I teach mathematics and computing; for example, our department's discrete mathematics course. As is customary for that course, it's used as the first foray into formal mathematics, including a heavy emphasis on reading and writing proofs throughout the semester (required for both our math & computing majors). In response to difficulties I've seen students wrestle with in past semesters, on the first day of the course I have a slide with this quote:

Writing Proofs in English

“The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that you are explaining the subject to a friend on a long walk in the woods, with no paper available; fall back on symbolism only when it is really necessary.” – Paul Halmos, How to Write Mathematics

Now, the last few semesters I've done this I find that I regularly get pushback. For example, this semester I had a student write in the live online chat, something like, "This seems unfair to students who have poor English skills". And I've been a bit stumped for a short, solid response to that.

Note that I'm at a community college in the U.S., which is part of a large urban university, and about half of our student population is foreign-born. We have a TOEFL requirement for enrollment, but some students indeed have very weak English skills (I've had a small number that apparently needed every course lecture translated by a friend, and couldn't communicate with me directly).

My initial instinct is something like, "You enrolled at a university where the instruction is in English, and has an English requirement for registration, so no one should be surprised at this." But (a) that seems wordy and byzantine, and (b) misses the underlying problem that people are led to believe that math work is entirely deterministic symbol-pushing, such that some people pursue it precisely because they think they won't need strong language skills (in fact, some of our advisors explicitly say this to students).

Obviously, replace "English" here with "natural language" at whatever institution and location you might consider.

What's the best, shortest response to student criticism that "It's unfair to expect writing in English"?

Related from SE Mathematics: Why there is no sign of logic symbols in mathematical texts?

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    In books by the Nicolas Bourbaki group the proofs have been written in French (with a lot of math symbols) Commented May 12, 2021 at 16:05
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    I just want to thank you for making explicit that mathematics, as usually taught, is over-grounded in obfuscatory symbology. For every student who bristles at the English issue, there are three who have been entirely denied mathematics education because the field would rather manipulate discrete Greek letters than present ideas in natural language. If the rest of your course emphasizes conceptual understanding the way that you seem to imply here, I am sure it is to the benefit of all your students.
    – Tiercelet
    Commented May 12, 2021 at 19:51
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    I remember this discussed on Mathematics years ago. I don't agree with "The best notation is no notation", imagine writing code in that way. Notation has concrete and well-defined rules and meaning, English words do not.
    – IS4
    Commented May 13, 2021 at 9:52
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    "This seems unfair to students who have poor English skills" Response: That is why we have a TOEFL requirement. and then move on.
    – NDEthos
    Commented May 13, 2021 at 14:18
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    Imagine writing code in that way, indeed. It would look like COBOL. Commented May 13, 2021 at 18:31

25 Answers 25


For writing proofs, full knowledge of English is not necessary. Most proofs can be written using a single tense. It is sufficient (and often advisable) to keep the structure of sentences very simple; that is very similar to how logic statements are written using math notation. Arguably, this exercise requires students to use only a small and simple subset of English level. Most students are expected to have it by the admission requirements of the course they signed on.

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    Funny how full knowledge is not necessary at research and at the top-level (see, ehm, hear the average English level at conferences), but how relevant is "local language" to get good grades in the modern teaching of mathematics at the lower level (high school, college, BSc and MSc ...). Source: personal experience at international conferences, personal bad english.
    – EarlGrey
    Commented May 11, 2021 at 15:01
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    @EarlGrey I find your comment interesting because (to me) it simultaneously suggests that 1) English skills do not need to be perfect, but 2) since English is the lingua franca of much professional communication there is a working skill level that is essential.
    – pjs
    Commented May 11, 2021 at 17:10
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    (+1) Especially for It is sufficient (and often advisable) to keep the structure of sentences very simple. Commented May 11, 2021 at 17:42
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    Nobody, at any conferece I attented, was unable to speak english. Yes some people had a strong accent but everyone was able to follow the talks which were held in english. Commented May 11, 2021 at 20:43
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    I think you guys missing the point. There is a big difference between "can", "reliably can" and "effortlessly can". For a native speaker, talking to a piece of wood in their mother tongue is easy, and Halmos's point is to encourage this natural, effortless state vs using symbols. For most foreign students, it is not an effortless, natural conversation, with free flow of thoughts.
    – Greg
    Commented May 12, 2021 at 5:34

Your experience resonates with me. In my classes I also have many students with imperfect English. I have often noticed (including as recently as a couple of weeks ago) that when I give lengthy word questions explaining a complicated real-life problem and asking to model it mathematically, some students struggle to understand what they are being asked to do due to language difficulties; and this is despite the fact that the question is written in grammatically correct, plain English, and I even make a conscious effort to edit the questions and “water down” the level of the English by removing complicated words and phrases to the extent possible.

Although I haven’t gotten pushback and don’t believe I have anything to answer for, thinking about this issue over the years has led me to conclude that there is a real problem of students who come to US universities with an incomplete mastery of English being at a disadvantage — often a significant one — because of their language skills. And I’ve wondered what I can do about it. For example, consciously trying to speak and write with slightly less complicated English words and phrases than I might be inclined to do if I knew I was addressing only native English speakers is one remedy. But it is a small and mostly (as far as I can tell) ineffective one. The students who were going to struggle are still struggling.

Now, whether or not this is “unfair” is a complicated ethical question and I don’t think it is my place to answer it. But one thing I’ve come to believe is that, at least, to the extent that it is unfair, the unfairness is a systemic kind of unfairness that is built into a university system that admits a large number of foreign students and expects them to handle an environment in which knowledge of (high-level, academic) English is assumed. In other words, it is not me who introduced the unfairness, and it is not me who has much power to do anything about it.

At the end of the day, my job is to teach mathematics, using English as the language of instruction, and that is what I do. If the subject I’m teaching involves writing word problems to teach people how to model and interpret real-life problems, that is what I will do. If the subject requires students to write proofs verbally, that is what I will require (though grading will be done based on the mathematical correctness of the answer and not based on correct grammar and spelling). There just isn’t any other way.

When a student says to you that what you are doing is unfair, tell them that you are simply teaching the subject you were assigned to teach as it should be taught by a qualified instructor. Ask them what they are suggesting that you do instead. My guess is they will not be able to make a coherent suggestion, and perhaps that will lead them to understand that the problem is more complicated than they realized.

Also: tell the advisors at your college not to tell students that mathematics does not require strong language skills! That is a horribly false and misleading thing to tell people and can obviously set them up for trouble and disappointment.

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    @AzorAhai-him- I’m confused, do you mean “a foreign national here by choice” in contrast to “a foreign national who is here not by choice”, or something else? I’m not seeing where I assume whether people are here by choice or not.
    – Dan Romik
    Commented May 10, 2021 at 21:11
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    @AzorAhai-him- okay. In case it was unclear, the main point I’m making is that students with poor English are at a disadvantage and that this puts instructors in an awkward situation of wanting to help those students to some extent but not having any practical mechanism for doing that without compromising academic standards. This applies equally to all students with poor English regardless of their particular status or life history. In practice, I suspect that foreign students are the main population for which this discussion is relevant, at least at my institution.
    – Dan Romik
    Commented May 11, 2021 at 1:35
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    "Ask them what they are suggesting that you do instead" I think that this is a key point that should be given more emphasis. If there are reasonable adjustments that OP can make to assist students who are less fluent in English, then I would argue that OP should consider making them. But it sounds like OP is already doing that by sticking to simple language. I don't think it's reasonable to expect an English-language course to entirely avoid the English language.
    – Lou
    Commented May 11, 2021 at 9:38
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    It seems to me that at least part of the problem with students coming to English speaking countries is that there seems to be widespread cheating in language exams. My experience is that three times I was closely involved in cases where a person had problems passing TOEFL or a similar exam, and in all three cases the persons were fluent in English and could even discuss their case with the faculty authorities. At the same time, I have often found students who come with a good grade in TOEFL, but cannot communicate, neither orally nor in writing. Commented May 11, 2021 at 16:13
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    to speak and write with slightly less complicated English words and phrases than... ifI knew I was addressing only native English speakers - I've actually been shocked at the large number of US domestic students I've had who get lost when I use polysyllabic words in my math classes.
    – Kimball
    Commented May 11, 2021 at 21:06

Being at a disadvantage due to a lack of necessary skills does not seem unfair to me. The ability to communicate clearly in written form is an essential skill for pretty well everything that we hope that our students will go on to do in their later life. Thus, it is a skill that we should actively encourage them to develop, by pretty much the same methods that we encourage them to develop, say, mathematical skills. That includes methods such as assessing them on that ability.

Just for reference, my department explicitly awards 20% of the available marks on each assignment (not including exams, because expecting people to re-draft work under time pressure is unreasonable) for "clarity of written communication". Neither this, nor a more implicit approach, is unfair: it is an assessment system rewarding better results to more able students, which is precisely what assessment systems are supposed to do.

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    I agree with you, but it is worth keeping in mind that someone with English as a Second Language is at a disadvantage for reasons outside their control (whether it is arguably fair or not). I can muddle through a basic conversation in Spanish, but I would have a hard time doing anything mathematically in Spanish more complicated than making change. Commented May 11, 2021 at 19:58
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    @TimothyAWiseman On the other hand, choosing to attend a university in which the instruction is in a language that you do not speak very well is generally within your control. At least in my country, quite strong English skills are (theoretically) required to study at university, so anybody studying here without such (and there are a surprising number) actively cheated on or otherwise bypassed the necessary tests. Commented May 11, 2021 at 23:36
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    @TimothyAWiseman it is absolutely in your control to not attend uni that uses a language you don't understand. it's not something they keep hidden from you and then woah surprise, they use english in the US. The english in a math course is, or should be, the least of someone's worries if they are weak in that area, because it's not like it gets any easier in the English lit course.
    – eps
    Commented May 12, 2021 at 15:26
  • @eps A lot of foreign students in the U.S. come for grad school. There are no English Lit course requirements for, say, a Master's or Ph.D. in Computer Science. Granted, you do have a point with regard to the case of this particular question where it's a Freshman-level undergrad class, which such courses would indeed be part of the required curriculum.
    – reirab
    Commented May 12, 2021 at 22:44

Frankly, I wouldn’t be surprised if changing “English” to “Plain Language” was sufficient to quell students’ concerns. It neatly sidesteps their manifest discomfort with the language itself. I doubt the manifest objection is the whole story—STEM folks always bristle at problems you can’t just calculate—so maybe you’ll even have more interesting and relevant conversations.

If you still hear objections, why not provide some resources? From the lowest level of personal effort to the greatest: You could refer your students to an on-campus writing clinic. You could offer a list of texts you found particularly illuminating when you learned to express complex mathematical ideas in writing. Or you could make sure students who were struggling knew you were available to discuss their work during office hours.

The responses others have offered are oriented towards winning your conversations with your students. It’s true there are larger structural problems in play for which it would be unreasonable to hold you responsible. The observation that Kids Today demand license and call it liberty is also hard to dispute. But your class isn’t about that, and your students aren’t an expedient means to proving a broader claim about society.

Your question makes it clear that you take your students’ education seriously and believe you do them a service by pushing them. Good! That’s demanding work, and too few academics are interested in doing it. But whatever else it may accomplish, making “Life is unfair” the hill you die on doesn’t teach them anything they didn’t know already.

(It also seems way more exhausting than pasting a couple hyperlinks into your PowerPoint, but that’s between you and your clipboard.)

The quotation is meant to be a little uncomfortable—counterintuitive, surprising, thought-provoking. Discomfort is an opportunity to learn. For that purpose, “English” is the single least interesting and relevant word you’ve got.

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    +1 as well, thanks for writing this. As a side note, I will say at the moment I'm hamstrung in some ways by distance learning -- formerly I was told my students could walk into the writing lab, whereas currently online writing-lab tutoring is gate-kept, requiring registration in an English course. Totally agreed that "life is unfair" is the about the weakest possible response, and I'd never say that. Commented May 11, 2021 at 12:38
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    (+1) for I wouldn’t be surprised if changing “English” to “Plain Language” was sufficient to quell students’ concerns, which is a nice and very easy fix to the sensitivity issue that might be the cause of many of the student complaints. Commented May 11, 2021 at 17:47
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    @DanielR.Collins That’s so frustrating—you’d think someone in the school’s administration would notice that students enrolled in an English class probably already felt pretty comfortable with their English. Sorry to have aimed that “life isn’t fair” bit at you. It was the most popular comment at the time I responded and that soured my sense of the tone. Commented May 12, 2021 at 13:08
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    @DanielP.Shannon: 100% with you on both points! Commented May 12, 2021 at 14:22

A short explanation of my grading philosophy in this instance often comforts my ESL learners:

While I will not take off points for general English mistakes (for example with grammar), I do take off points for mathematical English mistakes, that change/obscure the mathematical interpretation of your response.

While this does not put ESL students on the same footing as native English speakers, both groups struggle with technical language in their early proof classes, and so this puts the emphasis on what it's my job to teach all of them: good technical writing skills.

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    This is essentially the answer I would have given. The answers shouldn't be graded on the quality of the English, as long as they're understandable and express the correct mathematics.
    – Barmar
    Commented May 11, 2021 at 14:27
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    (+1) I didn't see this answer until I'd already written and posted mine, but this philosophy is behind pretty much everything I wrote in my answer. Commented May 11, 2021 at 17:49

The ultimate defence against 'It's unfair to assess us on skill X' is 'Skill X is listed in the published "Intended Learning Outcomes" document for the module/programme'. So, the question for OP is: does the "Intended Learning Outcomes" document, either for the module or for the programme, include something like "By the end of the module/programme, students will be able to make a case for a particular result, using verbal reasoning in English"?

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    This is if you have published intended outcomes. Commented May 10, 2021 at 20:23
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    @AzorAhai-him- In England and Wales, if we didn't have published intended outcomes, we'd have the regulator (OfS or its predecessor the QAA) on our backs about the lack of published intended outcomes, and we'd be too busy scrambling to deal with that to compose responses to this complaint. Commented May 10, 2021 at 20:41
  • The US/Canadian universities I'm familiar with, which I acknowledge isn't many, has only free-form text description of courses. But that's cool that it's regulated well in England/Wales. Commented May 10, 2021 at 20:52
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    @AzorAhai-him-: You may (or may not) find that's a rising trend in U.S. universities. Our college is in the process of hiring a new dean to oversee that requirement and enforce universal assessment of it. Commented May 10, 2021 at 21:49
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    E.g.: Today I'm informed that our provost has invited a speaker next week from the National Institute for Learning Outcomes Assessment Commented May 11, 2021 at 13:07

For my university at least, English proficiency is a requirement for enrollment to the point that students who learned English as a second language are required to take an English test like TOEFL as part of the admissions package. Therefore, although it may sound harsh,

You enrolled at a university where the instruction is in English, and has an English requirement for registration, so no one should be surprised at this.

is the assumption I make. (If I were to take classes or teach in a country that does not speak English, I would think it reasonable to have to learn the local language.) My class is not about reading, writing, or speaking English, so I will suggest students visit the university writing center if I notice them struggling.

(b) misses the underlying problem that people are led to believe that math work is entirely deterministic symbol-pushing, such that some people pursue it precisely because they think they won't need strong language skills (in fact, some of our advisors explicitly say this to students).

I think that math work definitely requires less language skills than some other subjects where writing is a larger part of the answer, but students still have to be able to understand the instructor's communication and explain their answers.

As a side note, I sometimes see students trying to derive a result algebraically without understanding what the math represents, and I usually suggest they describe their approach conceptually before going through the technical details.


Communicating academic work is a key part of doing academic work. Communicating in different forms (symbolic, graphical, natural language prose, computer code, ...) is essential to engage with different audiences. As an effective communicator of academic work, you don't get to choose what is best for your audience, so you have to learn to be effective in multiple approaches.

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    Who is 'you' in this answer? The lecturer or the students? Commented May 10, 2021 at 14:32
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    Both, but this answer should be read as a direct response to such a student comment.
    – Ian
    Commented May 10, 2021 at 16:10
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    You (Ian), as an effective communicator, can edit your answer to make it clearer. Commented May 11, 2021 at 22:18
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    I, as a lazy student, am accepting the poor marks received for failing to communicate clearly.
    – Ian
    Commented May 12, 2021 at 8:33

In addition to the other good answers:

There are two somewhat separate questions here, or maybe three. One is about the way "natural" mathematical questions arise in an English environment. Another is about old-or-new English formulations of direct mathematical questions in a classroom setting. Another is about responses in either case.

For complaints about having to respond coherently, as though it demanded an inordinate level of English fluency: well, no, it doesn't. A very rudimentary style is sufficient, and perhaps even better than long compound-complex sentences that self-interrupts, etc. Hemingway, not Faulkner. But, yes, English-style sentences. Not strings of various nouns and verbs with neurotic periods and capitalization.

Comprehension is a harder question. But for kids wanting to work in the U.S., "even in Math or tech", the "secondary" skill of fluency in English can be a decisive competitive edge. There are lots of people with good technical background, but with communication problems.


Each unnecessary mathematical symbol has a universally-applicable English equivalent

Replacing symbols with words is easy. Your students can just learn a single phrase to replace each symbol, and use that whenever they would use the symbol. Avoiding symbols isn't the hard part of writing mathematics in English.

The hard part of writing proofs is writing with proper structure. Students need to present their ideas in a logical order, signpost how each idea relates to the next, and have an appropriate structure for their proof. Importantly, these skills are only slightly language-dependent. If a student can write a good proof in their native language, a word-for-word translation of that would probably be better than many proofs written by native English speakers.

Your course notes or textbooks contain many examples of the language used in proofs

Native speakers as well as non-native speakers often don't know how to write proofs. I tutor a mathematics course with a writing assignment, and our standard first-instance answer when students ask how to write proofs is that they should copy the style in the notes. Our university devotes two weeks of one of its courses just to writing proofs, and I've heard of a university with an entire course dedicated to mathematical writing.

  • +1 If I were in this situation I would create a chart of symbols in one column and the correlated words and phrases in a second column so a symbolic proof could be replaced by words. A lot of the difficulties of spoken and written English do not really crop up in mathematical English. Commented May 11, 2021 at 13:08
  • This is so much more concrete than mine! And the focus on structure is very insightful. Thanks for articulating this. Commented May 18, 2021 at 15:14

I would answer:

"It is, nevertheless, an essential part of the skill you are here to learn. It is no different in principle to students with a weak mathematical background due to their previous education. As with any other challenging circumstance a student may face, specialist help is available for anyone having difficulties with any part of the curriculum and who needs to catch up on prerequisites."

If this is a common problem, there ought to be language tuition available specialising in remedial technical/mathematical English. Or they could get in touch with older/former students with a similar language/subject combination, find a study group, etc. But as regards "fairness", this is no different to the disadvantage faced by students who went to bad schools that didn't teach them calculus, or complex numbers, or whatever. Everyone has their own individual hurdles. For one student it's calculus, for another, English. Ask yourself, how would you answer someone who said they were not taught some prerequisite bit of mathematics at their previous school and it "wasn't fair"?

  • "Ask yourself, how would you answer" I'd say they are correct. It is unfair. Commented May 11, 2021 at 9:54
  • "Nevertheless" suggests that you accept the claim that it is unfair.
    – einpoklum
    Commented May 13, 2021 at 12:42

Consider the statement "Requiring students to exhibit a skill is unfair to students who don't have that skill."

The statement is true if the skill is not related to the course. It is unfair for a course in, say, discrete mathematics to require that students exhibit, say, the ability to swim. This is because one can do discrete mathematics even if one cannot swim.

The statement is false if the skill is related to the course. If is fair for a course in discrete mathematics to require that students exhibit the ability to count. This is because one cannot do discrete mathematics if one cannot count.

Is the ability to write mathematical proofs related to a course in discrete mathematics? Yes, if the focus is on theory; no, if the focus is on applications.

Is the ability to write in English related to a course in discrete mathematics? Yes, if the course is taught in English; no, if the course is not taught in English.

Tell your students that your course focuses on theory and is taught in English.

Edit: It has (correctly) been pointed out in the comments that even though one cannot do discrete mathematics if one cannot count, one can still do discrete mathematics if one cannot write in English. The reasoning I used in paragraphs 3 and 4 is not the same as the reasoning I used in paragraphs 2 and 3.
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    Good analysis. But as for the conclusion that it is not unfair, I think the feeling of unfairness comes from the fact that English is actually not required to do math: there are mathematicians who don’t speak any English. The requirement to speak English is a practical requirement, a bit similar to a requirement that a student needs to be able to walk down stairs because the lecture hall does not have an accessibility ramp. So one can argue that there is a bit of unfairness here, at least if the student wasn’t properly made aware of the importance of coming with good English preparation.
    – Dan Romik
    Commented May 11, 2021 at 14:09
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    @DanRomik Yes, English is not required to do math in the same way that, say, Swahili is not required to do math. But if the course is taught in English, then I would argue that English is required to do Math in that course. If a course is taught in Swahili, then a student who knows only English (and doesn't understand the teacher or the materials, and is unable to answer using Swahili words) should not complain that it is "unfair." What would be unfair is to require a teacher (who is hired to teach in English) to learn Swahili just so that their student could learn math better.
    – JRN
    Commented May 11, 2021 at 14:48
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    @DanRomik Anyway, you make a good point that my paragraphs 4 and 5 do not follow the reasoning in my paragraphs 2 and 3, and I thank you for that observation. I'm leaving the computer soon, but when I return, I'll try to edit my post to include your comment.
    – JRN
    Commented May 11, 2021 at 15:02
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    @DanRomik (By the way, I agree with what you wrote in your answer and I upvoted it.)
    – JRN
    Commented May 11, 2021 at 15:08
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    Thanks. I don’t have a strong opinion about the fairness issue. I said “one can argue” because I think it is a valid argument, but there are valid arguments also in the opposite direction (as you and others pointed out) so in the end it doesn’t seem clear-cut to me whether it is fair or not, or whether that’s even an entirely meaningful question.
    – Dan Romik
    Commented May 11, 2021 at 15:25

Maybe something like:

"It is important to understand the motivation of why something is done as well as the underlying beliefs/assumptions the writer has made. The role of mathematics is to allow us to understand complicated relationships between aspects of reality, the symbols' role is to help reduce mistakes in logical reasoning and allow us to deduce counter-intuitive relations with a high degree of confidence. However seeing a formula, and understanding a formula, are very different things. The second of which is important for extending our understanding to more advanced mathematics and applying these formulas in new or innovative situations."

If you are working with computer scientists or physicists this ability to explain in words rather than just symbols is important since the mathematics is only a means to an end rather than a thing unto itself. For mathematics students you could find the counter argument that mathematics is separate from the rest of reality and so symbol pushing should be enough. But that fails to take into account that the mathematics that they are working with is being worked with in a fleshy ball of meat, living deep within reality. Something that doesn't work well with just symbols disconnected from reality and has a limited amount of memory. This leads to the need for any student to be able to chunk ideas together when they want to learn more advanced topics, this is much easier to do with verbal and intuitive explanations than just a bunch of formulas and relations.

As for the students who are really bad with English, get them to first write their response in English, then a second time in their native language and run it through google translate. You're not going to be testing them on their grammar so the mistakes google translate put in probably won't obscure the underlying understanding that much if it was there to begin with (assuming its a fairly well represented language). And since your language of instruction is English, then if they can't understand that they should be spending time improving their English, and this gives them an opportunity to practice using English in a highly specialised case which they probably wouldn't otherwise get.


I first would recommend that you talk to the writing support services people at your institution, such as the writing center or the center for teaching and learning. They will have some suggestions for you about how to approach this topic in a way that is constructive and useful for students. I'm not in math, but I do have students who have challenges with standard written English, both immigrant and native born.

Here are some ideas I would suggest.

  1. I always say that one of my learning outcomes is that you will improve your skill in writing. This is not a grammar class, and writing mechanics will not be an important part of the grading, but what will be important is that you get a lot of practice in professional writing in our discipline.

  2. You can also add oral expression to this, and your example of explaining on a walk is a nice way of expressing this. I would even consider adding something about that in your learning outcomes.

  3. All the writing pedagogy people will tell you two things. One is that to get better at writing you have to write a lot. The second is that revision is essential. Create opportunities for your students to revise their writing.

  4. Don't try to teach a grammar or composition course. You're not actually qualified for that besides the fact that you have math content to cover. If you can help students make small steps in their writing be happy with that.

  5. Make sure you encourage students to take advantage of tutoring and other academic supports. Remind them every week that this is an option so that using them becomes de-stigmatized.

Since you are a fellow CUNY person I would strongly suggest that you see if you can get a WAC fellow to work with you. They are generally excellent and have good training.

Update I have been thinking a lot about your question. I think we were missing the point (largely) because the students weren't asking about writing but about writing in English. Here's what I've been thinking.

First I think you should reassure them that if they are doing math in their first language, that is normal. Being able to do math in a second language is the last or close to the last point of fluency.

What I might suggest is that you say they can write in their own language, but then translate into English if they prefer. I'd have them hand in both along with the Google translate version so that you know they went beyond that (which is the obvious first step). Your goal then, even though you won't exactly grade on it, is to help them get more fluent in English.I think you should stress to them how important this will be in the workforce. You can also encourage them to write directly in English and say that you'll grade it to the same standard as the translated submissions, not expecting perfection.

  • +1 Thanks for this; I'm glad you shared it, as it may be helpful to others. In my case, most everything you've suggested is stuff I've already added/tried. When I've touched base with my writing center in the past, they seemed trepidatious, yet said they'd accept my students in the lab -- but currently that's impossible, because online English tutoring is now gate-kept by the need to be registered in an English course. I fear a WAC fellow would be very frustrated by my technical situation. Commented May 11, 2021 at 20:17
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    Thanks for that. I've been thinking about your 4th bullet for a day now. I think I would pose that writing math (while mostly in English) is a specialized, technical form of writing (including its custom grammar and composition), and arguably I'm better trained and qualified for that than anyone else on the premises. Commented May 13, 2021 at 3:50
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    Correct. The WAC program is something I have used. But each campus organizes their programs differently, so you may not have that exact option. But I think if you have WAC coordinators you could ask them for a consult at least.
    – Elin
    Commented May 13, 2021 at 4:17
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    @DanielR.Collins Updated this with some new thoughts.
    – Elin
    Commented May 30, 2021 at 0:37
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    Related: academia.stackexchange.com/questions/127116/… Commented May 30, 2021 at 1:51

Rather than trying to respond to student reactions to that particular quote, it might be better from a sensitivity standpoint to simply give several examples with explanations, perhaps categorizing the examples into a few named types of poor exposition. Then, perhaps at some later time when students have had some experience in "writing mathematics" and getting feedback, you could present the Halmos quote in class and ask students to discuss what they think it means.

One type of poor exposition is excessive symbolism, which is discussed in Section 15 Resist Symbols of the Halmos paper (originally published in 1970 and reprinted many times since then). For example, one should resist the temptation to introduce notation in the statement of a theorem that is only needed to prove the theorem, such as "Theorem: Every point p in a bop-bee quasi-uniform space X has a proactive neighborhood." There is no need to include p or X into the statement of this theorem. However, I see this advice violated in just about every math paper I look at. Regarding excessive symbolism in general, see my answers to the Mathematics Stack Exchange questions Using the implies symbol in proofs and How formal or informal should math texts (written for different purposes) be?

Another type of poor exposition could be roughly identified with the linguistic term "structural ambiguity", and would include both incorrect and ambiguous uses of quantifiers, such as missing quantifiers, incorrect quantifier order, and what is sometimes called a hanging quantifier. It also includes more English-specific scope issues like "John sent a letter to each of his students" (Was the same letter sent to all students, or did each student get a personalized letter?), and "Every researcher of a company gave a presentation" (Did all of the researchers belong to the same company?), and "big truck driver" (Does this refer to a driver of big-trucks or to a big-driver of trucks?), although these more English-specific issues might be too tangential and overload your students with too much information (and thus handled on a case-by-case basis as they arise).

Still another type of poor exposition is the incorrect use of notation and terms, and I'm sure you can give many examples. One type of ambiguity that ordinarily doesn't cause problems, but which might be useful to mention in light of your course topics, is the use of "and" when an order is involved, such as "the ratio of p and q" and "the difference of p and q" and "the quotient of p and q" (example in my present line of work and my comments about it). However, I definitely recommend not falling into the rabbit hole of being obsessively correct for your students, and instead stick to the most significant concerns. Some things in the answers/comments to Abusing mathematical notation, are these examples of abuse? are what I consider as obsessively correct for your students, and very many of the things discussed in this critique of Rotman's book An Introduction to the Theory of Groups, written by the author of this book, are what I consider as obsessively correct for your students. Probably a better term than "obsessively correct" exists, but this is the best I can think of now.

For the most part the examples I've given are of a logical and mathematical character, and I think most of your focus should be on these types of examples, and not on grammar issues specific to English, although of course when grading student work you should point out grammar issues if you notice them (except those students who have a huge number of grammar issues in their writing, and for those students you may want to seek the advice of someone locally, as suggested at the end of @Elin's answer).


There's so many great answers here -- so I'm submitting my own so as to organize my favorite observations for future reference. My answer in future classes will be a combination of these insights:

  • English proficiency was required, and tested, to enter the college.
  • Writing in natural language is an integral part of the subject, as well as any academic work.
  • The writing we do is in fairly plain, simple English. If one can write a cogent, simple proof in another language, then it's a fairly easy job to translate it.
  • The hard part is really understanding the mathematics, not the notation.
  • If someone is weak at writing, then it's good to get practice, which we provide.

If students have been assigned to write in English and they complain:

This seems unfair to students who have poor English skills

The correct response is

  1. Some students learned English as a native language and other students learned some other native language, but everyone is expected to write in English. Yes, this is unfair.
  2. Most of the money and power in the world is associated with using the English language. This is unfair to people who speak other languages, but nobody knows how to change it.
  3. The best thing we are able to do about the situation is help students learn English.
  4. Completing this course will help you learn English. For example, writing proofs in English will help you learn English.
  5. Many students who are English language learners are successful in this course and in this discipline. (add examples)
  6. The resources available to help you are ...
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    +1 The purpose of education is for people to learn new skills and to improve the ones they have. A course that challenges students to practice and improve the skills they are weak in is the whole point of being there.
    – J...
    Commented May 13, 2021 at 11:05

I agree with the students, and think that quote is used in the wrong context. I don't know why you'd discourage using standard notation. In my teaching I wanted the students to use the correct technical terms. More importantly, I tried to get them to understand that jargon for a complex idea is natural and not show-offy at all. It seems to me you're going out-of-your way to add English proficiency where it's not needed.

Jargon-wise (which I realize isn't the same as notation, but close) I was glad when students were finally comfortable replacing "rules for how the symbols have to be" with "syntax" or "the input value in the call" with "the argument" and so on (I taught lots of intro programming). I had plenty of international students and quickly realized that most plain English is chock full of idioms, complex tenses and other challenging bits. They're already stuck listening to my colloquial English explanations, which is torture enough. And it's not as if jargon is a crutch -- you have to understand it to use it properly.

That quote -- I think I know what it's about. Most new notations are bad. Alternate ones spring up and it takes a decade before the "obvious" best way takes hold. I've read papers on new ideas where they spent time on a notation which wasn't all that clear and wasn't used much. They could have simply written "let A-prime be a compressed version of A which adheres to the previously discussed requirements". I think the quote is saying: "if you have a new idea, just say it. A good notation will come much later once the idea is explored".


Short and to the point

This is not unfair. The college has a TOEFL requirement for enrollment, and any student work will be graded in English only.

Algo, the guidelines ask for simple English, so it's beneficial to people with no mastery of English.

  • I don't think immigrants have to take TOEFL, this is not people coming from abroad to go to college.
    – Elin
    Commented May 13, 2021 at 3:40
  • Hopefully a professor commenting on the English requirements will not be writing "algo" instead of "also".
    – Nik
    Commented Oct 21, 2023 at 14:25

This quote is not about English but about avoiding symbolism when natural language can be used. Paul Halmos was born Hungarian, and the best way to understand this is to imagine him discussing math in Hungarian with another Hungarian born prominent mathematician of his era, such as Paul Erdős or John von Neumann. And this quote is generally misunderstood when applied to students.

Students (and mathematicians) should acquire proficiency both in using symbolism and in avoiding it. When one is unsure if an argument is valid one should be able to write it symbolically and that may help in verifying whether the argument is valid. Then when one wants to communicate an argument to someone else, avoiding excessive use of symbols helps.

Students should be encouraged to experiment with writing the same argument with more and with less symbolism. Students who are less proficient in the natural language required in the homework would have the added value of acquiring more experience in the use of that language.


I've had a small number that apparently needed every course lecture translated by a friend, and couldn't communicate to me directly

If they can't communicate in even the most basic English, I have a difficult time understanding how they could pass TOEFL to begin with, unless your university's entry mark for TOEFL is in negative territory.

Anyway, you seem to think that this is something that's your responsibility and something you can and should do something about. It's not. If students are coming into an English university, with courses taught in English, then they had better bloody well be able to keep up with the English that the instructors are using.

The simplest way to explain this is by turning it around on them, so:

What's the best, shortest response to student criticism that "It's unfair to expect writing in English"?

"If I had chosen to study at an institution that uses <student's native language> for instruction, and I lacked a fundamental grasp of that language, would it be valid for me to claim that it is "unfair" that I cannot understand the instruction being given?"

If they answer "no", they have defeated their own argument.
If they answer "yes", get them to articulate why, because either they are just being difficult, or their reasoning will be unsound and easily defeated.

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    @AnonymousPhysicist How is not having a choice "inappropriate"? Your argument makes zero sense.
    – Ian Kemp
    Commented May 11, 2021 at 9:54
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    That's not what I said. Your answer denies the truth, which is that most student's choice is between studying in English and not studying. Your answer, in the second to last paragraph, implies students chose to study in English when they could have chosen to study in their native language. Therefore, your answer is very inappropriate. Commented May 11, 2021 at 10:02
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    I deleted it because it was not written clearly. I am only criticizing the answer. Commented May 11, 2021 at 12:28
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    Regarding your rhetorical question “if I had chosen to study [etc]”, I would say that if the non-English speaking university admitted me despite my lack of necessary knowledge of the language of instruction, thereby misleading me by implying incorrectly that they consider me qualified to succeed in their course of studies, I would actually have some valid reason to complain. Whether it makes sense to use the word “unfair” in this context is a tricky semantic question, but the point is the answer to the rhetorical question is less obvious than you might think.
    – Dan Romik
    Commented May 11, 2021 at 20:35
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    This is certainly a response, but one that borders on the obnoxious, for reasons that are possibly not clear to you. I'm glad to elaborate in chat if you're interested. Commented May 12, 2021 at 11:36

The original slide doesn't push the reader to use English. Pretend that you are explaining the subject to a friend. This pretense could be done in the favorite language the two of you share. It doesn't have to be English. There are many languages that are just as well suited as English for this purpose.

Now the question becomes how to communicate with the professor. If the best language for this purpose is English, then that's the language to use. This may not be fair, but it's useful.

It may put some people at a disadvantage. They may have to work harder to polish their English communication skills. This could serve them well later on. Another solution is to find a different professor who teaches the same course, but in the student's first language.

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    as an aside, during part of my career, my assignment was to travel to Mexico or Puerto Rico and teach short courses in database design in Spanish. Concepts from relational mathematics are very useful in this context. I found that learning how to express these concepts in my second language gave me a deeper appreciation of the concepts themselves. Commented May 12, 2021 at 10:51

This is barely an answer, but having students with language issues is hardly new, and your college probably has resources for such students. You should locate them, and guide your students to them.

You should look into what language assistance resources are available for students, particularly ESL students, at your community college -- maybe there is a writing lab or something.

If you can locate the resource, you should have a discussion about this issue with the person that heads this resource, and work out a plan for how to bring these students to a higher level in your course.

  • I've done that in the past and have it on my syllabus. I believe that to date no students of mine have ever made use of those resources. And it's been made worse in the current distance learning situation: the online English tutoring is gate-kept and only available for people currently registered in an English course. Commented May 11, 2021 at 20:03

I would turn this on its head: 1) English is a basic skill you need in all paths of life - especially but not only in an English speaking country. 2) Some natural language is specifically needed to communicate about math and in your course that is English. So English is a two-fold requirement, thus it is fair to use it and request using it and in addition, using it also supports acquiring and improving the general skill of communicating in English. Languages are far better acquired if they are actually trained in different settings than solely in language courses. So you do your students a favour in helping them improve their English alongside learning math.

I've studied in a country where English is a foreign language and yet the same argument was provided there (university-level/college) for why the majority of classes was in English. Sure there were nay-sayers, too, and they were guaranteed enough courses held in the country's official language to pass their Bachelor's degree without having to go to an English language course, but few used that offer and there it was English or official native language of the country which likely in your case is English, too.


This is an overreach, but I suggest that instead of just formulating a counter-response, you spend a class or two familiarizing the students with the required terms/words in their English sense (rather than the strictly mathematical sense). Many non-native English speakers struggle to build an intuition for slightly abstract math, because they have never used the words manifold, convergence, degenerate etc. in their daily life.

Exposing students to these terms in the preparatory part of the course will help allay anxiety and also implicitly convey that appropriate terminology/syntax is required in the course. Maybe that can be made explicit as well. If the course schedule doesn't permit these extra hours, you could consider circulating handouts and having a non-graded quiz on the same.

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