There are some students in freshman calculus/even precalculus who compulsively use the three dots $\therefore$ in every single step: https://en.wikipedia.org/wiki/Therefore_sign

It's not "wrong" really, it just feels odd. I was never taught this sign in any class growing up. It seems like so much extra work and it's not necessary. Where do students get this idea from? Should it be discouraged?

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    $\begingroup$ For me it first arose in high school geometry (this was 1974-75), although I had probably seen it prior to this time in my personal reading. In thinking about the high school algebra and geometry texts I've seen over the years, I think geometry is mostly likely where your students learned it. Long ago I used it a bit more than I do now, but eventually I mostly stopped using it because it's easy to overlook unless you write carefully. About the only time I now use it is sometimes for quickly written handwritten notes just above the level of not-saved scratch-paper work. $\endgroup$ Sep 8, 2020 at 18:55
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    $\begingroup$ Can't tell you where it comes from and I was never personally taught it. I also very rarely see it, but a recent thread asked about the old book Calculus for the Practical Man by Thompson and I glanced through it. It looks like the last edition was from 1946 and Thompson makes very liberal use of the symbol. $\endgroup$
    – Thierry
    Sep 8, 2020 at 20:01
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    $\begingroup$ Should it be discouraged? Based on the quotes in my answer here, I'd say yes. $\endgroup$
    – Pedro
    Sep 8, 2020 at 23:45
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    $\begingroup$ I wonder if these are students who used to misuse the = sign in the way this question matheducators.stackexchange.com/questions/7964 describes, and have now learned that the three dots sign will not lose them points. $\endgroup$ Sep 9, 2020 at 3:28

1 Answer 1


I don't know the sources... but (at my R1 university in math in the U.S.) many students from abroad do tend to use this. My comment on it is that it is all too easy to misread this. That is, it's not stable/robust from an information-theory point of view. And I do advocate writing a narrative (in natural language, e.g., English) rather than a parade of formal expressions without explanation other than formal justification. That is, I do not just want to see a computation, I want to first hear the plan of the computation/deduction, and comments along the way about how the specific low-level details fit into that plan... etc.

Sure, at an early stage, when people have just first gotten the idea of actual deduction and proof, being very atomic-detail-oriented is not a bad thing. But, as I also tell my students, especially grad students, at a certain point no one wants to see all the small steps, because they're willing to trust you and/or can fill them in by themselves. Instead, a more high-level/top-down narrative, that prescribes the atomic steps (modulo irrelevant details) replaces the line-by-line, eventually.


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