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Over the past years teaching freshmen calculus I've repeatedly seen students make the following type of error:

Suppose they have to express some quantity $y$ as function of $x$, when the relation between $y$ and $x$ has been given by other means (like a graph, a table of values, verbally etc.). Suppose a correct answer is $y = x^2+1$ (I wouldn't mind if they add an $=f(x)$ in between.) A significant amount of students (I estimate 5%) will write $$ y=f(x^2+1) % \quad \text{ or simply } \quad f(x^2+1) $$ (The letter $f$ not having been used in the problem statement previously.)

These are engineering students in Switzerland and they do this in the first week of the first semester, but also towards the end of the first semester after I've addressed the issue in class. From school they should in principle be familiar with a correct use of function notation $f(x)$.

I don't understand where this comes from and how to fix it.

Questions:

  1. Is this error well-know? Has it been studied?
  2. Are there explanations for it and suggested ways of addressing it?
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    $\begingroup$ I also see this type of behavior, and I believe it stems from them trying to use function notation, but not really understanding its purpose. Yours reminds me of this question. $\endgroup$
    – Nick C
    Feb 24, 2020 at 15:58
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    $\begingroup$ I suppose it's possibly similar to when students write unnecessarily flowery prose in their attempt to be academic. They don't understand it, and they're trying to imitate their instructors. Unfortunately, because they don't understand it, they use it wrong. Just because they are "in principle" familiar with f(x), that does not mean they know what it means. $\endgroup$
    – Opal E
    Feb 24, 2020 at 18:44
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    $\begingroup$ Take points off for it. It's the only thing they'll respond to (and not necessarily even that). $\endgroup$
    – Daniel R. Collins
    Feb 25, 2020 at 2:19
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    $\begingroup$ Re: "It has happened that I spend 30 min in class discussing this mistake..." avoid that temptation at all costs! If only 5% of the students are doing that, it's not worth class time from everyone doing an autopsy. Offload that to an office hour. It's possible that (a) such students are simply too weak to ever correct it, and (b) often I find those students aren't even in the classroom at that point anyway. $\endgroup$
    – Daniel R. Collins
    Feb 25, 2020 at 2:21
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    $\begingroup$ Not comprehensive enough for an answer, but can I suggest getting them to read the notation aloud? I suspect they are saying “y is a function of x^2+1”. And in that case they are meaning the word “of” to mean in the sense “made of”. And that actually makes sense: they mean y is a function made of x^2+1. Finding the root idea can help you to combat it. $\endgroup$
    – DavidButlerUofA
    Feb 25, 2020 at 20:12

3 Answers 3

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For me, the standout problem here is this: relations are verbs. Without a relation/verb, you don't have a statement, you have the equivalent of a sentence fragment. What the student has written in this case isn't even wrong, it's just malformed nonsense.

I try mightily to get students to at least see, at first pass, that a piece of writing without any verb/relation is clearly malformed. For example, here's a quick quiz on reading chained relations where that's really the point I'm trying to get across (expect to see relations for meaning to be expressed, from which conclusions can be made).

However, in the last few days I was informed: (a) by a middle-school teacher that lessons on grammar are prohibited, (b) by a college speech instructor that they never correct grammar or syntax, (c) by a national educational advisor that no students can read textbooks anymore. So if that's true my desire to leverage grammar knowledge of natural language may not help.

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  • $\begingroup$ Sounds interesting, but I don't know how to translate what you wrote to this particular problem. Could you spell out in more detail how the missing link between verbs and relations might cause them to write $y=f(x^2+1)$? $\endgroup$
    – Michael Bächtold
    Mar 2, 2020 at 15:17
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    $\begingroup$ Yes! The parentheses in $f(x) = x^2 + 1$ are a preposition. This question is a grammar question, and the answer should be a grammar answer. $\endgroup$
    – Chris Cunningham
    Mar 2, 2020 at 15:23
  • $\begingroup$ @MichaelBächtold: Honestly, I was most directly responding to the question title, where the two options given were $f(x^2 + 1)$ and $f(x) = x^2 + 1$. $\endgroup$
    – Daniel R. Collins
    Mar 2, 2020 at 16:00
  • $\begingroup$ 30 seconds on this quiz? I spent more just on the first one, maybe I am just slow this morning. $\endgroup$
    – Rusty Core
    Mar 2, 2020 at 18:11
  • $\begingroup$ @RustyCore: Thanks for the feedback. Other testers have suggested it might have too much time on the clock. Personally I do the quiz in about 5 seconds. $\endgroup$
    – Daniel R. Collins
    Mar 2, 2020 at 23:03
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If I continually saw this mistake then I'd try to get the idea of $y=f(x)=x^2+1$ more solidified: "$y$ is a function of $x$ and that function can be represented by an equation." I'd probably try to incorporate functions and transformations into the first week of homework so that students know the difference between $f(2x)$ and $2f(x)$ and $f(x)=2x$.

One thing that I might try is briefly mention the idea of multivariable functions to clarify that $f$ takes certain variables, say, $x, y$ and this means it is $f(x,y)$. It not that $f$ takes the equation involving $x$ and $y$ itself.

I wonder if issue also occurs when you have polar functions? Would they consider $r=r(\theta)=2cos(\theta)$ to be something like $r(2cos(\theta))$? I'd also wonder what they'd think parametric equations would be like. How would they consider $x=x(t)=cos(t), y=y(t)=sin(t)$?

The reason I'd bring up polar and parametric equations is to see if they understand function notation for that. If they do, then maybe that knowledge could be applied to understanding $y=f(x)$?

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Suggest teaching students about composition of functions. Show them that if $f(x)=x^2+1$, then $f(2x)= (2x)^2 +1$ and $f(x^2 + 1)$ = $(x^2+1)^2+1$ If they do enough problems with composition of functions, they will be less likely to make this mistake again. This should be more effective than explanation about what the notation means.

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