I sometimes teach calculus in English whereas it's not my native language.

For example, during a course about antiderivatives, how do you (orally) pronounce $f$ vs $F$?

Which are the best?

  1. "the function small $f$" vs. "the function big $F$" ?
  2. "the function lowercase $f$" (or just "the function $f$") vs. "the function uppercase $F$" ?
  3. Is this example correct: "The function capital $F$ is an antiderivative of the function $f$." ?

When I need to distinguish $f$ and $F$, I choose - "small eff" vs. "big eff" / "capital eff". I find "uppercase/lowercase eff" to be a bit awkward; it doesn't run off the tongue as easily as the other options.

However, I might not distinguish them orally every single time. The flow of the lecture would determine when I would choose to emphasise the difference.

  1. If $f$ and $F$ occur in separate sections of the discussion, then I usually do not emphasise the difference. If needed, then I distinguish them once or twice when $F$ is introduced and $f$ is left behind (or vice-versa), and leave it at that.

  2. If $f$ and $F$ occur in the same section of the discussion, then I make sure to distinguish them orally whenever they occur together in a result or an equation. If $f$ has been playing the predominant role in the discussion, then I continue to call it "eff" and I call $F$ as "big eff" or "capital eff". Conversely, if $F$ has been predominant, then I call it "eff" and I call $f$ as "small eff".

It is also probably better not to overdo it, but where the line should be drawn depends on the needs of your students. For example, I might say something like

Okay, now take the function "small eff". We want to know where "eff" is differentiable.

If the students are keeping up with the discussion, then it should be immediately clear to them that "eff" means $f$ in this case. If some students are lagging behind (still figuring out what was written on board 1 whereas the discussion is on board 3), then they might get confused by such a sentence since they are juggling between two inputs (the board and my words).

Also, if the choice of notation is good and consistent, then after a certain point of time one can drop the distinction between $f$ and $F$ when speaking. As a related example, sometimes we need to talk about a family of functions, and we might use the notation $\mathscr{F}$ (which I speak as "script F" / "curly F"). Suppose we often have to choose a function from the collection $\mathscr{F}$, that is, we often write $f \in \mathscr{F}$ or some such variant. Then, after the first one or two instances of using "script eff", we can say

Let "eff" belong to "eff", or Let "eff" be an element of "eff", or Let "eff" be a function in "eff"

without any ambiguity, because the "grammar" is clear: none of the following make sense: $$ f \in f, \qquad \mathscr{F} \in f, \qquad \mathscr{F} \in \mathscr{F}. $$ So, it is clear what we really mean, even without distinguishing orally. If your choice of notation is good and consistent, then your students will gradually become aware of this grammar, and you will be able to drop the oral distinctions after a point of time. At what point of time—that will depend on your students and how quickly they become comfortable with the material.


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