Volume 3, Book 32, Number 228:
Narrated 'Aisha (radiallaahu `anhaa):
Allah's Apostle (sallallaahu `alayhi wasallam) used to pray (at night) in Ramadan.
Volume 3, Book 32, Number 229:
That he was informed by 'Aisha (radiallaahu `anhaa), "Allah's Apostle (sallallaahu `alayhi wasallam) went out in the middle of the night and prayed in the mosque and some men prayed behind him. In the morning, the people spoke about it and then a large number of them gathered and prayed behind him (on the second night). In the next morning the people again talked about it and on the third night the mosque was full with a large number of people. Allah's Apostle came out and the people prayed behind him. On the fourth night the Mosque was overwhelmed with people and could not accommodate them, but the Prophet (sallallaahu `alayhi wasallam) came out (only) for the morning prayer. When the morning prayer was finished he recited Tashah-hud and (addressing the people) said, "Amma ba'du, your presence was not hidden from me but I was afraid lest the night prayer (Qiyam) should be enjoined on you and you might not be able to carry it on." So, Allah's Apostle (sallallaahu `alayhi wasallam) died and the situation remained like that (i.e. people prayed individually). "
This indicates that praying Taraweeh in congregation is prescribed according to the Sunnah of the Prophet (peace and blessings of Allaah be upon him), but he refrained from doing it because he feared that it would be made obligatory for the ummah. When the Prophet (peace and blessings of Allaah be upon him) died, this reservation was no longer required, because the sharee’ah was established.
Volume 3, Book 32, Number 230:
Narrated Abu Salama bin 'Abdur Rahman:
that he asked 'Aisha (radiallaahu `anhaa) "How was the prayer of Allah's Apostle (sallallaahu `alayhi wasallam) in Ramadan?" She replied, "He did not pray more than eleven Rakat in Ramadan or in any other month. He used to pray four Rakat ---- let alone their beauty and length----and then he would pray four ----let alone their beauty and length ----and then he would pray three Rakat (Witr)." She added, "I asked, 'O Allah's Apostle (sallallaahu `alayhi wasallam)! Do you sleep before praying the Witr?' He replied, 'O 'Aisha! My eyes sleep but my heart does not sleep."
It is proven in al-Saheeh that the Prophet (peace and blessings of Allaah be upon him) used to pray eleven rak’ahs at night in Ramadaan and at other times, but he would make the recitation and other essential parts of the prayer so long that on one occasion he recited more than five juz’ (sipara) in a single rak’ah, reciting slowly and carefully.
And it is proven that he would get up halfway through the night, or shortly before or after that, and he would carry on praying until it was close to dawn, so he would pray thirteen rak’ahs in approximately five hours. This requires making the recitation and other essential parts of the prayer lengthy.
Note: the main focus of our night prayer (inside or outside Ramadan) should be to attain tranquility and to develop a special connection with Allah. There's no point in fighting about the number of rak`ahs of the Taraweeh prayer and there's no point in discussing what Umar (radiallaahu `anhu) did in his time and why he did it, if our night prayer does not bring us any closer towards Allah. Yes, we want to follow the Sunnah when it comes to the number of rak`ahs, but are we going to sideline the Sunnah of making the night prayer lengthy and slow? The latter is definitely more important, won't you agree?