philosopher > you will follow his minhugim and he’ll have the last word which moisdos your sons will attend,
shtark> literal definition of ימשל is to lord
While there is a notion that a man should follow his wife on the issues of gashmiyus and lead on ruchniyus, it does not mean that someone is having a “last word” on school selection, it is usually a mutual decision based on a lot of considerations. Just based on stereotypes, the husband might naturally know more about the derech of the mosdos, but the mother might know more about what really goes on inside based on talking to other mothers; might have better insight in the nature of the children and how they’ll be affected; and better intuition on how to evaluate personalities of teachers and principals. And, as mentioned above, stereotypes are not always true. Hopefully, both sides will bring all their info for mutual discussion.
As to leading, I once was zoche to be present at a summer retreat where a choshuve very traditional rav was giving a shiur sitting at the head of the outside table to several students, sitting on both sides. Rebbetzin was sitting at the end of the table, reading what looked like a paperback novel. At some point, Rav stopped speaking, search for a right word…. Rebbetzin lowered the novel, mouthed the word, Rav looked at her, and continued the shiur. Nobody noticed that, of course – I just happened to be coming from a side and seeing the whole scene.