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Several styles of writing about mathematics are appropriate in different contexts.

  • A mathematician writes something to be read by other mathematicians.
  • A mathematician writes something to be read by students at some specified level.
  • A student writes something in order to demonstrate that the student knows something.
  • A student writes a question addressed to an instructor.

Whoever writes an explicit account of the norms and standards governing the each of these or governing the choice among them, will disagree in some things with everyone else who writes such a thing. But that doesn't mean it's not worth doing.

Quite a lot has been written about the first bullet point.

I will be surprised if no one's written about the second.

I have an impression that the third is relatively neglected. And the fourth. But right now I'm mainly wondering about the third, and less so about the second. The third is something that possibly ought to be included in undergraduate math courses.

What has been published on the third item?

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  • $\begingroup$ Is something about the benefit or style of homework assigned for students part of what you are interested in for the third bullet? Or do you have something else in mind for what students write to demonstrate that they know something? $\endgroup$ – Clayton Aug 16 '18 at 22:10
  • $\begingroup$ Students (high school) usually have to do the third bullet point in exams (especially the smart ones). For example, in solving $x^2+3x+2=0$, a student should not immediately write out $x=-1,-2$. To get full mark, he or she should demonstrate the knowledge of factorization $(x+1)(x+2)=0$. $\endgroup$ – Szeto Aug 16 '18 at 23:16
  • $\begingroup$ @Szeto : That's a pretty trivial example. Toddlers are told to "show your work". Sometimes I marvel at the use of that expression by some people when addressing university students, as if they were three years old. "Explain your work" would be better. But in the question I had in mind secondary-school or university or graduate school students, not toddlers. $\endgroup$ – Michael Hardy Aug 16 '18 at 23:56

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