Many prescriptivists fear that the use of singular they (as in, “A student should take their time when considering prescriptive rules.”) is incredibly problematic. They are dissatisfied with the lack of plural agreement, though evidence suggests that the traditionally prescribed replacement (so called generic he) has ambiguous properties that are considerably more problematic.
Interestingly, there is no concern among prescriptivists that English you can be both singular and plural, but there are strong negative reactions among prescriptivists to regional attempts to clarify this pattern (e.g. y’all, all y’all, yous, yous guys, yinz, and yinzers).
Also ignored are ambiguous uses of we. Many languages have inclusive we (as in “We [=you, me and that other person in the room] should go”) and exclusive we (as in “We [=that other person and I, but not you] should go”). English doesn’t, and sometimes that seems like a real shame.
Initial attempts to change the generic he prescription (that which said one should use “A student should take his time considering prescriptive rules,” instead of the alternative sentence above) were initially met with strong resistance. However, accusations of political correctness gone amuck ignore controlled tests of the effect of this language pattern on both men and women. What’s more, complaints about wanting to maintain historical consistency (“that’s the way we’ve always done it”) ignore historical facts about the English language that most prescriptivists are unaware of.
For those who are interested in a more informed historical perspective on English personal pronoun usage, I recommend reading Sylvia Adamson’s research essay on English pronouns in the 18th Century and digging around for other linguistically informed histories of the English language.