my question is in the story of Rabban Gamlieil and Rabbi Yehoshua my students seem to feel that Rabban Gamleil is not being understanding to Rabbi Yehoshua points of view. I explained that the Bet Din must follow one law not two. They were okay with that. Then Rabban Gamlieil is removed from office and he apologizes to rabbi Yehoshua. And now he will reinstated as the Nasi. But why does the the Gemera think that he will not take a 50/50 split of the work? Why give rabban G. 3 weeks and REB"A one week?
Steven Penn, Teaneck, NJ
1) Your explanation about having one Beis Din and one ruling is correct. You should have a look at "Yesod ha'Mishnah v'Arichasah," Rav Reuven Margulies, Birurim #8,9. He has many wonderful insights into the Sugya. Among them, he points out that Raban Gamliel leaned towards Beis Shamai. (This was before the Bas kol that announced to rule like Beis Hillel always.)
Aside from what he writes, see Shabbos 30, and note the difference between the way Shamai and Hillel treated the would-be Gerim; Shamai believed in seeking sincerity, "Tocho k'Baro," before teaching a pupil. That is why they charged an entrance fee to the Beis Midrash (causing Hillel to once become buried in snow on the skylight). Raban Gamliel practiced the trait of "toughness" that Shamai taught for the same reason. He wanted to make sure that the Talmidim really came in order to learn l'Shem Shamayim. See Rashi top of Sanhedrin 11b.
(Besides this, it might have been necessary at that time period to emphasize the power of a Rabbinic ruling and that a person cannot choose to ignore the ruling and make his own ruling, since there was a major rift in the Mesorah at the time due to the two conflicting schools of Shamai and Hillel, see Rashi Bava Metzia top of 33b. That is why the Bas Kol that ruled like Beis Hillel "b'Chol Makom" eventually was necessary, in fact.)
The Talmidim favored Beis Hillel's approach, correctly assessing that in order to perpetuate the Torah in our day and age (that is, from their day and age to ours) we need to accept all to the Beis Midrash, and to show humbleness to the Talmidim. They will learn to respect the Torah nonetheless. That is why they ousted Raban Gamliel. (Rav Margulies points out that this explains why they did not consider giving the position to Rebbi Tarfon or Rebbi Eliezer, who were also alive at the time and undoubtedly were Gedolei ha'Dor. They, too, were from Shamai's Talmidim.)
2) Rav Margulies has an excellent explanation for why Rebbi Yehoshua denied having ruled contrary to Raban Gamliel, both in Berachos and in Bechoros. He also answers your question about why a 50-50 split wouldn't do Raban Gamliel justice. He shows that Rebbi Elazar ben Azaryah was only appointed "Reish Mesivta," not Nasi. Basically, they took one of the positions normally given to the Nasi, and appointed a separate person in his place. That is why he is never called "Raban" Elazar ben Azaryah, and that is why taking it from Raban gamliel did not involve "Ma'alin... v'Lo Moridin." Also, since Raban Gamliel was the real leader, he had to be given more honor than Rebbi Elazar ben Azaryah.
3) Why three weeks to one, and not two weeks to one (see marginal note in the Vilna Shas)? Explains Rav Margulies, because the honor of speaking was only given out during the "month of Kalah", when the nation would throng to hear the Rosh Yeshiva speak about the Chag, see Rashi Berachos 17b DH Shevacha. There are four Shabasos in a month. If you can't split them two-two, you've got to split them three-one!
Anyhow, that should answer some of your questions. Be well and l'Hitra'ot!
-Rabbi Kornfeld