Pam,
In a real way it would be a disservice to you to mention any specific names unless anyone here can identify current 2015 school situations. I say that because very often what was a good school in (say) 2014 has since gone under new owners/management and may already be a big problem school for foreign teachers.
But i say this seriously and with no exaggeration - if you are in downtown Shanghai you can literally find some kind of school/training centre on every block. I'm not exaggerating when I say a kindergarten or something is probably in view from your hotel window. Its not exaggeration you can find a Starbucks and strike up a convo with a few Westerners and walk out with directions/recommendations to 4 different schools in walking distance and contact numbers.
I dont actually know any good schools in Shanghai (never been there) but even if I did I'd wonder if they 1. still existed, 2. were taken over by horrible managers/new owners (and thats really common in China btw) or 3) 5 newer, bigger and better ones haven't popped up down the street from the one I knew.
Don't even worry - you arrive there with zero leads. In a week you have only one real problem: Trying to narrow it down to the 10 choices worth examining further and which of the 5 offers you should consider signing for.
Thank you. Do you know any good schools in downtown Shanghai that I can contact? Or just their names and I can google them.
Thanks!