Timeline for Why do we teach calculus students the derivative as a limit?
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| Nov 30 at 3:50 | comment | added | ASlateff | This old thread here seems to be open even more than 10 years later. I was also teaching first semester students at a Univ.of Technology. Functions were particular graphs. In engineering and sciences, graphs and diagrams are omnipresent. I also prefer to introduce relations before introducing functions, because relations are met easier in everyday life. A straight line is an "easy" graph, and for a tangent one has to fit a line to a graph. There exist many fitting criteria! Tangent is the limit of secants, and for the tangent computationally, the relative error should vanish in the limit. | |
| Oct 3, 2021 at 19:56 | comment | added | Thierry | I'm not a professional mathematician but I majored in math and the sin problem is completely easy to me. After all, I learned calculus from Hughes-Hallet! | |
| Mar 8, 2020 at 13:51 | comment | added | Tom Copeland | @Lspice, after coming back to America and getting some experience with 11 and 12-th graders studying calc, I understood how even functional notation is difficult for many, maybe most, students to handle. I attribute it to doctrinaire, opportunistic admin and overtaxed teachers who follow texts more or less blindly and promote students rather than deal with toxic parents ... . Don't get me started on the dumbing-down of America by big business (as educ. has become) and big gov't. E.g., witness the authors for Sci. Am. 30 years ago (researchers) compared to now (professional science writers). | |
| Mar 7, 2020 at 22:04 | comment | added | LSpice | @TomCopeland, students have a rough-and-ready understanding of concepts that mathematicians would recognise as applications of functions, but they don't understand functions themselves! As one illustration of this, witness the difficulty faced by students who, where $x(t)$ gives the displacement of a particle from the origin after $t$ time units, are asked to distinguish between the graph of $x$ versus $t$ and the trajectory of the particle. | |
| Mar 24, 2016 at 19:25 | comment | added | Deane Yang | By the way, viewing a derivative as a "sensitivity" is nothing but a way to describe the linear approximation of a function. But I believe it's a way that makes it easier to understand and use the approximation than the usual formula we teach. | |
| Mar 24, 2016 at 19:23 | comment | added | Deane Yang | No. The traders just know that there is a black box called the Black-Scholes formula (or more sophisticated variants) that, given a volatility and stock price, spits out the option price. They want to hedge their position by shorting the right amount of stock that will offset any changes to the option price. The derivative tells them the right "hedge ratio". It's all very simple and uses nothing but the basic concept of a derivative as a sensitivity. This simple view is useful in many other contexts. So I believe it's shame to flood students with so many other things but not this. | |
| Mar 24, 2016 at 14:17 | comment | added | Tom Copeland | So they have at least an implicit knowledge (and most likely explicit knowledge from books on quantitative vs. fundamental analysis of the market) of when and how to fit a curve (a function) to sections of massaged data, in agreement with my first point. Btw, I taught freshman calculus and physics without calculus during graduate studies at an engineering university. No motivated student had a problem with the courses, in particular with concepts involving ratios of changes. | |
| Mar 23, 2016 at 18:44 | comment | added | Deane Yang | I would hope that anyone who has learned calculus properly would know immediately (without having to think about the graph) that, under reasonable circumstances, the option price would change by approximately by $0.40. In fact, option traders, including those who have no formal training in calculus, do use the derivative in exactly this way. They even know that the second derivative tells them whether the estimate is an over- or underestimate. And they understand that there are extreme circumstances where the estimate is not useful at all. This is what I mean by working knowledge of calculus. | |
| Mar 22, 2016 at 21:46 | comment | added | Tom Copeland | I can only suppose that you would ensure that they think of the problem geometrically in terms of a graph of the dependent and independent variables of suitably averaged prices, which would be the natural line of thinking if one started from the physics approach first. I can appreciate how difficult it might be to come up with a derivative from the real data that would have predictive value. What should be the window for the time-averaging, ..., etc.? | |
| Mar 22, 2016 at 21:04 | comment | added | Deane Yang | That's not my experience at all. Here is a simple example from finance: The price of an option on a stock is a function of the stock price. Suppose that when the stock price is 100, the derivative of the option price with respect to the stock price is 2. Suppose today's stock price moved by \$0.20. Estimate the change of the option price. Will a student say immediately that the option price changes by \$0.40? Or will they try to figure out what rule or formula they should use to figure this out? | |
| Mar 22, 2016 at 20:50 | comment | added | Tom Copeland | Students have an implicit, working understanding of functions. They know that given any group of people they can assign a unique height (mod units in feet and inches or centimeters) to each person but given a height they might not be able to assign only one person to it. If they are confused, it can only be by an obfuscating formalism. | |
| Mar 22, 2016 at 20:33 | comment | added | Deane Yang | I view sensitivity as a way to indicate clearly why the derivative is a useful tool for analyzing functions outside the context of the motion of objects.. Also, I don't think the terminology is the crucial issue. I believe students stumble here, because we have not ensured that they have a working (and not just a formal) understanding of what a function is. | |
| Mar 22, 2016 at 5:56 | comment | added | Tom Copeland | The use of the term sensitivity for the derivative is incredibly silly to me, especially for physics majors. The natural introduction is through the Newton quotient, velocity, and trajectories of balls. Then generalize to limiting rates or ratios of changes of other quantities. | |
| Dec 22, 2014 at 17:04 | comment | added | Deane Yang | Ben, the ones who got it wrong just guessed that the derivative formulas would be the same for degrees as radians. The fact that they guessed instead of making some effort to work it out was in fact what flabbergasted me. I would also note that many (maybe most?) mathematicians are uncertain about whether the constant is $\pi/180$ or $180/\pi$. This is not such a serious issue, but it bothers me that any engineer or physicist would be able to answer instantly. | |
| Dec 22, 2014 at 15:39 | comment | added | user21349 | I've read through the whole dialog in comments about the sine function, and I'm still mystified. Among the mathematicians who got it wrong, what was the wrong answer that they gave? What was the reasoning that they offered? | |
| Oct 23, 2014 at 13:41 | comment | added | Deane Yang | Depends on what you mean. If you need it for your work, you'll learn it properly through what you do. That's always the best way, because you understand why it's needed. If you just want to learn it, you succeed either by having a really good teacher or studying it on your own (and not being satisfied until you understand it inside out). | |
| Oct 23, 2014 at 13:12 | comment | added | The_Sympathizer | @Deane Yang: With all this bad teaching, how can you hope to get a good education in math?! | |
| May 22, 2012 at 1:51 | comment | added | temp | Seems David Feldman just said exactly what I wanted to say in my last comment. I learnt math from books translated from Russian, they at least mention "the invariance of 1-forms" when they talk about the chain rule. People might found this pointless, as calculus students may not need that fact. But 1 year-ish later when I really get what that "the invariance of 1-forms" I was really happy. And for not so advanced people, that's a reliable way to memorize the chain rule, right? | |
| May 21, 2012 at 7:40 | comment | added | Deane Yang | David, you don't seriously believe that we should teach 1-variable calculus in terms of 1-forms, do you? Co-ordinate-free mathematics is important and useful to some of us (it certainly is for me), but, it seems to me, not for the vast majority of our students. And, yes, when I say "sin of x degrees", I do mean a different function than "sin of x radians". | |
| May 21, 2012 at 5:04 | comment | added | temp | Seems like this thread is still open to discussion. On the derivative of $\sin\theta$ and $\sin x$, I think the problem is, while most people are ok with functions, they sort of have formulas in their mind. It is ok if one just think this as a formula in this case. However, if one really think about $f$, the sine function as a function on the circle -- a map from the circle to the real line, defined independent of any coordinate, then he will soon realize that the "derivative don't make sense", what make sense, is the one form $df$. That's the real conceptual jump, which is done by E.Cartan. | |
| May 19, 2012 at 5:15 | comment | added | David Feldman | >In my department we audition all candidates for teaching calculus and often ask this question. So I might not get that job, but my answer would have been "it depends." If you really do calculus right, the derivative of a function is not another plain-old function, but rather a 1-form. So in that more correct framework the passage from a fixed function to its derivative is coordinate free and that means units don't matter. Of course if you really mean to change function instead of changing units, you have a different function and you get a different derivative. | |
| May 19, 2012 at 5:04 | comment | added | David Feldman | I like "sensitivity" instead of derivative, thanks for that. I also like "stability" instead of "continuity." I try to emphasize the issue of control...as in what would happen if you tried to drive a car where the position of the wheels didn't vary continuously with how you turned the steering wheel. | |
| May 4, 2012 at 4:27 | comment | added | Steve D | Personally, I don't think introducing $e^x$ as the inverse to the natural logarithm is the best way to do it. I actually wait until we've talked about derivatives, etc., then introduce $e^x$ as the solution to a particular differential equation. At this point my students have done some work with differential equations (via a CAS such as Mathematica), so this is not too foreign for them. I am a big believer that every calculus class should have a significant CAS component. So while I give an intuitive definition of a limit in class, most of my students see them mostly in guided calculation. | |
| Mar 9, 2012 at 18:27 | comment | added | Steven Gubkin | @Vectornaut - you are saying first year calculus students shouldn't understand the chain rule? | |
| Sep 29, 2011 at 14:48 | comment | added | Deane Yang | Agol, this is an issue only if you feel obligated to give a rigorous proof of everything. Defining the logarithm first and the exponential only as an inverse to the logarithm is for me pedagogically backwards, since the exponential functions are much more easily motivated. Heuristic arguments for the derivatives of exponential and trig functions are easily given. You can then assert the formulas as axioms. | |
| Sep 29, 2011 at 5:47 | comment | added | Ian Agol | One frustrating thing about teaching calculus is to try to prove (or give some idea of the proof) of derivatives of exponential and trig. functions. This is easily done if one teaches the definite integral first. Then one may define ln(x) as the integral of 1/u from 1 to x, and invert to get the exponential (this is done in Spivak and Apostol). Also, to differentiate sin(x), you need to use arclength, which is again done easily first using integrals. The alternative rigorous approach is to define the exponential and radians via approximation by rationals, which may be done before integration. | |
| Jun 24, 2011 at 6:19 | comment | added | Vectornaut | (--->CONTINUED) If you do pick a way of measuring angles (that is, a parameterization of the unit circle), you can think of the sine function as a map from the real line to the real line. The derivative of this "sine on the line" does depend on how you measure angles. Of course, this probably can't be explained to first-year calculus students... and if it can, it probably shouldn't be. :( | |
| Jun 24, 2011 at 6:19 | comment | added | Vectornaut | @Toby Bartels: "You're talking about the function that maps any x to the sine of x radians and the function that maps any x to the sine of x degrees, right?" For first-year calculus students who happen to have taken differential topology, there's a third option. The sine function is most naturally defined as a map from the unit circle to the real line, and its derivative turns tangent vectors of the circle into tangent vectors of the line. The derivative clearly doesn't depend on how you measure angles; in fact, it doesn't depend on the concept of "angle" at all. (CONTINUED--->) | |
| Apr 4, 2011 at 1:21 | comment | added | Deane Yang | It should be noted that when we first thought of asking the question about the derivative of sine of $x$ degrees versus sine of $x$ radians, the intent was not to test whether the potential instructor knew the right answer or not. We assumed that they did but wanted to see how well they could explain why to a student. It was a rude shock to discover that more than one person with a Ph.D. in math did not even know the right answer. | |
| Apr 4, 2011 at 0:24 | comment | added | Toby Bartels | Darn, I was kind of hoping that I'd misunderstood the question ... | |
| Apr 3, 2011 at 22:45 | comment | added | Deane Yang | Toby, the problem is easy, if you understand and use the actual meanings of the words "function" and "derivative". But it is a measure of badly we teach calculus that even people with Ph.D.'s don't always answer freshman calculus problems because they are going into a reflexive "freahman calculus mode", which does not involve or use the actual meanings of the word "function" and "derivative". If you think about it, most freshman calculus courses do not require students to know or use the precise meanings of these words. So most students don't. | |
| Apr 3, 2011 at 22:36 | comment | added | Toby Bartels | OK, that's still longer than saying, hey, $\mathrm{rad}^{-1}$ and $\mathrm{deg}^{-1}$ are different units, so of course the derivatives expressed in these units are different. (Probably not longer to guess the right answer, just to check it with confidence.) But how could it trip anybody up? | |
| Apr 3, 2011 at 22:32 | comment | added | Toby Bartels | I'm not sure that I understand the sine problem; it seems too easy. You're talking about the function that maps any $x$ to the sine of $x$ radians and the function that maps any $x$ to the sine of $x$ degrees, right? These are different functions, so they'll have different derivatives, barring some coincidence; intuition leads us immediately to the right answer. We still check for coincidence; two functions have the same derivative (if and) only if they differ uniformly by a constant. To rule this out, it's enough to notice that these two functions agree at $0$ but not at $1$. | |
| Jan 31, 2011 at 11:17 | comment | added | Gerry Myerson | Deane Yang, thank you for your thoughtful comments. | |
| Jan 30, 2011 at 4:23 | comment | added | Deane Yang | I do consulting in the financial sector, and first and second derivatives are widely used but never symbolic differentiation. The derivatives are always computed numerically and under the hood. What your finance students need to know is how to interpret and use these numbers. This is presented rather well in my view in the first few chapters of the Harvard Calculus text. | |
| Jan 30, 2011 at 4:20 | comment | added | Deane Yang | Gerry, I have two reactions to your comment. One is that if you are teaching a single semester "terminal" calculus course, then you definitely have to choose and compress your topics carefully. The second is that I still consider it much more important to teach such students how to use a derivative as a useful measure of "sensitivity", rather than how to compute derivatives symbolically. In fact, using the sensitivity approach (as often seen in physics and engineering courses), the product rule appears very naturally. | |
| Jan 29, 2011 at 23:49 | comment | added | Gerry Myerson | Deane writes about "the first semester." For at least half of my students, the first semester of calculus is the only semester of calculus they will ever see. If they don't see, say, the product rule first semester, they'll never see it. These are largely students in financial studies. | |
| Jan 24, 2011 at 11:36 | comment | added | Joel David Hamkins | Your description of a function as a box seems to miss the most important part: that whenever you put a given number in, you always get the same output. That is, the box behavior should be single-valued. (Otherwise, we might imagine a black-box that accepts a given input and outputs a random number, perhaps different every time, and although this accords with your description, it is not a function.) | |
| Oct 19, 2010 at 14:11 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Oct 1, 2010 at 3:36 | comment | added | Alexander Woo | We use the Harvard book, which I like a lot. I also like Ostebee and Zorn, if you manage to find a copy. | |
| Sep 28, 2010 at 2:07 | comment | added | Deane Yang | Alexander and Pietro, unfortunately I said "I would like to see...", which means I don't really get to banish symbolic methods for a whole semester. In fact, I advise being pragmatic and teaching in a fashion that will not alienate you from your department or school administration. That said, if you want to slip in more understanding (which I claim actually helps students learn the symbolic methods better), I recommend taking problems from the Harvard calculus textbook, as well as their precalculus text ("Functions Modeling Change"). Especially those where no formula is given for the function. | |
| Sep 27, 2010 at 19:47 | comment | added | Pietro Majer | Deane, I find very interesting your method, and I agree with your conceptual approach. still I prefer not to wait a whole semester before starting symbolic manipulations. The style I like is: start from the problems (even simple, but real); emphasize the need of abstraction to treat them better; introduce the theory (not too much) and prove theorems; then go back to applications, and, lastly, by another important step of abstraction, develop the useful notation and formalism, and the rules of calculus. We can follow this procedure at each chapter of calculus: series, derivatives, integrals &c. | |
| Sep 27, 2010 at 18:33 | comment | added | Alexander Woo | Deane - you have any decent materials for a semester of calculus without any functions defined symbolically? I do what you suggested for the first two weeks (not enough, but I have a syllabus to get through), and the third time around I finally got around to typing up some notes so students have something to read in preparation for class. That took a couple weeks out of my summer, and didn't turn out all that great. | |
| Sep 27, 2010 at 17:54 | comment | added | Deane Yang | I would add that when we used it as an audition question, we assumed that everyone do it the "hard way" and would see that each sine function could be written in terms of the other and apply the chain rule. We weren't looking for the right answer, just the right thought process. I was quite stunned to see people not only not finding the right thought process but also guessing the wrong answer. It was around then that I started to suspect that not only are we teaching calculus students very badly, we're also teaching Ph.D. students badly, too. | |
| Sep 27, 2010 at 17:51 | comment | added | Deane Yang | Harry, that is exactly how any pure mathematician, including me, would do it. But that's the hard way. For an engineer or physicists, who thinks in units and dimensional analysis and views the derivative as a "sensitivity" as I've described above, the answer is dead obvious. | |
| Sep 27, 2010 at 16:58 | comment | added | Harry Gindi | Granted, the thing about $\sin \theta$ is easy if you do a symbolic computation and consider the chain rule. | |
| Sep 27, 2010 at 16:41 | vote | accept | Steven Sam | ||
| Sep 27, 2010 at 13:41 | history | edited | Deane Yang | CC BY-SA 2.5 |
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| Sep 27, 2010 at 13:39 | comment | added | Deane Yang | Any changes to math courses should of course be done only in close collaboration with other departments who rely on the math courses. But I think you'll find that, with the computational tools available, many of them will quite sympathetic to a "concept first, hand computation second" approach. Besides, I don't argue against teaching symbolic computation, just delaying it. And I do also like the Calculus in Context book but have not had experience using it. I suspect it works best with students with a stronger background than the ones I teach. | |
| Sep 27, 2010 at 12:55 | comment | added | Thierry Zell | I emphatically agree that students don't know what a function is. But then again, it is a deceptively deep concept. As for modern treatments that emphasize other things than the standard, did you ever look at "Calculus in Context", the five colleges calculus? Available at math.smith.edu/Local/cicintro/cicintro.html My problem with this approach though is that even if you can convince me easily that it's the right thing to do mathematically, how will it mesh with the courses in other disciplines that students take, which will expect much more traditional material. | |
| Sep 27, 2010 at 12:39 | history | edited | Deane Yang | CC BY-SA 2.5 |
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| Sep 27, 2010 at 12:20 | history | answered | Deane Yang | CC BY-SA 2.5 |