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In the textbook I am using to teach mathematics to high school students I found the following illustration about composition of functions.

enter image description here

I do not agree with this illustration. For me $g$ is the slicer, $g(x)$ is the sliced potato, $f$ is the fryer and $f(g(x))$ is the bowl of french fries. I would introduce the notation (not on the diagram though) of $f\circ g$ for the process of slicing and then frying.

Am I being overly pedantic?

Ferenc Beleznay
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    No, you're not overly pedantic. Errors, like the labeling in that diagram, are especially dangerous when introducing abstract concepts, like function and composition, to students who have little or no previous experience with abstraction. – Andreas Blass Nov 15 '20 at 18:32
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    Whether you’re being pedantic is a broad question. An easier question to answer would be: “Can you supply non-contrived examples for which ignoring this distinction would lead to serious (i.e., hard to correct) errors?” – Steve Nov 15 '20 at 19:25
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    To play devil's advocate a little (because I broadly agree), the conflation of $f(x)$ and $f$ is fairly common in some uses: for example, we define the derivative of a function, not the derivative of an expression, but we would certainly write $\frac{d}{dx} x^2 = 2x$, and not insist that one says "Let $f : \mathbb{R} \to \mathbb{R}$ be given by $x \mapsto x^2$, then $f' : \mathbb{R} \to \mathbb{R}$ is the function given by $x \mapsto 2x$." – Reed Oei Nov 15 '20 at 20:49
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    I agree that you're not being overly pedantic but I also agree with Reed Oei that it's not a huge deal to sometimes conflate f and f(x). The illustration in that book is especially frustrating though because it's perfectly set up to make that very distinction in an easy-to-understand way. It just feels wrong for those potato slices to be sitting there with no label. If I were you I would make sure to at least mention the distinction between f and f(x) in class, but I wouldn't push it too much. – Thierry Nov 15 '20 at 22:19
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    Not sure if this was the asked-about issue, but the $f(x)$ on the fryer is bugging me. If $x$ was a whole potato, there should be a whole potato in the fryer. – Adam Nov 15 '20 at 23:12
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    The makers of the textbook should have spent more time thinking about the math of the diagram and less about how adding a real world example makes it so great... – James S. Cook Nov 16 '20 at 02:01
  • @Steve Confusing f for f(x) leads to Corinne's Shibolleth. https://mathoverflow.net/a/307990/745 But the establishment likes to argue that it is not a serious error. – Michael Bächtold Nov 17 '20 at 10:38
  • @ReedOei writing $\frac{d}{dx}x^2=2x$ is not a conflation of $f(x)$ and $f$. But I agree that it is not well defined according to modern standards. See: https://mathoverflow.net/q/115416/745 – Michael Bächtold Nov 17 '20 at 10:42
  • @MichaelBächtold Thanks for the example. Two ?s: 1. Would you consider the scientist’s answer to avoid this error if it were instead notated T(r(x,y)) instead of T(r) where r is the polar radius defined in the usual way with respect to Cartesian x, y? If so, does this still constitute a “serious” (hard-to-correct) mistake or a combination of sloppy notation and loaded terms? My guess is that the scientist and mathematician viewpoints would coincide if both were asked: “What is T(p,q)?” 2. Given the length and complexity of that other thread, how would you message this to a precalc student? – Steve Nov 18 '20 at 13:26
  • @Steve writing I think writing T(r(x,y)) is still giving a different meaning to the notation f(x), which does not coincide with the official meaning taught to students. (At least I don't know what it means) So I would either avoid doing that, or I would spend time explaining to students this different meaning. Concerning message to precalc students, good question. I don't know. I would not teach this alternative use at all, but thats ignoring the reality: they will at some later point encounter it, since it is so widespread. – Michael Bächtold Nov 18 '20 at 15:14

2 Answers2

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You are not being pedantic. The name of the process that slices is $g$, and the result of slicing $x$ is $g(x)$.

On the other hand, the textbook presentation seems to be for students who are just encountering function notation. In the language of the Dubinsky school of constructivism, students at this stage are not ready to distinguish between $g$ and $g(x)$. It will take some time and "action" before functions are encapsulated by the students as primary mathematical objects:

Action->Process->Object->Schema

So although you appreciate the difference, it might be too much to expect students at this stage to do the same.

But still, if the seeds are not planted, the development might never get started.

user52817
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    Is there really evidence, that beginning students are not able to understand the difference? I doubt someone would have problems distinguishing between sliced potatoes and the slicer. – Michael Bächtold Nov 16 '20 at 02:29
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    @MichaelBächtold-yes I think there is a strong body of evidence. Start with

    Breidenbach, Daniel, et al. “Development of the Process Conception of Function.” Educational Studies in Mathematics, vol. 23, no. 3, 1992, pp. 247–285. JSTOR, www.jstor.org/stable/3482775.

    Some references in this paper provide further evidence.

     – user52817 Nov 16 '20 at 02:51
  • That article is about college or higher level students. Is that the level where students in the US first encounter the function terminology and f(x) notation? – Michael Bächtold Nov 16 '20 at 04:34
  • @MichaelBächtold: in the US, students typically first encounter function notation in a course titled "College Algebra" or "Precalculus." The sequel to such a course is a course in differential calculus. Depending on the student, the precalculus course might be in the third or fourth year of high school (grade 11 or 12), or the first year of college. The OP says the illustration is from a high school textbook. One theme of the paper is that for calculus students, functions are often not yet encapsulated as first class objects, hence there is cognitive conflict with "differentiating a function." – user52817 Nov 16 '20 at 04:41
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    @MichaelBächtold: The illustration is from a textbook written for the International Baccalaureate Diploma Programme, so aimed for students aged like US grade 11-12 students. I looked up the US curriculum, the function notation is specifically mentioned in the High School ALgebra 1 part of the Common Core State Standard (http://www.corestandards.org/Math/Content/HSF/IF/A/1/). – Ferenc Beleznay Nov 16 '20 at 06:25
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I would argue that there is an issue with the diagram as labeled regardless of whether you are or are not being pedantic. For example:

  • Consider the case where $f$ and $f(x)$ can be used interchangeably, as is done in many US high school courses. In this situation, the labeling on the slicer and fryer are fine, because they denote the functions in question. Since $x$ is being used as an abstract placeholder, though, it should not be used as the label for the potato; the potato is a specific instance of a value of $x$ ($c$ is common in high school calculus courses, but $p$ might be more appropriate for potato). In this scenario, the labels would be (in order): $p$, $g(x)$, $g(p)$, $f(x)$, $f(g(p))$. You could then spend time discussing that $f(g(x))$ and $f\circ g$ are a process that was used to make the fries, rather than the fries themselves.
  • Consider the case where $f$ and $f(x)$ have separate distinct meanings, as some users have suggested. In this case, we are using $x$ as an input, $f$ as a function, and $f(x)$ as the output to that function. For this scenario, the potato can happily be $x$, and our labels in order would be $x$, $g$, $g(x)$, $f$, $f(g(x))$.

In either case, it seems reasonable to say that regardless of one's views on the finer points of function notation, the diagram uses inconsistent "mixed metaphor" labeling.

Vaekor
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