Dr. Muzammil Siddiqi, former president of the Islamic Society of North America, states the following:
Fasting six days of the month of Shawwal is mentioned in a Prophetic hadith. The Prophet (peace and blessings be upon him) is reported to have said: "Whosoever fasted in Ramadan and then followed up with six fasts days of Shawwal, it is like fasting every day." (Reported by Muslim)
The meaning is that the reward is like the reward of a person who is always in fast every day of his/her life.
It is highly recommended to fast six days of the month of Shawwal, but it is not obligatory. Those who want to fast can fast after `Eid Al-Fitr any six days during Shawwal.
It is not required to fast six days continuously without any interruption. One can fast according to convenience any time during the month.