For my write-up today of The Cambridge History of Judaism, Volume Three: The Early Roman Period, I will start by quoting something from pages 370-371, which is from William Horbury's "Women in the synagogue":
"Probably...Paul held that, 'in the assemblies' (I Cor. 14:34), women are free to pray and prophesy (I Cor. 11:5), but not to speak or to ask questions (I Cor. 14:34f)---that is...not to teach in public. This view would tally with the opinion current among Jews that women pray and hear the law, but do not teach...If, however, the passage I Cor. 14:34f is interpolated, it may have been intended to forbid women's prophecy. In either case, in early Christian worship some liberty of prophesying for women co-existed with a marked tendency towards constraint, characterized by the derivation of the subordination of women from the law (1 Cor. 14:34, I Tim. 2:11-15). Philo and Josephus similarly derive it thence, probably with Gen. 3:16 in mind."
I Corinthians 11:5 appears to imply that women in the church should pray and prophesy with their heads covered. In I Corinthians 14, however, a chapter in which Paul talks about prophesying, vv 34-35 say that women should keep silent in the church, that they should learn by asking their husbands at home rather than by speaking in church, and that women are to be obedient, according to the law (which Horbury says may be Genesis 3:16, which affirms that the man will rule over the woman). Is this a contradiction? Many argue that I Corinthians 14:34-35 is not authentically Pauline. Others, by contrast, seek to harmonize the passages. Horbury refers to the view that Paul allows women to pray and to prophesy in the assembly, but not to teach in the assembly, for Paul wants them to learn in silence. John MacArthur argues that women are permitted to prophesy to unbelievers, to women, to children, and to individuals, but not to do so in the Christian assembly. Under Rachel Held Evans' post, Complementarians are selective too, Calvinist Justin Taylor mentions the argument of Wayne Grudem and John Piper that Paul permits women to speak in the church, as long as they do not challenge or undermine the authority of men (which, according to I Corinthians 11, they were doing by not wearing a head covering). And, when John Piper was asked by a man if he was biblically allowed to listen to Christian teacher Beth Moore, Piper responded that this was fine, so long as the man did not submit to a woman as his spiritual shepherd and authority (see here).
My impression is that Horbury argues that the early church was reflecting Jewish practices on women praying and prophesying in an assembly. But, as far as I could see (at least in my reading so far), Horbury does not demonstrate that Jews permitted women to prophesy in temple or synagogue services. But, on page 379, Horbury does discuss examples of women leading song in the Jewish religion: Miriam does so (Exodus 15:20ff.), as do Deborah (Judges 4:4; 5:1) and temple choirs that include females (Ezra 2:65; Nehemiah 7:67; Psalm 68:26). In intertestamental literature, Judith leads with dance and song (Judith 15:12-16:1), and Job's daughters sang in an angelic language (Testament of Job 48-51). Could any of this count as prophesying?
(UPDATE: Here are some more quotes from Horbury's article, which shed light on the issue:
Page 389: "Hymnody verges, indeed, on prophecy, in which women spoke with authority...but prophecy is unlikely to have had a regular place in the constitution and worship of the synagogues." Page 398: "Prayer and hymnody...elements of worship in which women had some traditional prominence, are likely to have had their place in varying ways in the synagogue assembly from the Second Temple period onwards.")