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This is what motivated this question, but it's something that I've considered several times now. In the example presented, someone asked whether or not the derivative of a function is an exact quantity or just an approximation. This question was asked by someone who is just learning what a derivative is.

My answer gives reasoning to why the derivative is an exact quantity—at least, conceptually, why it is such at that level of learning. A comment posted (one that is, by no means, inaccurate) is:

In most models of "infinitesimal", it doesn't "cease to be an approximation": your error is still a nonzero infinitesimal.

To what extent should something like this be considered when answering a question of this nature?

Another, more extreme example, would be if, when someone asks "I don't understand multiplication. Is $4 \times 3$ equal to $12$ or not? One brings into the answer multiplication modulo-some number, or matrix multiplication, etc.

Should I add a little disclaimer of something like "in higher-level mathematics, [this]", leave the comment as an informative aside, or what?

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  • $\begingroup$ Related: meta.math.stackexchange.com/questions/11419/…. $\endgroup$ – user61527 Mar 4 '14 at 21:36
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    $\begingroup$ While I can't recall my original motivations, I stand by the (my) comment because I truly think it is relevant: there is some amount of cognitive dissonance when you think of two things that deviate by an infinitesimally small quantity are exactly equal; the 'more advanced mathematics' just makes a precise statement about where the disconnect is. The passage from "approximations to the thing" to "the thing" is an important concept, and IMO failure to acknowledge this passage is a large part of why people get stuck on calculus being unable to make exact calculations. $\endgroup$ – Hurkyl Mar 4 '14 at 22:33
  • $\begingroup$ ... so to answer this meta question, one thing one should consider from advanced mathematics is if they are actually telling you something about the intuition regarding a problem (whether or not it would require too much sophistication or technical detail to actually cite the advanced mathematics precisely). $\endgroup$ – Hurkyl Mar 4 '14 at 22:36
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    $\begingroup$ To make a more pertinent comment: I don't really like the linked to answer to the question so much, because it is very unlikely that in the OP's calculus class $\delta x$ is defined to be an infinitely small quantity, as the answer suggests. To me this sounds like you are alluding to content that the student should have seen before. The fact that there is actual content that you may be alluding to does not change the fact that the student has not seen it before: if I didn't know about NSA, "infinitely close" would be extremely confusing to me as an explanation. That's just my opinion. $\endgroup$ – Pete L. Clark Mar 4 '14 at 22:57
  • $\begingroup$ Just for clarification, my defining $\delta{x}$ as being infinitely small was purely for the sake of intuition for the (very shortly) afterwards statement of the limit definition of a derivative. Maybe I was taught differently than others, but when I first learned what a derivative was, it was through this definition. $\endgroup$ – AmagicalFishy Mar 5 '14 at 0:07
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    $\begingroup$ @AmagicalFishy: My point is that "let $\delta x$ be infinitely small" is not a definition for 99% of freshman calculus students, because "infinitely small" is not given any precise meaning: in fact, as far as the "standard real numbers" go, it is logically contradictory. This is not just an instructor's quibble: if you say that, then many students will flip back through their notes or rack their brains in search of what it means for something to be "infinitely small"...and they probably won't find anything. As I said, this "definition" would have been extremely confusing to me. $\endgroup$ – Pete L. Clark Mar 5 '14 at 0:26
  • $\begingroup$ I'm afraid I'm having a hard time imagining students who understand $\lim_{x\rightarrow \Delta{x}} x + \Delta{x} - x$ but don't understand "infinitely small." At the Calculus level, I was not learning mathematics via a series of precise, rigorous definitions—I was developing an intuition for things like the derivative. I can understand, as someone who is quite familiar with these things (and higher mathematics) how "infinitely small" is hardly a rigorous definition, but I think it is an intuitively beneficial idea. $\endgroup$ – AmagicalFishy Mar 5 '14 at 5:47
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    $\begingroup$ @AmagicalFishy: Since I am describing my own experiences with calculus (as a high school student), I would respectfully ask you to make room in your imagination for such students to exist. If you had told me that two unequal real numbers were "infinitely close together" I would have explained to you why that was impossible using decimal expansions. I do remember learning the epsilon-delta definition of a limit as a calculus student, and I remember that I didn't understand it completely, but felt that eventually I would understand it better. $\endgroup$ – Pete L. Clark Mar 5 '14 at 6:01
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    $\begingroup$ In general I remember often being frustrated by "intuitive explanations" as a very young student of mathematics: it seemed like they were calling on knowledge that I didn't already possess. In particular I remember being told about spaces being topologically equivalent in terms of "stretching but not tearing" and wondering what on earth the actual definition might be. $\endgroup$ – Pete L. Clark Mar 5 '14 at 6:04
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    $\begingroup$ I don't claim that my experiences are typical, but I know from teaching that whenever you want to call upon intuition you need to be careful because not everyone will have that intuition. When I taught multivariable calculus to engineering students I learned that their physical and geometric intuition was very uneven, and that I couldn't count on it to the extent that I had expected to be able to. Finally, my main point is that it is good to try to give intuition and it is good to try to give definitions -- some of both -- but one should distinguish carefully between them. $\endgroup$ – Pete L. Clark Mar 5 '14 at 6:07
  • $\begingroup$ I agree with this. Perhaps I took your prior statement too literally—I don't mean to sound as if rigorous definition is bad or detrimental to a student's understanding. I mean, specifically, that the idea of two points between which you consider a secant line coming infinitely close together (to give you a tangent line) is a valuable idea for developing the intuition in dealing with the derivative. I do not think it is something that most Freshman Calculus students will struggle with, and the ones who do will benefit from the definition (undoubtedly presented shortly after). $\endgroup$ – AmagicalFishy Mar 5 '14 at 6:13
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    $\begingroup$ @AmagicalFishy: I think that when you say "infinitely close together", that calls up something in your mind: a picture, and more words. You should not assume that what that phrase calls up in your mind is the same as what it calls up in everyone else's mind: it won't be. If "infinitely close" is just your shorthand for "as h gets arbitrarily close to 0, the secant line slides closer and closer to some fixed line, which we call the tangent line", then fine. But "infinitely close" by itself suggests something static to me, whereas the limiting process is dynamic. $\endgroup$ – Pete L. Clark Mar 5 '14 at 6:18
  • $\begingroup$ Also: what you suspect that most Freshman Calculus students won't struggle with is one of the things they struggle with most. I've seen it many times over many years. I've recently taken an entirely different approach: limits and continuity are manifestly equivalent concepts, but for some reason limits are introduced first and continuity is explained in those terms. But all our intuition is for "nice unbroken curves", not "what happens to a curve when you remove the point and then get close to the point which isn't there..." That sounds like a zen koan to many calculus students. $\endgroup$ – Pete L. Clark Mar 5 '14 at 6:21
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    $\begingroup$ @AmagicalFishy: I spent three undergraduate years as a paid "drop-in math tutor", during which time I explained limits, including epsilon-delta, to hundreds of students. I had it down so that given half an hour I could get almost any student to understand something about epsilon-delta proofs. Being an instructor though I go on to give them the exams and learn what students don't understand as well as what they do. $\endgroup$ – Pete L. Clark Mar 5 '14 at 6:37
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    $\begingroup$ The limit concept is really one of the hardest things in all of undergraduate mathematics to get students to write and think coherently about. It is even a bit unfair in that it appears at the beginning of freshman calculus but almost no freshman calculus student is equipped to understand it thoroughly. It is something that undergraduate math majors struggle with as well: one often needs to study beginning topology to get it. I saw from one of your questions that you are just beginning this yourself, so you can probably look forward to viewing limits differently in the future. $\endgroup$ – Pete L. Clark Mar 5 '14 at 6:46
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This has come up before, but it's one of the real key points so let's take it up again.

Any answer that you write on this (or any other SE) site should be motivated by a sincere desire to be helpful to someone. This does not imply (in my opinion) that your answer has to be appropriate for the OP, because for most questions on this site the OP is just one of a much larger number of mathematics students who will be reading it. It is a matter of judgment when to leave a "higher level" answer; after a certain point I think you have to revisit the idea of having a sincere desire to help. If your answer to what seems to be a standard freshman calculus question involves measures and distributions on infinite-dimensional manifolds, then maybe you should look for a better spot to expose this knowledge: the people who would actually benefit from it may not be the ones who are reading the question at all.

What you are talking about seems slightly different to me: for pedagogical reasons you are leaving an answer which is more intuitive than precise (and in fact, when made precise may not be literally true). I think this is certainly part of giving good answers to a certain level of question, and more generally, part of being a good math teacher: part of the balancing act of knowing advanced mathematics (e.g. having a PhD in the subject) and teaching freshman calculus is that students can benefit from the depths and rigor of your subject knowledge --- but only if you show a lot of restraint, otherwise you end up snowing them altogether.

Where the right balance is depends on essentially everything in sight, so it is hard to say objectively, and well-meaning experts may disagree. I try to never say anything that from a more advanced perspective would turn out to be wrong (thus I am usually either very precise or really vague in what I tell students). Not everyone believes this is a good way to teach. In fact, I sort of believe that I would be a better teacher if I were able to lie a little more.

I remember once attending a talk for graduate students by a fellow young faculty member. He was talking about homotopy theory, and he mentioned that if two spaces had the same homotopy groups they had to be homotopy equivalent. I interrupted to ask whether I had heard him correctly, because at least the isomorphisms on homotopy groups should be induced by a map on the spaces. His response to that was memorable: "Shut up, Pete." I asked him whether I had heard his response correctly. I had [my last question had been in the way of a joke] and the talk moved on happily. This person was hired into a position that took advantage of his especially strong teaching skills, and his research expertise was algebraic topology [and mine is not]. In other words, he made a different pedagogical choice than what I would have made, and both his teaching skills and subject area knowledge were such that I have every reason to suspect that his choice was correct and mine would not have been. (But I still would never say that, and I can be the guy who pipes up on this site to mention that some pedagogically useful answer has a minor technical flaw. Sorry; we are who we are.)

To bring things back to the example: I would never say "infinitely close" in a calculus class, because it sounds like it should mean something, and it could, but I don't mean it to mean what it could in the sense of infinitesimals (or at least, not necessarily). Instead I would give the epsilon-delta definition at least once -- again, pretty much bad teaching that I can't get away from -- do my best to explain it geometrically at least once, and then use very vague language like closer and closer from then on. (Or in response to a persistent question I would volunteer to give the formal definition again, which is pretty much a showstopper.)

By the way, the correct answer to "Is the derivative [or the value of some other kind of limit] only an approximation?" is surely no, right? To say that it is seems to me to be committing a much larger teaching sin: to be overly wedded to an overly formalistic interpretation/understanding of a more advanced mathematical topic. (This is roughly the sort of sin that some calculus texts commit when they try to prove a big theorem: they expose their own knowledge of advanced calculus to be a bit more fragile and formalistic than I would like it to be.) Yes, you can do a nonstandard analysis calculation where at the end you take the standard part of a nonstandard real number, and in that precise technical sense the derivative might actually be an infinitesimal approximation. But then you take the standard part function and get the exact (in the standard sense!) answer. Unless you are actually teaching a nonstandard calculus class in which you are letting the students in to this technicality, you shouldn't hold onto it yourself when you are teaching calculus.

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  • $\begingroup$ (+1) The only thing I would disagree on here is that the primary desire should be to help the specific OP. If anything can be added so that it is also beneficial to a greater audience, that is great too. $\endgroup$ – Thomas Mar 4 '14 at 23:01
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    $\begingroup$ @Thomas: Most questions here get multiple good answers, which is a positive aspect of the site. If one is leaving an answer that is probably not appropriate for the OP, it would be very helpful if that were clearly indicated (otherwise some poor student could spin his wheels on something that he is not presently equipped to understand). It also couldn't hurt to wait for at least one straight-ahead answer to be left in order to leave a higher level answer. If one follows these practices, I see no negative consequences to leaving higher level answers. $\endgroup$ – Pete L. Clark Mar 4 '14 at 23:04
  • $\begingroup$ I completely agree with you on this. I was imagining the situation where no other answer has been given. So, of course, if a good straight up answer has been given, it would only be great if another answer could dig a bit deeper. $\endgroup$ – Thomas Mar 4 '14 at 23:06
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    $\begingroup$ I think small didactic lies are more dangerous on aplace like MSE where people on all levels show up. The didacic lie might be helpful to beginners who want a rough intuition, but it might seriously confuse people who are just learning the formal definition and trying to grok it. $\endgroup$ – Michael Greinecker Mar 5 '14 at 7:17

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