able to discuss philosophy and literature at the cocktail party. Darn, it's hard! Next time you see an engineer/mathematician/scientist, you can at least say- wow, that girl/guy made it through all those unbelievably difficult math classes- must be one smart cookie.</p>

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She is likely going to drop down to an Honors level pre-Calc and frankly, I'm thrilled.

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I am puzzled how she got into Calculus without having had a pre-calc course, but every school seems to do math differently.</p>

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<p>It is quite unfortunate that almost all the popular science books out there, such as those on BIG Bang, can be read by readers who are essentially math illiterate. That shouldn't be so. </p>

<p>One of my favorite book of all time, ranking up there with works by Dante, Proust and Shakespeare, is Feynman's QED, the strange theory of light and matter. It brings out the basic properties of light in a way that is as puzzling and as satisfying as the best literary descriptions on what make us human. It is, however, not a book that can be read without some understanding of simple calculus. While someone can distill the concept down so that one can state the properties without the math component, it is not unlike reading Dante's Paradise in translation, which can hardly be translated without losing almost all the thrills. I would think that most here would agree that "light" is one physical property that most educated person should know something ablout. How do we explain to our children why is it that we can see through a window, to see ourself in the mirror, to correct our eye sight with glasses?</p>

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<p>JHS--I think it's exactly the proof/logic approach that my daughter is enjoying. The game of it. But it's her teacher that's keeping it from being boring. He is also a man on a mission to help kids be more comfortable about the concept of making a mistake and not getting everything exactly correct all the time. He wants them to think, take chances, not be timid about playing with the ideas. This is a radically liberating idea for his class of (mostly) over-achieving perfectionists.</p>