# Revisiting topics from previous courses

I teach calculus to students who have almost all taken calculus before. (Primarily first-year college students who took calculus in high school but didn't perform well enough to skip the course.)

For many topics, my students think they know the subject already, but all they actually know are a bunch of short-cuts or special cases. The problem is that when I try to illustrate the new ideas with simple problems, they fall back on their short-cuts rather than using what we've covered. Then when they get to the harder problems, they're completely lost, because the short-cuts don't work, but they haven't gotten any practice with the new material.

I can tell them "you have to solve this problem using this method", but that feeds into the lesson that math is an arbitrary pile of methods which we choose based on the professor's whim, and it causes them to resent the new method.

My question is what are good tactics to break through this: to either convince students that it's in their interests to rethink how they approach the simple cases, or even better, a way to make the parts they know useful rather than an obstacle.

(The original version of this question gave about a concrete example of a subject where this happens, but it attracted a lot of off-topic answers focusing on the example rather than the question, so I've removed it for clarity.)

• Also, to further clarify the question, I'm not asking why they can't solve harder problems (abstract thinking is indeed a big issue at this level). I'm asking about how to overcome a specific obstacle to getting there. Usually we work our way from more concrete problems to more abstract problems. But in this case, students insist on solving the really concrete problems an unhelpful way, and then have no stepping stones to get to more abstract ones. – Henry Towsner Mar 17 '14 at 17:54
• The notation $\sin^3 x^2$ may not help, especially since then $\sin^{-1} x^2$ invites confusion. In a calculus class, $(\sin x^2)^3$ or $(\sin(x^2))^3$ is better. – user173 Mar 17 '14 at 18:02
• @MattF. I thought most professors used $\arcsin x$ to prevent such confusion. I'm kind of confused with why $\sin^3 x^2$ isn't a breeze. What technique did they learn that would make $\sin x^2$ easy but not $\sin^3 x^2$. – David G Mar 17 '14 at 18:33
• @Skytso: As far as I can tell, there's no underlying technique. They basically learned the chain rule the way one learns to speak a language: they practiced simple examples until they developed an instinct for the right answer. (I think it's telling that they can't handle $\sin \frac{x^2+1}{x+1}$, since no real technique should treat that differently from $\sin x^2$---the difference is that they can only do the chain rule when the two pieces have easy derivatives themselves.) – Henry Towsner Mar 17 '14 at 18:42
• I think this general question really have been posted as a new question, rather than an edit of the original, year-old question. The responses about the chain rule were not "off-topic" -- the original question was about the chain rule (it was right there in the title!). Now we have 13 'orphaned' answers that are no longer responsive to the question. – mweiss Jul 24 '15 at 17:32

Sometimes it can help to take things to a ridiculous extreme, like finding the derivative of $\ln(\sin(\cos(e^{x^2})))$. Tell them that the step by step process is like peeling off onion layers:

1. Take the derivative of the outside the function (the first layer), and leave the inside completely alone:

$\frac{1}{(\sin(\cos(e^{x^2})))}$.

2. Now multiply that by the derivative of the second function (the second layer), and leave everything inside of that function completely alone: $\frac{1}{(\sin(\cos(e^{x^2})))}\cos(\cos(e^{x^2}))$
3. Now multiply by the derivative of the third layer', and leave everything inside of that alone. Have students try it first, separately, then reveal the answer and see what they got: $\frac{1}{(\sin(\cos(e^{x^2})))}\cos(\cos(e^{x^2}))(-\sin(e^{x^2}))$
4. Have them complete the last few steps then check their answers: $\frac{1}{(\sin(\cos(e^{x^2})))}\cos(\cos(e^{x^2}))(-\sin(e^{x^2}))e^{x^2}2x$

Doing a long problem likes this helps them see how mechanical and straightforward it is. Once you do this, have them try: 1. Find the derivative of $\sin(x+e^x)$, then 2. Find the derivative of $\sin(xe^x)$.

If they get stuck, tell them to remember: Take the derivative of the first layer first, leaving everything inside alone, then multiply by the derivative of the inside, using any rules necessary.

I would leave powers of functions for last (like $\sin^2(x)$), because they look so different from other functions, being applied on the right. If you taught them about layers, you can explain that the power is the outermost layer. Otherwise, one option is just to teach it as a completely different rule: $$\frac{d}{dx} [f(x)]^n=n[f(x)]^{n-1}f'(x)$$

• This seems like a perfectly reasonable way to teach the chain rule for the first time, but I don't see how it addresses the question, which is about getting them to connect the chain rule in multi-step examples like these to the method they learned in high-school. – Henry Towsner Mar 17 '14 at 19:31
• This is the method I use to reteach students who have seen it before. I've found that starting from scratch can be very effective. Other people may have different methods, which I look forward to seeing; I've learned a lot from this site already. – Brian Rushton Mar 17 '14 at 19:32
• But how do you get your students to start from scratch? The problem I'm running in to is that I can teach them this way (and this is roughly what I do), but as soon as I turn my back they go back to trying to use the old way every chance they get. – Henry Towsner Mar 17 '14 at 19:34
• It sounds like the old way is simply learning to recognize certain problems and how they derive. I think that simply giving them longer derivatives that require the chain rule properly will eventually help it to sink in. – David G Mar 17 '14 at 19:57
• @HenryTowsner, give them a few homeworks/quizzes/exam questions where you specifically ask to do it step by little step for a monster like the one in this answer. – vonbrand May 2 '14 at 20:36

Edit: Since another response mentions using APOS theory to analyze what happens in learning about the chain rule, I thought I would direct you to a source you might find of interest:

Clark, J. M., Cordero, F., Cottrill, J., Czarnocha, B., DeVries, D. J., St John, D., & Vidakovic, D. (1997). Constructing a schema: The case of the chain rule? The Journal of Mathematical Behavior, 16(4), 345-364. Free PDF link.

Some of the sources contained therein or subsequent articles that cite this paper might aid you in your revisiting of the chain rule. Here is an image of the abstract; below the break is my original answer:

You close by parenthetically remarking:

Of course, general answers are helpful, as are ones specifically concerned with the chain rule. The broader question here, if you want to give a more general answer, is not "how should the chain rule be taught", it's "how should we deal with students who have previously covered topics in unhelpful ways".

I will attempt to provide a more general answer, which could be applied to the chain rule in particular:

George Polya's (1945) How to solve it contains a list of strategies ("heuristics") for mathematical problem solving. One of these strategies is to "think of a related problem." In the case you are describing, it seems as if students are not realizing that the problems that they are faced with are chain rule problems; more precisely, they are not able to see the relation between something like the latter examples you give and earlier problems solved with the chain rule.

One of the reasons Polya's book was so successful is that its content resonated with mathematicians, and students who do particularly well in mathematics classes. More generally, though, one might ask whether mathematicians (including experienced mathematics teachers - not just those who do research) and students would interpret related problem in the same (or at least a similar) way.

This was a topic of research around 1980, and the result was that mathematicians were inclined to relate problems by looking at deeper structural features, whereas students were more inclined to "relate" problems according to superficial features. (I will provide a few related references below!)

A way to intervene (generally or with the chain rule) could be as follows: Give your students a collection of problems and ask them to sort the collection by the approach they would use to solve the problems. Examine the sortings, and see if those of the students accord with your own.

If they do not (e.g., problems solved with the chain rule are not grouped together) then discuss the problems and their methods of solution, and explain why you sorted the problems as you did. Follow-up by having them classify another collection (or even the initial collection...) and see whether their approaches to sorting are closer to what you would expect/hope.

[If you should carry out such experiments, I would be interested to hear the results!]

References:

1.

Schoenfeld, A. H., & Herrmann, D. J. (1982). Problem perception and knowledge structure in expert and novice mathematical problem solvers. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8(5), 484. Link.

2.

Silver, E. A. (1979). Student perceptions of relatedness among mathematical verbal problems. Journal for research in mathematics education, 195-210. Link.

3.

Silver, E. A., & Smith, J. P. (1980). Think of a related problem. Problem Solving in School Mathematics, NCTM, Reston, Virginia, 146-156.

• Sorry but the fact is that this kind of approach is precisely what makes mathematics inscrutable to most students. Specifically, it encourages guesswork and conformity to social expectation (including what the teacher says and how the teacher writes). This invariably produces students who will be utterly confused and scared if their teacher one day keeps telling them that their correct answers are incorrect. This is because they are unable to determine with absolute certainty whether they are correct or not (barring carelessness) but instead rely on the teacher's judgement. [continued] – user21820 Jul 24 '15 at 12:09
• [continued] Thus they cannot even identify mistakes that their teacher makes, and even when they do not understand something it will hardly occurs to them that their teacher could be wrong. Furthermore, even if they are sharp enough to realize that something must be wrong, they do not dare to challenge the teacher based on pure reason, but rather they always rely on appeals to authority. If everyone says that they are wrong, they will go with the crowd. In contrast, students properly taught will be more than capable of correcting and even teaching their own teachers. – user21820 Jul 24 '15 at 12:13
• (In marked disagreement with the premises of the earlier comment that seem to make this answer very different from what it is...) I strongly support "identification" tasks, for the reason, among others, that when done well it can get students to operate at a somewhat higher conceptual level (rather than lowest-level mere-execution of lowest-level operations...) As an "amateur" educator, I'd suspect that there's enough inadvertent disconnect within students' own minds that if induced to "identify/relate" they can even inform themselves... [cont'd] – paul garrett Jul 24 '15 at 20:15
• ... [cont'd] of information they didn't know they had. In the best situations, this can get them out of "mental ruts" (a.k.a. collapsing back to old habits that will not scale...) The dangers perceived by the other comment(s) to this question are potentially real, but exist in any case. I think the notion that there's a readily accessible "absolute" that is all we need to do math is widely held, but profoundly mistaken in complicated, subtle ways. But, yes, some parents (and students) would rebel/react when given any but execution-of-algorithm (or "problem-solving") tasks. No concepts needed? – paul garrett Jul 24 '15 at 20:19

I no longer teach, but I used to delay the general chain rule for a while by making use of special cases. For example, after introducing the rule $$\frac{d}{dx}x^{n}=nx^{n-1}$$ (the power rule) and giving some examples with polynomials (and what I used to call "polynomial-like functions", which includes things like $$\frac{2}{\sqrt{x}} – 2x\sqrt{x} + 5x^{1/3} + x^{3}),$$ I would introduce the general power rule, which is $$\frac{d}{dx}u^{n}=nu^{n-1}\frac{du}{dx}$$ where $$u$$ is some function of $$x.$$ This was usually abbreviated as $$(u^n)'=nu^{n-1}u',$$ and I would indicate how the power rule is the special case when $$u(x)=x.$$ Each of the functions $$e^{x},$$ $$\ln{x},$$ $$\sin{x},$$ etc. come with their own general version --- the general $$e^{x}$$ rule, the general logarithm rule, the general sine rule, etc.

On tests (especially on gateway type tests) and on 5-minute quizzes I would include what I called abstract types, such as find the derivative with respect to $$x$$ of $$\sin{f(x)}$$ or find the derivative with respect to $$x$$ of $$\;x^{2} \ln{(f(x) + x^{3})}.$$ Of course, these types were done in class and given in homework, the latter being supplementary problems assigned (this code for that dirty word, worksheets).

It was only after students were (for the most part) fairly comfortable with these rules, both the specific versions and the general versions, and they were comfortable with evaluating derivatives of "abstract types", that I would bring up the even more abstract idea of having a general function on the outside. Thus, the derivative with respect to $$x$$ of $$f(x)$$ is $$f'(x)$$ (no surprise), and the corresponding general version is $$\frac{d}{dx}f(u) = f'(u)u'.$$ But before giving this, I would make a column on the board to show the pattern of all our general rules. The column would have 5 or 6 functions like $$e^{u},$$ $$\ln{u},$$ $$\sin{u},$$ etc. lined up so that in one vertical line we'd see all the $$f'(u)\,$$'s appearing and just to the right of these we'd see in another vertical line all the $$u'\,$$'s appearing.

At some point I would give one or more of the standard "analogy justifications" (e.g. bicycle gears), as well as various alternate notations such as the "little circle" composition notation and the Leibniz version where fraction cancellation seems to occur. However, all this came well after when I felt most students had fully internalized the formal technique. In this way the focus could be on the forest, without the tree leaves getting in the way.

What follows is a slightly edited version of my 4 November 2006 post the Chain Rule thread in the Math Forum discussion group AP Calculus.

Here's what I used to do regarding the mechanics of carrying out the chain rule.

If the function was, for example, $$f(x) = \sin{(3x^{2} + 2)},$$ then I'd write $$f'(x) = \cos{(3x^2 + 2)} \cdot 6x,$$ with a semi-circular arrow drawn over the top. The arrow's tail began just above (and roughly centered on) the $$3x^2 + 2$$ expression inside the cosine. The arrow curved in a concave down fashion (like the graph of $$y=-x^2)$$ to where the $$6x$$ was. I drew the arrow's head so that it pointed right at the $$6x.$$ When I was actually doing this, I drew the arrow after writing $$\cos{(3x^2 + 2)}$$ and before writing the dot and the $$6x.$$ I drew the arrow to "remind us" that there was more to do --- there's the derivative of the inside that still needed to be taken care of.

Every "inside" got its own arrow.

$$\frac{d}{dx}\sin{(3x^2 + e^{2x})} \;\; = \;\; \cos{(3x^2 + e^{2x})} \ldots$$

got $$2$$ arrows -- one arrow was for the sine (tail started at the $$3x^2 + e^{2x}$$ expression inside the cosine) and another arrow was for the exponential (tail started at the $$2x$$ that was inside the exponential which showed up after I had written down the term that the first arrow pointed to).

$$\frac{d}{dx}\exp\,\{\,\sin\,[\cos\,(x^2)\,]\,\} \;\; = \;\; \exp\,\{\,\sin\,[\cos\,(x^2)\,]\,\} \ldots$$

got 3 arrows -- one for the $$\exp,$$ one for the $$\sin,$$ and one for the $$\cos.$$ In this case, the arrows humped horizontally across like the graph of $$y = |\sin{x}|$$ does on the closed interval $$[0, \, 3\pi].$$

Almost all of my students wrote their answers this way on tests, which was fine with me. I didn't require them to rewrite the derivative leaving out the arrows or require them to erase the arrows.

Incidentally, when I first introduced the product rule (or when a student later had difficulty), I found it helpful to insert a parentheses template before going further. Thus, when showing the use of the product rule for

$$f(x) \;\; = \;\; (3x^2 - 2x)^{3} \cdot \cos{3x},$$

I'd first write

$$f'(x) \;\; = \;\; (\;\;\;\;\;)'*(\;\;\;\;\;) \;\; + \;\; (\;\;\;\;\;)*(\;\;\;\;\;)'.$$

The idea was to get the students to see the big picture before we got into the details. Some teachers do this using rectangular boxes.

Then I'd write the appropriate factor inside each pair of parentheses, not rewriting or doing anything else -- just copy the factor exactly as it was written.

$$(\,(3x^2 - 2x)^3\,)' * (\,\cos{3x}\,) \;\; + \;\; (\,(3x^2 - 2x)^3\,) * (\,\cos{3x}\,)'$$

$$(\,\text{one derivative goes here}\,) * (\,\cos{3x}\,) \;\; + \;\; (\,(3x^2 - 2x)^{3}\,) * (\, \text{other derivative goes here} \,)$$

Then I told them: "Now we're $$3/4$$ done!" Most of the time we really weren't $$3/4$$ done, but this often generated laughs (especially when it was clear there was quite a bit involved in taking care of the two indicated derivatives), and I think it helped to put the more math-phobic students at ease. The "$$3/4$$" comes from having half of the problem taken care of with the setting up and filling in of the product rule tinplate, and having half of the parentheses parts already done (i.e. $$2$$ of the $$4$$ parentheses parts were freebies). If necessary, I'd go to another part of the board in order to work out each of the indicated derivatives and put the results where I wrote "one derivative goes here" and "other derivative goes here" in the last displayed expression above. I usually only did this (calculate the derivatives of the factors on another part of the board) the first day we covered the product rule, or when someone was having a lot of trouble with it later on, or when the function itself seemed to be scary looking to the students.

• Plus 1 for the "parentheses tinplate." – Frank Newman May 20 '15 at 12:46
• @FrankNewman: If you think that parenthesis templates help students, you should actually think about what it means. It means that students do not even understand the substitution rule, namely that if $q=i$, then substituting $q$ for $i$ in an expression keeps its value the same. If the product rule is actually understood, then using it is an utterly trivial application of substitution. – user21820 Jul 24 '15 at 11:58

I suggest to have a look at your students' concept of a function. Typically, they see a function as a process but not as an object that can be part of a larger object. (They will identify terms as parts of a larger term; however, they don't realize them to be functions in a process sense.) There is a nice piece of theory by a group led by Ed Dubinsky, they call it APOS theory, referring to Action, Process, Object and Scheme as levels of understanding:

Dubinsky, E., & Mcdonald, M. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. The teaching and learning of mathematics at university level, 275–282.

Maybe you could try to make a bit more explicit that the use of the chain rule requests two functions to be chained. One could try to simply identify a "chain of functions" in terms like $\sin(x+1)$ or even $x^4=(x^2)^2$. As a powerful application you may let them try to work on $x^x$. Unfortunateley, the transition from process to object state of understanding has no clear way that always works.

There are many problems rising from this problem, e.g. they might struggle with seeing a differential operator (or any other operator) is linear, since they get confused because there are two functions at different level: the operator itself and its input are both functions.

When helping friends with this who don't understand the chain rule, I actually show them chain rule from the very beginning and make them apply every derivative rule to things that don't necessarily need it such as $f(x) = x$.

$$f(x) = x^1$$

$$f'(x) = (1)x^{1-1}\cdot(1)$$

$$= x^0 \cdot (1)$$

$$= 1 \cdot 1$$

We then move on to $f(x) = x + 1$, and then to an $f(x) = 3(x+1)$, and then to a $f(x) = 3 (x^2 + 1)$, slowly getting more complex. They'll hate you for it at first, but then will start to see that you should show every step every time, no matter what.

By showing them that it works for even the easiest problems, I've found that something seems to click in their heads letting them know that they should use the chain rule for everything. I think the hardest part about chain rule is that people don't understand why they use it sometimes and not others, so showing them that they should use it all the time makes it become habit as problems become more complex. So if you start out saying "This is the chain rule. Use it on every derivative problem, even if you don't think you need to," it should become habitual rather than a daily struggle.

• This reminds me a little of this answer to a question on MSE. – J W Apr 8 '14 at 17:58
• Glad someone else thinks like me :) I've found it very effective. I've taught people this 6 or 7 times now and they've all understood completely. It's awesome seeing their test on it and how well they did – meer2kat Apr 8 '14 at 18:19
• Thanks @quid ! I haven't learned the Latex stuff yet :) I'll work on it! – meer2kat Apr 9 '14 at 19:27
• You are welcome, and welcome to the site! Most of the edit was actually done by @Mike If you have not yet seen it meta.matheducators.stackexchange.com/questions/93/… could be useful if you want to learn about math formatting possibilities here. – quid Apr 9 '14 at 19:57

In France this problem is made especially acute by the fact that the concept of composed functions are not part of the high-school curriculum, so that student learn a bunch of separate rules to differentiate special cases of composed functions (one for $u^2$, one for $u(ax)$, one for $\ln(u)$, etc.)

What I try to do is to use student's natural lazyness by telling them I will teach them a rule that contains all of these rules, so that they will be authorized to forget them.

One thing that could help them grasp it is to list a lot of functions they learned to differentiate using the chain rule, and ask them to find the common pattern between the formulas instead of giving it yourself. This may take a little time, but that would be well-used time.

With problems such as $\sin^3 x^2$ I tell them to explicitly introduce the helper function $u(x)=x^2$ and rewrite it as $(\sin u(x))^3$ or, if necessary, more helper functions. That seems to help!

"For many topics, my students think they know the subject already, but all they actually know are a bunch of short-cuts or special cases. The problem is that when I try to illustrate the new ideas with simple problems, they fall back on their short-cuts rather than using what we've covered. Then when they get to the harder problems, they're completely lost, because the short-cuts don't work, but they haven't gotten any practice with the new material.

I can tell them "you have to solve this problem using this method", but that feeds into the lesson that math is an arbitrary pile of methods which we choose based on the professor's whim, and it causes them to resent the new method."

Talk to them and explain why you are doing what you are doing!

At the beginning of the course, say many of you have had calculus before but didn't do well enough to place out of it. This can help you at times, but at other times can hurt you. If you already know some of the material, that is great, makes it easier to study. But it can lead you into false confidence and into goofing off and not studying material hard enough (both old and new). Note, I think this is the bigger danger than using isolated shortcuts! Be up front and tell them that you don't want them to take calculus three times! This is the chance to learn it right, so let's get 'er done! [Also mention that sometimes they may only know a shortcut or one method and not pay enough attention to rest of the course for problems that don't work with that shortcut or particular method.]

In terms of learning different methods, I would tell them that YES, you do want them to learn and drill particular methods using THAT method at the time. Tell them you will allow them in tests to use whatever method works (if there are multiple approaches). But it is important to have many tricks in the bag, many tools in the box. So when learning a method, go ahead and use that method in the homework drill (and recitation). Only way to learn new things. Otherwise you rely too much on old methods and don't ever get the benefit of new.

Compare sports for instance. In wrestling practice, you will learn different takedowns (single leg, double leg, hip throw, etc.) When you are learning that trick, you need to practice that trick. In a match, you can use whatever seems best at the time. And you may have some tricks you are better at and lean more towards. But there will be situations where only one trick works or it works easier. The more you can get good at different tricks, the better a wrestler you will be. You won't be as good if you only have one takedown that you are good at.

I always thought compacting notation helps in such cases (it helps me not to make mistakes in complicated cases). To consider the "ridiculous example" of @Brian Rushton (to whose answer I believe mine is just an extension):

$$y_1=\ln(\sin(\cos(e^{x^2})))$$

I would make the following definitions

$$y_2 \equiv \sin(\cos(e^{x^2})),\qquad y_3 \equiv \cos(e^{x^2}),\qquad y_4\equiv e^{x^2},\qquad y_5 \equiv x^2$$

So now I would be looking at

$$y_1=\ln(y_2),\qquad y_2 = \sin(y_3),\qquad y_3=\cos(y_4),\qquad y_4= e^{y_5}$$

and so I could write the "direct" derivatives

$$\frac {\partial y_1}{\partial y_2} = \frac 1{y_2}\qquad \frac {\partial y_2}{\partial y_3} = \cos(y_3),\qquad \frac {\partial y_3}{\partial y_4} = -\sin(y_4),\qquad \frac {\partial y_4}{\partial y_5} = e^{y_5},\qquad \frac {\partial y_5}{\partial x} =2x$$

Visually this helps connect with the abstract chain rule

$$\frac {\partial y_1}{\partial x} = \frac {\partial y_1}{\partial y_2}\cdot \frac {\partial y_2}{\partial y_3}\cdot\frac {\partial y_3}{\partial y_4}\cdot\frac {\partial y_4}{\partial y_5}\cdot\frac {\partial y_5}{\partial x}$$

and then we insert the simple derivatives we calculated

$$\frac {\partial y_1}{\partial x} = \frac 1{y_2}\cdot \cos(y_3)\cdot [-\sin(y_4)]\cdot e^{y_5}\cdot 2x$$

and then substitute for the $y$'s from the definitions we have made in the beginning.

1. Explicitly tell students how you make the decision of what rule to use. There are features of the problem that sing "chain rule" to you, so explicitly tell them what those features are. Also include in this information a list of things to try in order of most likely to produce a helpful result. It's a bit like a doctor in the emergency room of a hospital. The patient presents with various symptoms, which they learn how to associate with various medical conditions. Those possible conditions are also sorted in their minds into an order of likelihood and they start with the most likely first.

2. Give them a "mantra" to state. Something they can repeat to themselves to help them focus. The one I use with students is "I would know what to do if THIS part was x." This helps them choose the chain rule AND focus on which part needs to be differentiated when. I should point out what I do for the rest of the procedure: I say "I'll pretend it IS x to start with. But it's NOT x, so I have to multiply by the derivative." I cover up the part that is pretending to be x until I differentiate it, and I also use the "arrows" approach mentioned by Dave L Renfro to help them focus.

• I like the comment about explaining what sings out to you to use a method. Seems very collegial and in the trenches together. How you have to overcome problems. I also like the pretend it is x. It's simple intuitive language to connect with students. – guest Mar 10 '18 at 19:38

This seems like an issue about learning objectives. What is it exactly you are wanting students to show you? It sounds like what you want is for students to show that they can approach an exercise or problem from different perspectives and using multiple methods. Is that right? For the sake of argument I'll assume it is.

With that assumption I'd recommend two things.

1. Make this expectation an explicit goal of the course in your syllabus, and use it as an organizing principle for your students' work. If a stated goal of the course is to be able to solve calculus-related (or whatever-related) problems using a variety of methods then you are allowed to restrict and direct their work to specific methods. For example if you state in the syllabus that a successful student will be able to solve calculus problems using graphical, numerical, and symbolic methods -- and not just one of these -- then you can give problems that insist on a particular method, and it won't be arbitrary but rather a way of assessing how well they have mastered that particular goal.

2. This goes with the previous point, but you can give problems that specifically involve multiple methods and then make students do the same problem more than once, using different methods. Problems that emerge from data sets are perfect for this in calculus. For example:

Suppose the temperature of a cup of coffee ($T$, in degrees celsius) has the following temperatures $t$ minutes after it's been brewed:

Time 0   5   10   15   20
Temp 98  63  40   29   21
`

Part 1: Estimate the instantaneous rate at which the temperature is changing at exactly 15 minutes by using the data in the table to make an estimation. [Ideal approach is a central difference estimate.]

Part 2: Estimate this instantaneous rate again, by plotting the data in scatterplot, fitting the function with a curve, and using the resulting graph. [Ideal approach would be to draw the tangent line to the curve they got.]

Part 3: Estimate this instantaneous rate again, by using technology to get a formula for the curve of best fit and using derivative rules. [You could specify that they must use an exponential model, for instance, if you want something other than a polynomial.]

Part 4: Compare the three methods for estimating the instantaneous rate of change you used. Make sure that the results are all fairly similar. What are the pros and cons of each method? Which method seems to give the most accurate result? How can error or uncertainty propagate in each of these methods?

I bolded step 4 above because this is IMO where a university-level approach to calculus needs to be reaching -- not only making computations but thinking about the relative merits and demerits of computational methods, and understanding the limitations of one's tools. By having students do problems that force them to do the same thing in different ways, you give students an opportunity to think critically about what they're doing.

In some cases students have severe limitations on their short-cut methods and aren't even aware of them. For example if student are trying to find the area under a normal distribution curve and have the formula, they literally cannot answer that kind of question by hand calculations. So many students who are whizzes at taking derivatives and antiderivatives don't really get the fact that symbolic techniques are not always the best approach or even a possible approach and so they need to have a full repertoire of methods and the intellectual ability to compare them and select the best one.

• This is great advice (and something I should be doing more of), but to answer your question, all I actually want students to do in this case is learn the new techniques in place of, or at least alongside, whatever they learned in high school. If the best way to do that is make it an opportunity to compare methods, that may just be an opportunity to do two good things at once, but in many of the specific cases I'm thinking about, the short-cuts aren't really a coherent method, they've just (essentially) memorized the solutions to the most common problems. – Henry Towsner Jul 24 '15 at 18:29

Wolfram Alpha Pro will take them through any derivative step-by-step for \$5 per month. See here for your example of$\ln \sin x^2$, with the light blue rectangle saying "step-by-step solution". • How does this help teach students who are already using shortcuts? – David G Mar 17 '14 at 19:23 • @Skytso: It means they can take any example, try it themselves, and check their work. (I'm surprised by the downvotes. If I suggested that they get a tutor for \$50/hr rather than \$5/month, would that be better?) – user173 Mar 17 '14 at 19:27 • This is not that unreasonable. But I worry it opens a slippery slope for the students. What's to stop them from just using this service for all of their homework? For a truly conscientious who just wants to check their work while they learn, this could be fine. But I don't think this is a good, general solution. – Brendan W. Sullivan Mar 17 '14 at 19:34 • @brendansullivan07: What stops them from using WolframAlpha is the same thing that stops them from hiring someone to do their homework, or copying their friend's homework. Or perhaps it doesn't stop them from either. Perhaps the next question is: Given that students are using this, and will have access to it even after the class ends, what changes should we make to calculus instruction? – user173 Mar 17 '14 at 21:18 • My students use wolphram alpha freely (well, they dont pay), and there is no problem. They use it to check their work, they still try to understand. (maybe it helps they know they cannot use WA at the exam). – kjetil b halvorsen Mar 19 '14 at 18:57 I think the problem is partly because we often use the established notation like$\frac{dy}{dx}$but throw away the intuition, and hence differentiation becomes a meaningless task rather than a motivated procedure. It starts with considering any two variables$x,y$that depend on some parameter$t$(usually position or time in the real world). Then$(x,y)$traces a path through the state space as$t$changes, and we are interested in the points where the path is smooth (not the mathematical sense), namely where the path is unbroken and has a direction. At any point, for any change in$t$, denoted$Δt$, there will be the corresponding changes in$x,y$, denoted$Δx,Δy$. The path being smooth at a point is then the same as saying that, at that point,$Δx,Δy \approx 0$as$Δt \to 0$(Of course this can be made precise, but in fact we can manipulate it axiomatically without using the usual epsilon-delta definitions!), and also the ratio$Δx:Δy$tends to some ratio as$Δt \to 0$, where the ratio depends only on$t$and not on$Δt$. In the specific case of$\frac{dy}{dx}$, we must have$Δx \ne 0$and we define$\frac{dy}{dx} = r$where$\frac{Δy}{Δx} \approx r$as$Δt \to 0$, if such$r$exists. (Note that we do not need non-standard reals but just ordinary asymptotic analysis.) With the above underlying framework, it is trivial to prove the basic facts of differentiation and much more. In particular, the chain rule says that given any variables$x,y,z$that depend on a parameter$t$, whenever$\frac{dy}{dx},\frac{dz}{dy}$are defined, then$\frac{dz}{dx} = \frac{dz}{dy} \frac{dy}{dx}$. Note that the above definition of$\frac{dz}{dy}$does not allow the case where$y$is constant, but this case never occurs in practice since we then do not need the chain rule. For example given variables$x,y$such that$y = (x^2+3)^4$, if we let$z = x^2+3$, then$z$is a variable and we can immediately find$\frac{dz}{dx} = 2x$and$\frac{dy}{dz} = 4z^3$, and then conclude that$\frac{dy}{dx} = (4z^3)(2x)$. Even better still, without creating$z$one can write$\frac{dy}{dx} = \frac{d((x^2+3)^4)}{d(x^2+3)} \frac{d(x^2+3)}{dx} = (4(x^2+3)^3)(2x)$. In both cases, it is simply a matter of a trivial instantiation of the chain rule, and the second direct method also reminds the chain rule user of the reasoning behind the rule, since the$d$s were indeed originally motivated by the$Δ$s, which cancel because they are actually nonzero. At an undergraduate level one might choose to define everything in terms of the modern epsilon-delta definitions, but in my opinion that is usually not done well. In most cases that I've personally observed, students are unable to justify the epsilon-delta definition, and many of them even produce haphazard guesswork when asked to prove a limit or derivative via the epsilon-delta definition. I would say that the latter is due to not teaching logical reasoning (basic first-order logic) before other subjects, but the former is due to the lack of motivation of defined concepts. For example, why on earth should we consider Cauchy sequences or the Bolzano Weierstrass theorem? Most students are just taught them as if they were unexplainable facts like the fact that our planet has only one moon. Finally, I do not believe in just teaching students to follow some procedure to get the answer, otherwise we are just teaching them to become a computer, and computers have already taken that kind of job. If they do not understand why they should do something (not just what they are doing), I consider that a failure in mathematics education. • Clearly people don't like hard truths about mathematics education, since the framework I propose is both rigorous and intuitive (and in fact similar notation is used in some applied mathematics textbooks albeit without the rigour). My post also answers the question, because any other approach that does not address the conceptual understanding completely will succumb to the same issue of students memorizing what to do for what problem, since their teachers are not teaching them the important reasons behind the mathematical objects and techniques used. – user21820 Jul 25 '15 at 8:20 It seems as though the real problems, your students have, are not the chain rule but analysing algebraic expressions and doing derivatives based on this analysis on the one hand and coping with unknown objects (like$g(x)$) on the the other hand. For the first problem, your consequence shouldn't be to go back to simple derivatives, but to algebraic expressions. Let them analyse the expressions to be derived. You can support this by using colors either for the different layers or for the different operations/derivation rule prototypes. Another support are Klammergebirge (parentheses mountain ranges). You can do this with LEGO or magnets on a black board. Then, let them give you the derivatives of every single layer first, and mark them in the respective color/at the respective level and finally put them back together. Yes, this will look silly to your students, but you can show them, that what they need to learn, is something very basic, very elementary, very abstract. For the second problem, give them exotic functions (maybe like$\arcoth$), they don't know the derivative of, and use them in easy examples (like$f(x)=\arcoth(x^2+1)$. Then give them problems like: Let$f'(x)=\sin(x)/x$. Find$(f(x)f'(x))'$. Force them, to treat$f(x)\$ as a mathematical object and not as a task/action/process. (See the other answers' reference to APOS.)

• @BenjaminDickman Thanks for pointing out the typo. For the rest: Firstly, I have never said, that APOS is a sufficient framework, and secondly, some researcher having difficulties with it in thinking about the chain rule is no scientific criteria for other people whatsoever. – Toscho Aug 21 '14 at 19:20