Multiple students writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ -- why?

Question

I'm currently teaching a couple of courses that have a calculus prerequisite, and within the last week I've had two students make notational mistakes that amount to writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ (although in terms of different variables than $y$ and $x$). E.g., they might write something like

$$ x^2 \frac{d}{dx}=2x.$$

At least one of them actually seemed relatively fluent with calculus, in the sense of knowing facts like $\int x^n dx=x^{n+1}/(n+1)$. I pointed out his mistake and made the analogy with the ungrammatical expression $4\sqrt{}$. I mentioned that I had just had another student make a similar mistake recently, and I asked him if he could explain more about what he was thinking when he wrote it. His response was that that was just the way he had gotten used to seeing it in his calculus book! (I assume this is not actually true.)

Can anyone provide any insight into why this would be a common enough mistake that I would see it this frequently? I wonder if there is some confusion because the $f'$ notation involves a postfix operator, or because in an integral we usually write the $dx$ at the end...?

It would seem obvious to me that their notation wouldn't make sense if you think of the derivative as the ratio of two infinitesimals, but presumably many of them haven't been exposed to that way of thinking. Part of their confusion may also be because they are used to everything being $y(x)$, never $\Psi(x)$ or $v(t)$. They also never seem to have been asked to think about the meaning of notation and why it makes sense, and don't seem to understand attempts to elicit discussion of this kind of thing. E.g., if I ask them whether they've been taught that $d/dx$ is an operator, like $\sqrt{}$ or $\sin{}$, they generally look at me blankly, as though this is not the kind of thing that was ever discussed in their calculus class.

Answer 1

When a student writes incorrect notation, ask them to read it out loud. I would say something like:

Something here doesn't look right, but we can fix it. Could you read this work out loud? I think probably you are not super familiar with this topic, and that's okay, but this can help us fix it.

I had success with this when dealing with a student who wrote "$\sqrt 4 = \sqrt 2$". After some prodding and reassurance, they read it to me: the student said "The question is the square root of 4. The answer is the square root is 2." The key here was that the student didn't understand what the "$=$" meant (which is a whole topic of discussion on its own) nor did the student really understand what the "$\sqrt{}$" meant (see also this question which is the source of my understanding).

But in this case you may be surprised by something similar. Maybe the student will say that "$x^2 \frac{d}{dx} = 2x$" says "$x^2$ prime is $2x$"? Or slightly better "$x^2$'s derivative is $2x$". If the student reads "d d x," don't accept that: get them to translate what it means to them inside the sentence. Maybe they think "$\frac{d}{dx}=$" is a group of symbols that separates a question and an answer, like my student above.

No matter what they say, the correction is relatively easy: you can explain that $\frac{d}{dx}$ actually precisely means "The derivative of." Then you can read their work back to them for effect to help them see why it looks strange: "$x^2$ the derivative of equals $2x$." It's important that you emphasize this is not your opinion, but that these symbols have an accepted precise meaning.

I hope this helps.

Answer 2

I see the same mistake in my Calculus students' work a lot. And from my observations, I think there's a very simple reason for that — for typical students (here in the U.S.) this is multiplication, and in multiplication order doesn't matter. In fact, for typical students (…) almost everything is multiplication, including function notation; and they simply extend their writing habits to this new symbol without giving it much thought. And I'm going to demonstrate below that I mentioned function notation for a reason.

Disclaimer: described below are my personal observations and opinions; I can't provide any supporting literature. But then, I have years' worth of these observations, which are consistent across hundreds of students I've taught and tutored.

Imagine a middle school student needs to solve an equation like $$\frac{x}{5}=3.$$ Of course, the approach is to multiply both sides by $5$. And the typical way students are going to show their work on paper is by writing the fact that they are multiplying by $5$ by adding some markings in the same line, and then the result in the next line: $$\begin{align} \color{blue}{5\times}\frac{x}{5}&=3\color{blue}{\times5} \\ x&=15 \end{align}$$

As part of the reason for students developing this habit, I also blame the tradition (habit? expectation?) of showing what they are going to do in the same line. There's no room between the equals sign and the $3$ to fit "$\color{blue}{5\times}$" in there, so writing this multiplication from the left on the left and from the right on the right is basically the only way to fit it. And at this point it's fine. Even though pedagogically it would be much more preferable to indicate multiplication from the same side on both sides of the equation, in the spirit of doing the same thing to both sides of an equation, arithmetically this works and leads to the correct answer. (Of course, then we really have to deal with this problem in our abstract algebra classes when solving or manipulating equations in non-abelian groups.)

But then we move on to more advanced classes and encounter equations such as $$e^x=10.$$ Don't forget that for most U.S. students (deep sigh…) $f(x)$ means "$f$ times $x$", $\sin(x)$ means "sine times $x$", etc. Yeah, function notation is another sore topic… So, by following the same recipe — and because there's no room between the equals sign and the $10$ anyway! — students "multiply" both sides by $\ln$: $$\begin{align} \color{blue}{\ln}e^x&=10\color{blue}{\ln} \\ x&=\ln10 \end{align}$$

And then this pattern continues. One of the first instances when students have to write $\dfrac{d}{dx}$ themselves (as opposed to a given problem with this notation) is in the topic of implicit differentiation. It's the same situation again: given an equation, perform some steps to get an answer, and so that's what they do. For example, starting with something like $$x^2+y^2=10$$ they "multiply" both sides by $\dfrac{d}{dx}$: $$\begin{align} \color{blue}{\frac{d}{dx}}(x^2+y^2)&=10\color{blue}{\frac{d}{dx}} \\ 2x+\cdots&=\cdots \end{align}$$ (where lots of different things may appear in place of those dots), and then continue to the subsequent steps.

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Of course, as others have already said, a deeper reason is not understanding the differential operator $\dfrac{d}{dx}$ as an operator, not treating mathematical notation properly in general, etc. But I wanted to address the specific question of where this particular mis-notation comes from, in my opinion.

Answer 3

I didn't like f(x) — that looked to me like f times x. I also didn't like $\frac{dy}{dx}$ — you have a tendency to cancel the d's. So I made a different sign, something like an & sign. For logarithms it was a big L extended to the right, with the thing you take the log of inside, and so on.

I thought my symbols were just as good, if not better, than the regular symbols — it doesn't make any difference what symbols you use — but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, "What the hell are those?" I realized then that if I'm going to talk to anybody else, I'll have to use the standard symbols, so I eventually gave up my own symbols.

~ Richard Feynman

So, explain to your students that this is an accepted notation for differentiation, nothing more. Explain, that it should be taken as a single operator, not a fraction; that d's cannot be canceled. Show them other notations and allow using them even if your course and the textbooks stick to Leibniz's notation.

I think that Lagrange's notation is much cleaner for single-variable derivative than Leibniz's:

$$ (x^2)'=2x$$

Answer 4

A good way to start is setting clear that

$$ \frac{d}{dx}$$

as a block, is an operator, and

Operators are things that act on their right side.

So $\frac{d}{dx} y$ means "derivative of what is at the right", i.e. derivative of $y$.

I suggest you to highlight that sentence in yellow.

So $y \ \frac{d}{dx}$ just makes no sense because it is "$y$ times derivative of... nothing", as there's nothing at the right side.

I also like starting with $\frac{d}{dx}$ and not $\frac{dy}{dx}$, so that it is clearer that it is not a fraction, but a whole-block operator.

And finally, I'd suggest to read it as "derivative of", so that the students are tempted to complete the sentence in their minds. It's more serious than it seems haha.

Hope this helped.