One of the marks of a great person is the ability to change one’s mind. While such may end the career of a politician[1] it demonstrates intellectual maturity. The willingness to integrate new material and ideas into our worldview brings vitality and increased creativity to the issues at hand. 
Openness to other ideas especially those of one’s frequent “opponents” is far from common. It may be uncommon but it is to be treasured. It is the key reason that Jewish tradition follows the view of Beit Hillel in the over three hundred disputes with Beit Shammai. The Gemara actually notes that the students of Shammai were sharper than those of Hillel. However that sharpness came with an unflinching conviction that they were correct and other views could be ignored. Such a view not only displays a form of arrogance, it makes mistakes much more likely. Those open to the viewpoints of others are not only “nicer” they are capable of seeing the issues in a much broader context and are thus more likely to be correct. The more one thinks they may be making a mistake the more we can be confident they are not.
“Why was the halacha established according to Beit Hillel? Because they were kindly and modest, they studied their own rulings and those of Beth Shammai, and were even so  [humble] as to mention the teachings of Beit Shammai before theirs.” (Eiruvin 13b)

Studying the words of Beit Shammai was much more than an intellectual exercise. “One who is half a slave and half free works for his master one day and for himself one day; these are the words of Beit Hillel.” (Gittin 41a) Two people partner together to purchase a slave with the slave alternating his working days between his two masters. When one of his two masters frees him the half slave works every other day day for his remaining master with the other day off for himself. This is all fine and good except for an ancillary problem, one that may not concern a slave master or an employer but must concern society, that of work-life balance. In this instance the balance would be 100% work and 0% life.
“Beit Shammai said to them: ‘You have repaired his master, but himself you have not repaired. To marry a maidservant he cannot do for he is half-free. [To marry] a free woman he cannot do, for he is half-slave. And was not the world created for the sake of pru urevu, for our children as it says (Isaiah 45:18) "Not for emptiness did He create it [the world], but for settlement He formed it." 
This half-slave may be half free but being a half-slave he cannot marry - neither a slave nor a free person. The Mishna continues with the solution: “Rather, due to tikkun haolam, fixing the world, we force his master and he makes him a free man, and [the slave] writes a document [of debt] for half his value. Beit Hillel retracted and ruled in accordance with the words of Beit Shammai.”
A situation where one cannot marry brings destruction to the world. Fixing the world means ensuring all - even the slave - can marry, even if that means his master must free him[2]. Once Beit Shammai pointed that out Beit Hillel reversed course.
This Mishna originally appears in Masechet Eduyot as the first of three cases where Beit Hillel reversed course and ruled in accordance with the view of Beit Shammai. Tellingly we have no recorded cases of Beit Shammai changing their minds and ruling in accordance with the view of Beit Hillel.
Yet we must distinguish between Shammai himself and Beit Shammai, the school founded by his students. Mishnayot Eduyot, the first compilation of Mishnayot[3], begins by recording (the) three halachic arguments between Hillel and Shammai themselves. Yet in all three cases the Mishna rejects the views of Hillel and Shammai recording a third and decisive position. If such is the case “why do we mention the words of Hillel and Shammai for naught? To teach future generations the one should not [rigidly] stand on their words, for the fathers of the world did not stand on theirs.” (Eduyot 1:4)
This humility is all the more striking when one considers that in the third debate - that involving the kashrut of a Mikvah - Hillel and Shammai followed the testimony of “two water carriers from the dung gate”.
That Shammai is not to be equated with his followers Beit Shammai, becomes clear from the fact that this same opening chapter of Masechet Eduyot lists no less than five debates in which Shammai and Beit Shammai take different halachic positions[4]

The very first chapter of the Mishna, chronologically speaking, stresses the importance of being open to the view of others and the intellectual independence of students vis a vis their teachers. Following in their path is one of the ways of tikkun olam.
[1] I often heard Rav Schachter quote Rav Soloveitchik that politics is kulu sheker, complete falsehood.
[2] As I was writing these words I could not help but think of the “destruction of the world” caused by ex-husbands who refuse the grant their unable to be ex-wives a get, thereby keeping them enslaved. Freeing agunot literally fixes the world. 
[3] The Tosefta (Eduyot 1:1) describes how the Rabbis upon arriving at Yavneh, fearful of people “seeking words of Torah but unable to find them” began compiling a list of teachings “starting with Hillel and Shammai.” As the list of teachings is based on the author and not the topic eduyot is the most subject diverse of the mishanyot. Serving as the first compilation of what later became 63 masechtot most of these same mihnayot appear in the masechtot devoted to the particular topic at hand.  
[4] As far as I know this the only time that we are witness to such a phenomenon.