You are arriving in winter, some public schools inside Capital that don't even have functioning heat. The ceilings are often falling down, and the floors are barren, the buildings are in disrepair. Most just don't even have close to the equipment and facilities that you would see in a public school in the States. We say that the public system in the States is broken, after seeing the system here you will never say that again.
There are some decent public schools in certain barrios, so be careful with your decisions. Do you have any local contacts in BA that can help you out? Unfortunately I can't help you with that since all the Argentines I know send their kids to private schools. My boyfriend went to a public school but that was 18 years ago now and he said he'd never send his kids to one these days. Most foreigners choose private institutes so that their kids can participate in a bilingual or trilingual programme.
There are some private institutes that function on the American school calendar, but most run on the Argentine calendar which means they will be arriving mid-year.
Sorry to completely slam the public school system here, but it is in a sad state of neglect. If more people paid their taxes, and if those taxes were distributed legitimately, maybe they'd function better.
The teachers are very dedicated considering their ridiculously low salaries, but there have also been innumerable strikes, so be aware that a public school could be closed at numerous points during the year with no notice.
Private schools on the other hand, are not regulated, so they are able to increase their tuition fees without warning at any point during the year. So when you sign on do consider that your monthly fees could jump 20% or more at any moment.