In our house, we love to tear the challah on Shabbat. My mother insists it is not dignified and that the challah should be sliced.
The rabbis have a different conflict.
Raba insists that for any loaf of bread, the benediction (HaMotzi) should be said first and then loaf is broken. Rabbi Hiyya ben Ashi says that the loaf should be broken as the benediction is concluded. So is the blessing concluded with a broken piece or with a whole loaf which is only subsequently broken? Some follow Raba, some follow R. Hiyya. All argue.
A compromise is described:
A Tanna [repeater of texts] recited in the presence of R. Nachman b. Isaac: One should place the broken piece under the whole loaf and then break and say the benediction. He said to him: What is your name? Shalman, he replied. He said to him: You are peace [shalom] and your Mishnah is faultless [shalem], for you have made peace between the scholars.