Joseph, the Aruch HaShulchan says before quoting the Gemara, that “Even though they recieve a reward: (quote from Sotah 20a, followed by Sotah 21b, which is the phrase about “one who teaches his daughter Torah (shebaal peh) it is as if he is teaching her matters of transgression,) connotating that it is surely unadvised for one to teach his daughter Gemara, but if one’s daughter learns Gemara, she still gets a reward. The Aruch HaShulchan says that “Because their minds are ??, and they find in the Torah meaningless matters, because their minds are inadequate (for Torah study, even though they get a reward). But like most poskim, the Aruch HaShulchan rules that one’s daughter can learn Torah Shebichsav. In the Tur it says the opposite of what the Shulchan Oruch says, that Torah shebaal peh is not tiflus, and Torah shebichsav is. The Birkei Yosef notes that it is most probably a printer’s error.
I looked in the Shach and the Taz, and they both say that one’s daughter should not learn Torah shebaal peh, the Taz in the beginning noting the same observation of the Birkei Yosef, the printer’s error in the Tur, and that the Tur intended to say the same thing as the Shulchan Oruch, of course intending to say as well, like the Birkei Yosef, that he is with the Shulchan Oruch in this case. The Taz also notes that Lechatchila one’s daughter can learn Torah shebichsav, because the king would read Sefer Devorim to every Jew,, including women and children, and not only the men. The Shach on ???:? says nothing but an explanation of “One who is commanded and does is greater than one who is not commanded and does,” thus also deciding like the Shulchan Aruch and the Tur.
The Birkei Yosef notes that in the Machloikes with R’ Eliezer and Ben Azzai in Sotah 21b that if one is obligated to teach his daughter Torah shebaal peh, that we pasken like R’ Eliezer, and so for anyone who says that Ben Azzai says te opposite, we pasken like R’ Eliezer due to the opinions of R’ Yehoshua and R’ Elazar ben Azariah, if I understood the Birkei Yosef correctly.
The Rambam says on the matter: ????? ????? ???? ?:?? A woman who learns Torah, she has a reward, but not as big as a man’s, due to the fact that she was not commanded to. The one who is not commanded and does, his reward is not as big as the one who is commanded and does. Even though she gets a reward, our Chachamim commanded that one should not teach her. So we can see that the Rambam is of the opinion that it is an issur to not teach one’s daughter Torah shebaal peh, as consistent with his idea, that is stricter than other posikm’s, that lechatchila one should not even teach one’s daughter Torah shebichsav, but bedieved it is not like teaching tiflus.
The Shulchan Oruch HaRav quotes the Gemara in Kiddushin, that a woman is not obligated to learn Torah, because it says to your sons, not to your daughters, in ????? ?:?. Later he quotes the Rambam in what was mentioned in the previous paragraph. It seems like a contradiction, but in reality the unobligation is for Torah shebichsav, and the quote from the Rambam is for Torah shebaal peh. So, basically the Sulchan Oruch HaRav says that Lechatchila one can teach Torah shebichsav, but holds by the strict opinion of the Rambam that it is an issur to teach Torah shebaal peh. however, still the Shulchan Oruch HaRav says women need to learn Hilchos Nidah, Tvilah, and other things that apply for women.
The Bach says the lechatchila teaching a woman Torah shebichsav is OK, because if she dosen’t know Torah, how would she know how to fulfill all the mitzvos? Also, the Bach says that a woman can learn by hearing, but can not teach by stating, asserting. Hearing is merely at the time of its command (she can hear a command or public reading, but not learn on her own, like the public reading of Sefer Devorim. But a woman should still learn the things that women need to learn, and so she should say Birchos HaTorah every day.
The Rema says that a woman should learn the things applying to a woman, and she is not obligated to teach her son Torah (presumably both shebichsav and shebaal peh). But if she does help her son or husband so they can dwell in Torah part of their reward is given to them. This last sentence of the Rema can either refer to a Yissachar- Zevulun relationship, one works, and other learns, benefit divided, or when she helps: for example answering a question or learning together. The Rema also says not OBLIGATED. Not obligated, means you don’t have to do it, but you can. A woman will still get reward for learning even though she is not commanded, as R’ Chanina says: “The one who is commanded and does his reward is greater than the one who is not commanded and does.”
The Beis Yosef says on the Tur: With reference to this it is taught: If her merits suspend her (from dying from frinking the waters) (Sotah 20a). It is said in the Gemara (Sotah 21a) If you say, the merits of Torah, she is not commanded and does! The explanation of Rashi on this is that she is not commanded and does, and so her reward is less than one who is not commanded and does. And what is written (to support this)? One who does something without being commanded, his reward is not like the one who is commanded and does. (This all implies that even if a woman learns Torah she gets a merit, but not big enough to save from death, as the Gemara says: “A merit can save for 1, 2, or 3 years.”
Last of all, what is the context of R’ Eliezer’s machloikes with Ben Azzai? Ben Azzai says: One is obligated to teach his daughter Torah: in regard to what? In regard to her merits from the learning saving her from immediately dying from Mei Sotah, suspending her death for 1, 2, or 3 years. R’ Eliezer only says it is folly, in regard to it is worthless. It will not save her. According to Rashi’s interpretation of promiscuity, it advocates promiscuity, for she can do any immoral acts and not die for a couple more years.
*This is all according to my understanding of the text.
My entire post is intended to prove that women are unadvised to learn Torah Shebaal peh, but it is not a strict issur.