You put up the slide. You explain, carefully, that proofs are written in sentences — that mathematics is a language, and English (or whatever natural language you're using) is its syntax. And then, from the back row or the course evaluation or the office hours doorframe, it comes: "It's unfair to make us write English in a math class."
If you've taught an intro-to-proofs course — discrete math, foundations, transition to advanced mathematics — you've heard it. Maybe not in those exact words. Sometimes it's "This is a math class, not an English class" or "Why are you grading my grammar?" or the particularly stinging "My high school teacher never made us do this."
It's frustrating. It feels like they're missing the point entirely. But here's the thing: they genuinely don't see the point. And that's not a character flaw. It's a predictable consequence of how math has been presented to them for the previous twelve years.
What they've learned math is
Think about the typical K–12 trajectory. Arithmetic. Algebra. Geometry (maybe two-column proofs, maybe not). Trig. Calculus. At every stage, the answer is a number, an expression, a graph, a simplified form. The work is computation. The notation is symbolic, compressed, deliberately stripped of natural language. "Show your work" means "show the algebraic steps," not "explain your reasoning in complete sentences."
So when they walk into your discrete math course, their mental model of "doing mathematics" is: manipulate symbols correctly to get the right answer.
Then you hand them a definition of divisibility and ask them to prove that if a | b and b | c then a | c. And they write:
a | b so b = ak. b | c so c = bm. So c = akm = a(km). So a | c. QED.
And you hand it back with "This needs more explanation" or "Write in complete sentences" and they think: I did the math right. Why are you penalizing me for not writing an essay?
They're not being difficult. They're being consistent with everything they've been taught.
So how do you respond?
Not by dismissing the complaint. Not by saying "This is a writing-intensive course" like it's a bureaucratic requirement. You respond by reframing what the course is actually about — and doing it early, often, and concretely.
1. Name the shift explicitly, on day one
Your slide with the quote — "The best notation is no notation; the best proof is a clear explanation" (or whatever the full quote was) — is a good start. But don't just show it. Talk about it.
Say something like:
"Up to now, most of your math classes have been about computation: here's a problem, here's a technique, execute it correctly. This course is different. Here, the goal isn't just to get the right answer — it's to convince someone else that your answer is right, using reasoning they can follow. That requires words. Not because I like grading essays, but because mathematical reasoning doesn't live in symbols alone. The symbols are shorthand for ideas. The ideas need sentences."
Say it in your own voice. But say it. And then show them what you mean.
2. Show, don't just tell: the "two proofs" demo
On day one or two, put up two proofs of the same simple statement. For example: "If n is an odd integer, then n² is odd."
Proof A (symbols only):
n odd ⇒ n = 2k+1. n² = (2k+1)² = 4k²+4k+1 = 2(2k²+2k)+1. ∴ n² odd.
Proof B (words + symbols):
Assume n is an odd integer. By definition, this means there exists an integer k such that n = 2k + 1. Then n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1. Since 2k² + 2k is an integer, n² is one more than twice an integer, which means n² is odd by definition.
Ask them: "Which one would you rather read? Which one would you trust if you were grading it? Which one could you explain to a classmate who missed today?"
Let them sit with it. Don't rush to declare Proof B better. Let them articulate why the second one communicates more reliably. That moment — when they say "The first one skips steps I'd have to figure out" or "The second one tells me why each step follows" — that's the moment the requirement becomes theirs, not yours.
3. Separate "English mechanics" from "mathematical communication"
This is where a lot of instructors lose credibility. Students hear "write in complete sentences" and think "I'll lose points for comma splices." Be explicit: you are not grading grammar. You are grading clarity of reasoning.
Put a note on your syllabus and repeat it in class:
"I don't care about fancy vocabulary or perfect grammar. I care that a reader — me, a TA, your future self — can follow your logic without guessing. If your sentences are clunky but the reasoning is clear and complete, you'll get full credit. If your grammar is flawless but the logic has gaps, you won't."
Then prove you mean it. When you grade, comment on the structure of the argument, not the prose style. "This step needs justification" not "Run-on sentence." "What definition are you using here?" not "Avoid passive voice."
4. Give them a scaffold, not just a standard
"Write in complete sentences" is vague. Give them a concrete template for the first few weeks. Something like:
- State what you're proving. ("We want to show that if a | b and b | c, then a | c.")
- State your assumptions. ("Assume a | b and b | c. By definition, there exist integers k and m such that b = ak and c = bm.")
- Do the algebraic work, but narrate it. ("Substituting, we get c = (ak)m = a(km). Since km is an integer...")
- Conclude by referencing the definition. ("Therefore a | c by definition of divisibility.")
Call it a "proof skeleton" or "argument structure" — not an "essay outline." Let them use it on early assignments. Grade partly on whether they followed the structure. Over time, they'll internalize it and stop needing the scaffold.
5. Use peer review — carefully
Once or twice early in the semester, have them swap proofs and answer: "Could you follow this? Where did you get lost? What question would you ask the author?"
Not "Grade this." Not "Check for grammar." Just: "Be a reader."
This does two things. First, it forces them to read proofs critically — a skill they rarely practice. Second, it makes the audience real. They're not writing for you, the all-knowing grader; they're writing for a peer who doesn't already know the answer. That shifts the motivation from "satisfy the professor's arbitrary rule" to "be understood."
6. Acknowledge the legitimate frustration
Sometimes the complaint isn't "I don't see the point" — it's "I have good ideas but I'm slow at writing them down" or "I have dyslexia / ADHD / a language background that makes this harder."
Have a private conversation. Offer accommodations: extra time on written exams, the option to record an oral explanation for some assignments, a grader who knows to focus on logic over mechanics. But don't lower the expectation that mathematical reasoning must be communicated. That's not gatekeeping — it's the discipline.
You might say: "I get that writing isn't your strongest mode. But in this field, if you can't explain your reasoning to a colleague, the reasoning doesn't matter. Let's figure out how to make this work for you."
The long game
You won't convince everyone in week one. You might not convince everyone by week twelve. Some students will leave the course still muttering "unfair" on the evaluation.
But the ones who do get it — the ones who start writing "Since km is an integer, a | c by definition" without the scaffold, the ones who come to office hours and say "I rewrote this proof three times because I couldn't make the logic flow" — those students have learned something that no computation-only course could teach them.
They've learned that mathematics is a social activity. A proof isn't a certificate of truth; it's an argument. And arguments require language.
That's worth the pushback. Keep the slide. Keep the standard. Just make sure they understand why before you hold them to it.