Alhamdulillah, our kids did fasting sunnah in the tenth of Muharram. They reminded me one day before the tenth of Muharram that they wanted to fasting. They learned and practiced our religion in daily life.
The Messenger of Allah (sallallahu alayhi wasallam) has been reported as saying when asked which prayer is better after obligatory prayer: "Prayer in the middle of the night." He was asked again which fast is better after Ramadan. He said: "In the month that is called Al-Muharram."
(Ahmed, Muslim, and Abu Dawud)
The Prophet (sallallahu alayhi wasallam) however, emphasizes the fast of the tenth day, 'Ashura. The word "`Ashura" is derived from `Ashara, which means ten. The observation of this day goes back to Prophet Musa bin 'Emran (alayhi sallam).
In a hadith related by Ibn Abbas (radiallah anhu) when the Messenger of Allah (sallallhu alayhi wasallam) came to Medina, he found the Jews fasting the day of 'Ashura. He inquired why do they did so. They replied that it was a good day, the day which Allah delivered Prophet Musa (alayhi sallam) and the children of Israel from their enemy. Prophet Musa (alayhi sallam) fasted that day.
The Prophet (sallallahu alayhi wasallam) responded: "I have more right to Musa than you." He fasted the day and commanded the believers to fast." (Agreed upon.)
May Allah accept our ibadaah. Ameen.