I am in high school and in the morning my principal makes announcements before אשרי at the end of שחרית. The announcements often have absolutely nothing to do with davening. For example, he will tell us the scores of a game that the school’s basketball team played recently. I know there is no halachik issue of hefsek at this point in davening, but it still doesn’t seem right to interrupt שחרית to tell us this information. In elementary school, I don't thinks the announcements were as unrelated to davening as they are now, but they were also given before אשרי, which leads me to believe that it is accepted to do this. Why?
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There is precedent to interrupt mid-davening for a mitzva or for needs of the community. These announcements were actually made right after Yishtabach. Examples in the Shulchan Aruch and Rema (54:3) include:
Our custom is not to interrupt after Yishtabach, but there are many places that do so before ashrei for important matters if they are concerned some might leave.
Your concern was also with interrupting for matters such as basketball scores. I can only think of 2 possibilities:
Not-so-good: The announcements started as proper, but sometimes we forget the "heter" and need to be reminded. If this is true, don't be embarrassed to ask the announcer in a respectful way why he does this.
Better: We all know how much high school students enjoy davening ;) Perhaps the announcer (or whoever sanctioned the announcement) is trying to pump up some adrenalin and hoping to channel it to the rest of davening, but feels that that would be inappropriate prior to davening.
Let me list the issues with this that you may want to bring up.
1) The Shulhan Aruch (28:1 37:2) implies clearly that you are not allowed to be Mesiah Daat (lose focus) from the Tefilin for even one second.
2) Well this is more of a Humra: the Arizal wouldn't even talk Divre Mussar or anything in the Bet HaKeneset because he suspected he might speak things that aren't related to Torah.
3) This is for those who recite Leshem Yihud: Kaf HaHaim Palachi (as well as the Zohar) warn very much against talking in the middle of an action if one recited Leshem Yihud. The Zohar even says that one who stops in the middle of the action he made Leshem Yihud on Has WeShalom his life will stop (Rahmana Lesilan).
4) Rebbe Nachman of Breslev (this source is also from the Zohar and Arizal seen in Orhot Sion) states that one should avoid speaking at all from waking until the end of Shacharis.