The schools of thought of Shammai and Hillel are known for their vigorous debates throughout the Talmud. Apart from a select few instances,
Beis Shammai, the school of Shammai, is generally stricter in its interpretation and application of the law, while Beis Hillel is more prone to leniency. When it comes to their debate concerning the pattern of lighting the Chanukah lamps—whether to kindle eight lights on the first night and gradually decrease (Beis Shammai), or to kindle one on the first night and gradually increase (Beis Hillel), ultimately the same number of lamps are kindled regardless. Nevertheless, here too Beis Shammai’s opinion is more “demanding” than Beis Hillel’s.
The Talmud explains the debate (according to one answer) as a question of whether the Chanukah lights should demonstrate how many days of Chanukah have already passed, or how many are still to come. Taken a step further, this debate can be said to reflect our Sages’ perspective on the original miracle of Chanukah, when a cruse of oil sufficient for only one day burned for eight days instead. On the one hand, we could say that the miracle grew from day to day with the oil burning yet another day. On the other hand, when considering the miracle after the fact, we could say that all eight days were contained in that single jug of oil from the get-go. From that perspective, the amount of days of “miraculous potential” contained in that single cruse was greatest on the first day, gradually diminishing as the days passed and fewer miracle days were necessary.
Therein lies the “leniency” of Beis Hillel and the “stringency” of Beis Shammai. The average person, argued Beis Hillel, can only be expected to acknowledge and give public thanks for G-d’s miracles once they have visibly materialized. As such, Beis Hillel recommended lighting the menorah in a way that reflects how each additional step was seen as the miracle unfolded. Beis Shammai, however, demanded that the average person recognize G-d’s miracles even when they are still in potential form and have not yet fully materialized.
--Likkutei Sichos, vol. 30, p. 301
