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This question is essentially communal, and its ramifications for the community’s identity, cohesion, and avodat Hashem play a major part in any decision. Arriving at a balanced decision entails consultation with a local religious authority who is in position to judge what will be of most spiritual benefit to the specific community. In cases like these, communal customs and norms can have greater halachic weight than written sources. Our response is intended as a preliminary discussion of the general issues.
On a strict halachic level, when tefilla is not in session, members of each gender can enter the other’s synagogue space. Common practice is to maintain gender separation during the pauses between tefillot, as well, though a mechitza might be adjusted to increase visibility during a derasha. But this still leaves room for different approaches to women speaking in the synagogue.
Many communities will not have women speak even after tefilla. While Rav Moshe Feinstein (OC 5:12:3) maintained that a woman could teach Torah to a mixed group after tefillot on an occasional basis, he added that the teaching should be in a private home rather than in a synagogue space, and only with the speaker sitting down. Unfortunately, he did not explain these caveats or their halachic force beyond mentioning “greater modesty.” Rav Aharon Lichtenstein z"l was not in favor of women speaking directly after synagogue. His son, Rav Mosheh Lichtenstein, Rosh Yeshivat Har Etzion, shared with Deracheha that his father's objection was based on a view of speaking after tefilla as remaining within the flow of the synagogue prayer service, as opposed to more of a beit midrash type setting, and that we cannot be sure of how his father would relate to this matter today.
Many other communities sometimes have women teach a communal shiur in synagogue following prayer. Where that reflects the values of the particular congregation, it is considered halachically acceptable, and can be important. Some communities pause slightly or take a concrete action (such as making adjustments to the mechitza or to seating that enable women to hear the shiur more optimally) after tefilla ends, in order to make it clear that the formal prayer service has ended and the communal shiur is beginning. Some synagogues also have a woman speak from a different location in the synagogue than the rabbi. (In a synagogue in which speaking by the Aron Kodesh is strongly associated with the rabbi, a slightly different location may be preferred for any lay speaker.) These choices, too, depend on the specific community involved. Rav Mosheh Lichtenstein does not rule out a woman speaking following tefilla.
If a community chooses to include women in the roster of speakers, but after prayer, then it would be desirable to move all speakers to after prayer.
A small, but growing, number of communities have women speak during pauses in prayer. When a woman speaks after tefilla, the distinction between the two communal endeavors of prayer and Torah study can be made clear. In contrast, if a woman speaks during a pause in tefilla, even just before Adon Olam, the distinction between prayer and Torah study may be blurred. Many rabbis consider this to be especially problematic, because the requirements for gender separation are stronger during tefilla, and they view a pause during tefilla as akin to tefilla. For this reason, it would be strongly preferred that women speaking during a pause in tefilla do so from within the women’s section, to maintain some level of gender separation.
For a number of halachic authorities, optimal gender separation is not maintained when a woman gives a derasha during a pause between tefillot, even from within the women’s section. For example, Rav Baruch Gigi, another Rosh Yeshiva of Yeshivat Har Etzion, considers it undesirable for women to speak during a pause in tefilla, even from the women’s section, though he does support women speaking from any area of the synagogue after tefillot are complete.
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