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To me, first-semester calculus is a "big ideas" course, whereas second-semester calculus is about a certain bag of tricks. How much memorization is it appropriate to require in the first semester?

If one gives open-notes and/or open-book exams, then a clear message is sent to the students that it's not all about memorization, and I think that's a good thing. I think instructors often use an overemphasis on memorization in order to keep students happy, since students universally accept a requirement to memorize, but may get very angry about a requirement to think.

But for example I often encounter students who don't remember, one semester later, that the derivative of $e^x$ is $e^x$. This complete lack of technical facility would seem to make it impossible for them to use their calculus for anything. It could be like turning out piano students who can't locate F-sharp on the keyboard.

Is there some basic body of knowledge that a calc student should be expected to memorize? If so, what does it consist of, and what are effective methods for enforcing this expectation? I have at least one colleague who has a short test on calculus facts that requires memorization and that is completely separate from the other tests in the course. Is this a good idea?

To put this another way, suppose that a student who got a C from me in first-semester calc shows up in your second-semester course. Is there a particular fact X such that, if the student didn't know X, you'd curse me for not having high enough standards for a C? In the case where X is $d(e^x)/dx=e^x$, I would think that an A student would understand that this is essentially the whole idea behind the exponential function, and therefore wouldn't have to memorize it -- but this might not be a reasonable expectation for a C student.

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5 Answers 5

up vote 12 down vote accepted

All the answers so far are very general, so here's a very specific, practical answer.

Students entering a second-semester calculus course need to be very good at taking derivatives. This includes memorizing the following parts of the differentiation algorithm:

  • The power rule (including the meanings of fractional and negative exponents)

  • The product rule

  • The chain rule (specifically, they should be good at using it)

  • The derivatives of $\sin x$, $\cos x$, $e^x$, and $\ln x$.

Of course, they must also understand the structure of complicated formulas well enough that they can figure out which rule to apply.

The reason students have to know this is that most of Calculus 2 is integration, which means doing differentiation backwards. You literally have to know the details of differentiation off the top of your head if you want to do it backwards. It is vital for checking the results of integration, and it's also used during both substitution and integration by parts. It is also used during subsequent topics, including differential equations and power series.

A few more notes:

  1. It's usually ok if students don't memorize the quotient rule. This rule mostly not relevant for integration, which means that it rarely comes up in Calculus 2. (Also, I happen to think that students are better off using the $A/B = AB^{-1}$ method than the quotient rule.)

  2. Depending on what techniques of integration are covered in Calculus 2, the students may need to know the derivatives of the other trig functions ($\tan x$, $\cot x$, $\sec x$, and $\csc x$) as well as the inverse trig functions. While students who have these memorized coming into Calculus 2 may be slightly better off, it's usually not necessary, and they can always pick them up along the way.

    Incidentally, of these rules, the most important is the derivative of $\tan^{-1}x$.

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I agree with these conclusions pretty much 100%, but I would add an extra justification. Even if I knew that none of my students were going to take second semester calc, I would want them to know more or less these facts, because I feel that these facts also have conceptual importance. The chain rule, for instance, is closely tied to the fundamental meaning of a derivative: it's what you multiply a change in the input by to get a change in the output. The quotient rule, on the other hand, is not much more than a formula that follows from the other more conceptual rules. –  Mike Shulman Apr 4 at 3:36
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This seems like a very good list. My only quibble is that maybe the chain rule should not be presented to students as something to memorize, since it's so trivial to justify (non-rigorously) in the Leibniz notation. Personally, I don't have the signs in the derivatives of sin and cos memorized; I recover them by visualizing $f'(0)$ for each function. –  Ben Crowell May 6 at 15:47
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My experience, even with grad students, strongly indicates that students do not realize that "passing the course" (or even "getting an A") is irrelevant, insofar as we want to demand something like perfect recollection of all details from all prior math courses. Apart from gaming-the-system issues, kids just can't fathom this, as it is quite contrary to other threads in their education.

If those remarks seem not to be responding to the question... this incomprehension on the part of the students has the consequence that they certainly will not review, and are not receptive to in-class review, of courses in which they received a passing grade. It is mildly amazing to me, but I have witnessed it many times.

This seems to be an argument in favor of substantial memorization in lower-level courses, because otherwise the most rudimentary facts will have slipped out of students' heads. I think that to some extent this is correct, insofar as memorization is a dubious approximation to understanding, but is an approximation, and does imbed things in peoples' heads. In contrast, an incomprehending disengagement visibly (in my experience) does not stick things in peoples' memories.

A common device to reward having-things-in-their-heads is to emphasize speed. Indeed, having things in one's head speeds things up. However, "speed" is also not what we should want to declare a "mathematical virtue".

The most conscionable answer I've come up with is in-class repetition of critical riffs. The idea is to do the repetition for them, without declaring that they should "do drills". This is especially relevant at the graduate level, I think! Yes, somewhat less material is covered, but maybe a much greater fraction is remembered.

That is, it is obviously not intellectually sensible, and not humane, to demand lots of memorization, even though it would benefit students in the long run. Rather, the repetition that will ingrain things in their memories might be created in-class, ... without announcing that that's what happening? :)

(I've even had students complain that they "see what I'm doing", and are somewhat annoyed that I have caused them to remember things that they didn't really want to remember after the exam was over... A strange objection.)

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One other aspect I wonder about is how to deal with things people can derive. Not calc, but for instance I derive "special" triangles like 30/60/90 from easier-to-memorize components (e.g. equilateral); or do work rate problems from the "velocity" equation. This puts me at a distinct disadvantage on timed tests from people who did memorize that, but when we see the problem we both know exactly how to solve it in the same amount of time. How would you deal with that sort of thing? –  Jsor Mar 30 at 7:18
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Basically, how would you not penalize the student who says "okay, I 'll remember that this is a special case of general equation B and rederive it when that time comes" over the student who says "I'll remember this specific equation and apply it when I see a problem like this." Both are different types of memorization in a way, but it's hard to construct a way to not reward the latter student even when both are equally capable at solving the same problems. –  Jsor Mar 30 at 7:22
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@Jsor, your points are entirely valid, and relevant to the ideal intellectual aspect of mathematics courses. Thus, I avoid penalizing for "lack of obedience" or "lack of quickness", or superficial virtues. The two complications are that the established grading system "insists" on a certain grading appearance, and, the vast majority of students behave in a certain fashion (not irrational at all, given the general game-to-be-played). Ideally, I'd like to persuade people of the virtues of certain approaches, whether or not they have alternatives. Persuasion is good, as opposed to bullying. –  paul garrett Mar 31 at 1:55
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As little as possible. We do have technology (books, Wikipedia, CAS, ...) which are more than able to replace the "learn by rote" that was critical for a medieval scholar, who could call himself lucky to have access to a dozen books.

Besides, I've heard the phrase "but that is the subject of a course we already passed!" too many times already when asking for something simple like the derivative of a simple rational function or a trivial integral in my second year discrete mathematics class...

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I would tend to agree with "as little as possible." But how little is possible? –  Ben Crowell Mar 30 at 0:56
    
@BenCrowell, that will depend on the later use. Formulas for derivatives of powers, natural logarithm, exponential function. Chain rule, derivative of product. That's it, as far as I'm concerned. Need also basic theorems on limits. –  vonbrand Mar 30 at 1:09
    
That seems like a reasonable list. So you have the derivative of $e^x$ on your list, but not the derivative of $\cos x$. How does this play out in your exams? How do you enforce the requirement that they memorize the one derivative but not the other? –  Ben Crowell Mar 30 at 1:30
    
@BenCrowell, it so happens that trigonometric functions are rare as generating functions... and can be looked up as needed. Point taken, anyway. –  vonbrand Mar 31 at 3:55
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I don't quite know if this answers the question, but I ask no more memorization of my students than I find myself capable of.

Perhaps due to (ahem) chemical indiscretions in late adolescence/early adulthood, I find myself not to be the greatest memorizer. Things like the sum-of-angles formula for $\sin$ and $\cos$ I find myself hopelessly unable to retain for more than about twenty minutes.

I know, however, that this is idiosyncratic. Some people are capable of remarkable feats of memorization. So I often end up teaching two tracks: I tell people in my calculus courses that, if they are only interested in gaining knowledge (as opposed to understanding) and are capable memorizers, then knowing e.g. this or that list of facts will likely suffice for passing (almost certainly without an A) my course. On the other hand, for those who want understanding or are (like me) incompetent memorizers, here is a bag of tricks for rebuilding all the information in this course from very little. Yes, you still have to remember the very little, but after using the bag of tricks for some time, you develop "muscle memory" that can get you from this to any given thing you might need.

An example of this is the use of De Moivre's formula to figure out the aforementioned sum-of-angles formula. It's the only hope for people like me.

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I let this be up to the student. I tell them, "these things will be provided on the exam, ... then proceed to give them a list of a few formulas, .... If it's not on that list and it's in the relevant chapters in the book or was presented in class, you will need to be able to derive it or have it memorized." I try to keep memorization to a minimum. I am often pleasantly surprised at how many students actually choose to derive some of the formulas. If I told them strictly to have memorized that formula, perhaps they would have gotten less out of the course.

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