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What is the role of women within Judaism?

Orthodox Judaism views men and women as having complementary roles in life. As a consequence of the different roles, men and women have different obligations (positive ways of fulfilling the will of our Creator). Jewish men, for example, have an obligation to pray at three fixed times each day. Jewish women on the other hand, don’t need have this obligation. There’s an important word usage to note in the above paragraph: obligation. Traditional Judaism looks at actions in terms of duties and obligations, not the modern socio-political notions of rights. Thus men and women have different duties and obligations; the question of rights never arises.
– Excerpted from the Soc.Culture.Jewish FAQ, Section 8. Woman and Marriage.
Women were traditionally exempted – and often banned – from any Judaica study beyond a basic understanding of the Torah, and the rules necessary in running a Jewish household. Women were discouraged from learning Talmud and other advanced Jewish texts. Women are exempt from having to follow much of the set daily prayer services, and most other positive time bound mitzvot, such as wearing tefillin. (There are a number of notable exceptions).
Some statements in classical rabbinic literature strongly forbid women from learning Torah (including the Bible, as well as all rabbinic literature.) Yet other statements in classical rabbinic literature are egalitarian, and allow women and men to study Torah. Today, it is understood by practically all Jewish denominations that the restrictions on Torah study by women were the results of sociology, not theology. As such, women today study Torah, in its widest sense, as men do. The sole exception is the world of Haredim (Ultra-Orthodox Jews) who maintain significant bans on woman studying Torah, and indeed, on women’s rights in general.
For details see Women and Torah study

Traditionally, women have not been eligible to be counted in a minyan, as a minyan is a quorum of those who are obligated. Although the Talmud and Shulkhan Arukh technically allow women to have aliyot to read the Torah, this is not done in Orthodox shuls due to the principal of Kavod HaTzibur – the honor of the congregation.
How has the traditional role of women changed?
One of the first major breaks with the traditional role of women came from the Chofetz Chaim, Rabbi Yisroel Meir HaKohen (1838-1933). He overruled traditional prohibitions on the basis that in the modern world it is now important for women to have an advanced Jewish education. Soon after this, the Bais Yaakov network of Orthodox Torah schools for women was built.

Recently many rabbis in the Modern Orthodox community have set up schools that bring advanced Jewish studies to women, including Stern College at Yeshiva University, and the Drisha Institute (both in New York City).

A small but growing number of Modern Orthodox Jews have proposed that it may acceptable for the Orthodox movement to ordain women as rabbis; a few Modern Orthodox rabbis have ruled that if the congregation does not object, women may have aliyot to read from the Torah. This phenomenon is still an anomaly within Orthodox Judaism. Most Orthodox Jews reject these ideas as unacceptable deviations from tradition.
Can Jewish women receive Aliyot?
In 1955 the Committee on Jewish Law and Standards (CJLS) of the Rabbinical Assembly (RA) ruled that a synagogue may give aliyot to women. This psak (decision) was based on a teshuva by Rabbi Aaron Blumenthal, based on these points:
Talmud Bavli, Megillah 23a, says “Anyone may ascend for the seven honors [of reading the Torah in public], even a minor, even a woman, but the sages have said that a woman shall not read in public because of K’vod hatzibur [dignity of the congregation].”
This rule is repeated in the Tur and the Shulkhan Arukh. The Tosefta reads “Anyone may ascend for the seven honors, even a minor, even a woman”; then it says “One may not bring a woman to read in public. ” According to Prof. Saul Lieberman this meant that women in the congregation were allowed to read the Torah; however, it was considered wrong to bring in a woman from the outside to do this; that would imply that the congregation had no man competant to read from the Torah. Many non-Orthodox Talmud scholars believe that some Palestinian synagogues resorted to inviting a woman for this purpose; this verse reflects the Babylonian practice which rejected this custom. Thus, this concern is not relevant today when all synagogues have members (of both sexes) capable of reading the Torah scroll.
Rabbi Jacob Emden (18th cent.) wrote “It seems to me that a woman may not read the in public wherever this is possible. But the first sentence (granting her permission) would apply when there are not seven men (among the ten who constitute a minyan) who can read, and there is a woman who can read, that they cannot do without her.”
The Ran (14th cent. Toledo) commenting upon the Alfas on Megillah 23a. notes that “All may ascend …for the seven honors; This means to complete the number of seven but not that -all- may be minors or women.” Thus, women may read the Torah, but some aliyot have to be reserved for men. Rabbi Akiba Eger (18/19th cent. Posen) writes that women may ascend for an aliyah on Shabbat or the festivals.
In 1975, the chairman of the Committee on Jewish Law and Standards (CJLS), of the Conservative movement’s Rabbinical Assembly, wrote that congregations that do not count women to the minyan may nevertheless call women to the Torah, as the two issues are distinct.
In recent years the same rabbinic texts have been cited with similar reasoning in growing segments of the Modern Orthodox community have come to the same conclusion. See the papers on this subject in the 2nd issue of “The Edah Journal” (Volume 1, issue 2, Sivan 5761) by Rabbi Mendel Shapiro and Rabbi Yehudah Herzl Henkin.