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Scholarship on women in the medieval Islamic world is still in its infancy, and this is going to affect my answer in two ways beyond the immediately obvious. First, the subfield hasn't had its "Holy Feast and Holy Fast" moment: that is, we haven't had a focused study on how to interpret the bulk of sources on medieval Muslim women that survive, especially biographical entries. We're left trying to adapt methodologies from studies of Latin women to apply to parallel yet different types of sources like legal and Church/mosque records.

Second, we haven't had the chance to see how a more comprehensive knowledge of Muslim women's activities affects the overall understanding of medieval dar al-Islam--essentially, we have to try to fit women into a historical narrative defined by and consisting almost solely of male perspectives and retrospectives. (A comparable example: scholars used to talk about the Western twelfth century renaissance as a Paris/northern France/maaybe a little English phenomenon, but mostly Paris. When we examine women's literary and epistolary activity of the time, suddenly this vast intellectual network spanning western Europe emerges. Paris might be a nexus, especially in developing scholastic theological method, but it's not the whole story or even most of it.) I'm going to try to shift the paradigms a little in this answer as an indication of the potential for future scholarship.

By "golden age of Islam," I take to mean in particular the intellectual and cultural life of the Abbasid caliphate in the east and the Cordoba caliphate emerging from the Umayyads in the west, ranging into Fatimid courts across North Africa and even Muslims' participation at Christian courts in Sicily and Iberia. That is, I won't be looking at women whose activities we might consider primarily political/ruling.

One emerging and very profitable strand of research has been religious patronage. Since women in Islamic law control their own finances, we have abundant records of donations made by women/under women's own names to religious foundations. Women patronized and founded hospitals, mosques, and madrasas. They were also major benefactors of the hostels and other support infrastructure along the pilgrimage routes from Syria and Baghdad to Mecca.

From biographical dictionaries, we also get a glimpse of women religious teachers. It's not an accident that illuminations of the great Persian/Arabic romance Layla and Majnun show Layla in madrasa right beside her future lover. Women taught groups of students in mainstream schools; it also seems likely that Sufi mysticism provided a path for women to exercise religious instructional authority. Certainly the poet Rabi'a (eighth century) developed a massive hagiographical tradition of hosting and instructing students, male and female alike. And for famous ascetics like Rabi'a, "teaching" was not limited to formal book instruction and those with the time/money/freedom to acquire it. As in Latin and Greek Christianity, Sufi saints were popular tourist attractions and counselors.

What we don't have, unfortunately, is a canon of texts written by these women teachers, with the exception of some verses by Rabi'a and apocryphal texts attributed to her, Lalla, and a couple others whose provenance is uncertain. Although women poets in the Jahiliyya era of pre/early-Islam Arabic has been lovingly preserved since then for guidance into the nature of Quranic Arabic, no parallel grace or exception to the rule that "men write theology" was extended to these women or to compilations of their teachings gathered by others.

However, there is one relatively sizeable and almost entirely untapped storehouse of medieval Muslim women's writing: the poetry and songs of the so-called singing girls, slaves and manumitted slaves turned noblewomen at the elite courts in Baghdad and Cordoba. To understand the significance of enslaved and emancipated women's participation in musical and intellectual life, it's important to say a few words about that environment.

We have a tendency to pull scientific/mathematical/related developments out of their historical context. Science is important because, well, it's science! Tautological. Self-evident. But as current political climates show, infrastructure for scholarship is essential for its conduct, and is anything but inevitable. In fact, in the Middle Ages, scholarship in what we might call the proto-sciences--theoretical math, astronomy/astrology, alchemy/metallurgy, medicine/anatomy--became a crown jewel of rulers' prestige! Support for scholars was as much a demonstration of power as maintaining a menagerie of the most exotic animals--the fiercest beasts, the rarest creatures.

And this situation opened up a sliver of a window for a few women to dazzle their contemporaries and leave legacies of themselves, if not always their original teaching. In tenth-century Cordoba, al-Hakam II invited an exceptionally renowned astronomer/mathematician to come teach a handpicked student: an enslaved woman whose brilliance was recognized even above a court of women (qiyan, "singing-girls" in Anglophone useage) educated in philosophy, literature, and mathematics. Sure enough, the biographical encyclopedia entry tells us, the anonymous katiba soaked up the new instruction. Within three years she had mastered astronomy and was apparently demonstrating how to use an astrolabe back at al-Hakam's court.

This story, I think, shows very well the possibilities and limitations on women in the course of helping understand Cordoban court life more broadly. Scholarship was not an end in and of itself, nor only a means of scientific "advancement." It was a rich part of everyday culture and a vital tool in the display and exercise of power. A few women were not excluded from this life in their own day.