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As mathematics educators, we all have come across students using mathematical notation incorrectly (looking at you, $\frac{d}{dx}$ vs $\frac{dy}{dx}$ or $\frac{\infty^2}{\infty}$). My question focuses on "verbal notation." For example, my hackles go up when I hear the following:

  • "take the prime of $f$" or "$d$-$dx$ the function" or "derive the function" instead of "compute the derivative of $f$" or "find $f'(x)$" (edit: or "differentiate the function"). Double chalkboard-fingernails for "the prime of the prime" and it's ilk.
  • "anti-derivative the function" instead of "integrate the function" or (even better) "find the indefinite integral of the function"
  • "minus/minusing $a$ from $b$" instead of "subtract $a$ from $b$" or "compute $b$ minus $a$"
  • "plus/plussing $a$ and $b$" or instead of "add $a$ and $b$" or "find $a$ plus $b$"
  • "take the inverse of a fraction" instead of "take the reciprocal of a fraction" (debatable, the "multiplicative inverse of a fraction" does appear in sources)

The list goes on from there--I would be curious to hear your pet peeve phrases! My question is this:

Is it overly picky and pedagogical to correct such phrases? If it is appropriate to correct these phrasings, is it situation dependent (tutoring/recitation/lecture) and how would you do so?

I would like to emphasize that this is a question specifically about phrasing and verbalizing mathematical operations. Assume that the hypothetical student is generally performing the correct operations, is capable of reasonably proper written notation, and "plussing $a$ and $b$" would be the correct step.

Edit: Running list of other phrasing

  • "vertexes" vs. "vertices", probably applying to pluralizations of many other words as well (axises vs axes, ...). Credit to kcrisman
  • Opposite of the above, "vertices" or "vertice" to refer to a single object (c.f. $x$-axes etc). Credit to Andreas Blass
  • Misuses of mathematical verbs such as "Solve 16 + 58" or "Prove the integral." Credit to Jack M
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    $\begingroup$ I don't see how using the term "Anti-Derivative" is such a peeve. It appears in textbooks often enough. $\endgroup$
    – Weckar E.
    Mar 16, 2017 at 14:02
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    $\begingroup$ @WeckarE. It is a case of verbing the noun. The action is usually called anti-differentiation, which results in the anti-derivative. $\endgroup$
    – Adam
    Mar 16, 2017 at 14:57
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    $\begingroup$ "Plussing" and "minussing" physically hurt me ears when I hear them. $\endgroup$
    – Feathercrown
    Mar 16, 2017 at 20:45
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    $\begingroup$ I'm personally much more put off by misuses of mathematical verbs such as "Solve 16 + 58" or "Prove the integral", which show actual conceptual confusion rather than just being ignorance of standard terminology. $\endgroup$
    – Jack M
    Mar 16, 2017 at 23:08
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    $\begingroup$ It would be interesting to see whether students who say things like "taking the prime of $f$" and "d-dx the function" are less likely to know that $f'(x)$ and $\frac{d}{dx} f(x)$ mean the same thing. $\endgroup$
    – Daniel Hast
    Mar 17, 2017 at 14:31

5 Answers 5

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Personally, I don't think we attend to this sufficiently in lower-level mathematics (where it's actually needed most). Students need that vocabulary to interface with books, future teachers, tutors, other students, etc. I run questions on it in weekly quizzes; and if I had my druthers, it would be a major component of all tests (in addition to application-level stuff).

In my experience, you've got to jump on that stuff as directly, firmly, and as soon as possible to make a difference. Really lead by example that it's a priority for you that students know how to interface with that language for their next step. I never let it go by if it comes up in class; I always address the class with, "Can anyone help me? What's the correct word for this?". At least by the level of college algebra and above my students definitely respond positively to this, and it gets better rapidly.

Some of my non-native English speakers express outright fear the first day when it becomes clear that this is the emphasis, but I do try to reassure them that in some sense we're all in the same boat, and prior students in that situation have rapidly improved and done extremely well. They're usually thankful for that emphasis by the end of the semester.

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    $\begingroup$ When did "interface" become a verb? ;-) But I love the point on knowing what the vocab is for future use. Sort of like how if you don't know both prime and Leibniz notation, you are in deep trouble later on. $\endgroup$
    – kcrisman
    Mar 16, 2017 at 5:04
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    $\begingroup$ @kcrisman: Interface is listed as a verb in every dictionary I can find (e.g., merriam-webster.com/dictionary/interface). I don't have the OED to look up full history -- but at least since my Webster's New World, printed 1988. $\endgroup$
    – Daniel R. Collins
    Mar 16, 2017 at 10:14
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    $\begingroup$ @kcrisman Around 1940. At least according to google's ngrams: books.google.com/ngrams/… $\endgroup$
    – Kevin
    Mar 16, 2017 at 13:06
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    $\begingroup$ @kcrisman Funny thing that it was a verb before it was a noun (in the modern context that doesn't solely revolve around aviation). $\endgroup$
    – Weckar E.
    Mar 16, 2017 at 15:01
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    $\begingroup$ I found it much worse that students didn't understand what $=$ meant than that they would "d-dx it" (which by the way is mathematically absolutely fine as it's an operator even if verbally I suppose it's strange since you don't "f 5" if you want to find $f(5)%$). Unfortunately I found that the semester I would have with the students wasn't long enough to break the 8+ years of teachers using $=$ to mean "compute" and not punishing $3x+4=5=1/3$ and $f(x+3)=\sin(x+3)=\sin^2(x+3)$ when computing $h(f(x+3))$ $\endgroup$
    – DRF
    Mar 21, 2017 at 11:05
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I wish to give a slightly different answer compared to the others.

Strict and Standardized Notations is Very Important

They not only help us communicate better, they also help us think. They prime us to remember things and understand things better. For example, if I see $a^2 + b^2 = c^2$, I think Pythagoras Theorem and right angle triangles. If I see $k^2 + y^2 = t^2$, I don't.

It also allows you to be more accurate, and make sure your logic is not flawed.

Non-standard Notation Could also be helpful

Though rare, not using standard notations could help with thinking about a problem in another way, or coming up with a different sub-field of math.

There are Different Standards/Language is Evolving

Like any other language, mathematical language is evolving. If enough people uses a phrase, it is a correct phrase. Different mathematical papers uses different standards.

Commonly used Standard Notation could be suboptimal

The first notation for a subfield is usually made by the guy who ventures into this subfield. Being the first, he is exploring in unfamiliar territory, and his standards ends up suboptimal. Then more people come in, and each tries to invent a better standard, or a more universal notation, and it ends up like xkcd 927.

Furthermore, it should be noted that different notations are more useful in at different times.

Verbal Notations are often much more flexible than written ones

People often don't speak in completely correct sentences. Things are shortened. Words are changed.

Verbal Math is often an attempt to translate a formula to English?

How would you say $(2a + b) \times c$? There is no guide to speaking formula. Do you say "The product of c and the sum of two-a and b"? That clearly got the multiplicands in reverse, and what is a two-a? Or do you say "Open bracket, two times a plus b, close bracket, times c.

As another example, take "three x plus four b over seven all over nine". What does that mean?

So I'd say saying "d d x" or "d over d x" is perfectly fine. And if we can say that $f'$ is f prime; why can't be say that it is the prime of $f$, or that the action of differentiating is taking the prime of f?

Not Everyone Uses the Standard Notations/the same standard notations as you do

Unless your students will only be talking to you/other people that strictly follow the standards, they'll need to be flexible.

Are you sure you are right?

Are you sure that the things you find incorrect are actually incorrect, and not just using a particular standard?

Are you sure vertexes is not a allowable pluralization of vertex? Are you sure that the word vertices is not an appendage that is being/has been phased out? Will you insist that data must be plural, and one must use datum for the singular?

Are you sure you are pronouncing $\Omega$ correctly. Do you pronounce it like this or this? The former is more Greek, and is often used by people from certain areas in Europe (And, sometimes if taken to the extreme, sounds like "OH MY GOD"). Something like the latter is more used in America. The common pronounciation for me and my peers is something slightly different from the latter.

Are you sure the word "derive" cannot be used to mean "differentiate"? I cannot support this with evidence, but I remember some sources using derive in that manner, and some sources claiming that derive can indeed mean "differentiate".

You've already mentioned the debatability of inverse. I'm going to claim that using inverse of a fraction instead of reciprocal is perfectly allowed. And I would argue that "minus a from b" is perfectly allowed as well. "Plus a and b" is slightly more awkward. However, without consulting a mathematical grammar guide and dictionary, can you tell me why "plus" can not be used that way in math?

The Bottom Line

Everything considered, notations are important. You should seek to introduce your students to the different types of verbal notations. They definitely should be able to fluently use the word "differentiate". You should impress that some notations are more proper than others, and should be used most of the time. If called upon, they should be able to use proper notations.

However, it is also important for them to understand and use other "less proper" notations. In general, it is fine to use these "less proper" verbal notations. However, if it leads to a situation where the students are unable to use proper notation, or when the usage of certain verbal notation is hindering communication or thought, proper notation should be emphasized.

Finally, you can simply use the "correct" notation in your speech, and in general, the students will follow. You can also explicitly note, every so often, that while "derive" can be used sometimes, there are other notations, and "differentiate" is in a generally better and more clear word.

Addendum

I feel the need to add onto this answer.

I would first like to draw attention to this question, which has great answers.

To quote some of the quotes given:

"The student of mathematics has to develop a tolerance for ambiguity. Pedantry can be the enemy of insight." - Gila Hanna

As far as possible we have drawn attention in the text to abuse of language, without which any mathematical text runs the risk of pedantry not to say unreadability. - Bourbaki

Also linked in the answers to that question is an article by Terence Tao, who describes the progression of mathematical education in three stages: "pre-rigorous", "rigorous", and "post-rigorous". I'd argue that any sub-field in math is learnt kind of in this manner. I would say that the student should be only be steered toward correct notation in the pre-rigorous stage, and that if notation is to be emphasized, it should be during the "rigorous" stage.

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  • $\begingroup$ Many excellent points; I very much appreciate the devil's advocate point of view (a good portion of why I asked this question). I'll have to think about a few of them to give a reasoned response. A first response would be: why do you feel that "plus $a$ and $b$" is more problematic than "minus $a$ and $b$"? From the perspective of ambiguity, fewer things can go wrong with "plussing" numbers due to it being a symmetric operator... $\endgroup$
    – erfink
    Mar 16, 2017 at 7:53
  • $\begingroup$ I only feel it is more problematic because it sounds weird. It probably sounds because the word plus is often used in other situations more often. "Minus a from b" sounds like a complete sentence ("minus a and b" would be wrong), but when I hear "plus a and b", I expect something in front of it. I'm ultimately not sure. Most likely it has to do with the frequency I hear these things. $\endgroup$
    – Fluidized Pigeon Reactor
    Mar 16, 2017 at 8:11
  • $\begingroup$ I think that's my point--I can't necessarily define why "plus $a$ and $b$" is wrong from a strictly mathematical perspective, but I can tell you that it feels awful to say out loud. Points to ponder before we fall too deep into a conversation for linguistics.sx =) $\endgroup$
    – erfink
    Mar 16, 2017 at 8:18
  • $\begingroup$ Yes, that is true. It does indeed feel bad to hear sometimes. However, that is a function of our own education, and our own standards, which we got from those who taught us and from those we interacted with. However, what sounds bad to us may not actually be wrong. On the one hand, we have to be careful about saying things like "You should never start a sentence with "because"". On the other hand, we are supposed influence our students to do more proper things. $\endgroup$
    – Fluidized Pigeon Reactor
    Mar 16, 2017 at 8:29
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    $\begingroup$ This leads us to 2 solutions. One is to try to learn about every standard and notation. The other is to make soft suggestions. If something sounds wrong and awkward, but might not be strictly wrong, we might say "It might be better if we rephrase what you said this way". If something is definitely wrong, and should definitely be corrected, I think we can say something much more firm. What do you think? $\endgroup$
    – Fluidized Pigeon Reactor
    Mar 16, 2017 at 8:31
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The standard verb is "(anti)differentiate", right? That's quite a mouthful. Probably okay to correct but with a light heart - make it into a joke, if the context is right. It is useful to be able to use standard terminology, so I hear you.

As an example, I had a graph theory class once where one student consistently said "vertexes" rather than "vertices" - I never once had to correct him after the first week, another student and he made it into a running game. For all I know he tells this same story in his career as a jazz musician (not kidding!).

What you shouldn't do is find ways to shame students who are struggling with the computations, let alone concepts. (I'm not suggesting you are doing this. But it's easy to come across this way, as many of us have experienced.) Bonus points for first person to use correct terminology? Or pie for the first one to come up with a reason why "prime the prime" would be ambiguous? That last one seems pretty unambiguous to me, by the way - it's more annoying because it focuses too much on algebra than the idea of acceleration than because of the wording.

As a side note, "verbing the noun" seems to be more and more common, and is probably a normal linguistic change within English in general. This discussion may seem quaint a hundred years from now (imagine smiley emoji/emoticon here).

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  • $\begingroup$ I agree that "prime the prime" is unambiguous in meaning, but sounds really grating. I look at it as there is a symbol ' (prime) that denotes a derivative is being/has been taken, but the operation we performed was not "priming." I would find it similarly strange to hear "take the Sigma of a sequence" instead of "take the sum of a sequence." $\endgroup$
    – erfink
    Mar 16, 2017 at 3:31
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    $\begingroup$ Also, "verbing the noun." Good point. So much for the Queen's English. $\endgroup$
    – erfink
    Mar 16, 2017 at 3:31
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    $\begingroup$ Maybe that question is for linguistics.SX.com :) see e.g. bbc.com/culture/story/… and Bill Watterson's take at gocomics.com/calvinandhobbes/1993/01/25 $\endgroup$
    – kcrisman
    Mar 16, 2017 at 4:58
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    $\begingroup$ "So much for the Queen's English." - or as Shakespeare wrote (in a previous Queen's English) "but me no buts!" Verbing and nouning conjunctions weirds language even worserer ;) $\endgroup$
    – alephzero
    Mar 16, 2017 at 8:44
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Here's my stab at a self-answer:

I think we would all agree on the need for precise written notation is important within mathematics. Unless the context is specifically reverse polish notation, a student writing $+~2~~ 2$ would be bizarre and incorrect. As such, I feel that it is also important to emphasize precision when verbalizing mathematics.

Using the analogy of mathematics as foreign language, it would be strange to learn French with strict emphasis on proper spelling and grammar but to never have pronunciation corrected. While mathematics is primarily a written language, more emphasis is justly placed on written notation. However, I feel that we should also place value on spoken mathematics by correcting such phrasings.

My personal approach follows advice of what I've heard to do when a colleague is using a fancy vocab word incorrectly: try to use the same word in a proper context as soon as possible, rather than a direct "I do not think it means what you think it means." My goal is to point out the mistake while not coming off as nit-picky. My personal approach also tries to be sensitive---humiliating a student, even unintentionally, in front of their peers can be quite damaging.

For example, if a student used one of these phrasings while offering a suggestion or asking a question during class, I would try to parrot the statement back correctly and placing a slight emphasis on the correct phrasing:

  • "Good---in order to find the critical points, we'll need to compute the derivative of $f$ and ...
  • "I agree, subtracting $b$ from both sides of the equation will ..."
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    $\begingroup$ The "parrot the statement back" strategy is dependent on the audience population. With high-functioning, engaged and interested listeners, it is indeed helpful and polite. But with low-functioning students (not engaged, language and listening problems) it is in my experience too subtle, and only explicitly addressing it makes a difference. $\endgroup$
    – Daniel R. Collins
    Mar 17, 2017 at 2:50
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    $\begingroup$ Note that + 2 2 is Polish notation. Reverse Polish is 2 2 +. en.m.wikipedia.org/wiki/Reverse_Polish_notation $\endgroup$
    – paw88789
    Apr 19, 2021 at 1:29
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Personally I feel like there are much more important issues for all students I've encountered while TA'ing/teaching in the US than whether they verbalize math correctly. I've generally taught lvl 300 courses (basic calc) and I've uniformly found that the students have incredibly poor notation with very few exceptions.

While verbalizing math badly isn't great, unless your students are drastically different to the ones I've seen, it feels like focusing on that is a bit like making sure the aspiring cook who doesn't know how to turn on the stove is great at naming the ingredients for beef wellington.

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  • $\begingroup$ I think we're generally referring to similar students in lower-division under-graduate courses. I will agree with you that written notation is more important than verbalization, but this doesn't excuse poor verbalization of mathematics. This is part of why I asked the question---how do we steer and correct verbal mathematics without spending an entire lecture on the subject? $\endgroup$
    – erfink
    Mar 22, 2017 at 3:13

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