If the school has purchased the app (even for a specific student's use) on a school owned iPad, the licence belongs to the school and cannot be shared with any individuals. The child's family has to purchase a separate copy of the app, to be able to use the app at home.
However, the content created on the school iPad, can be transferred to the home iPad, via a backup, using Dropbox / iTunes / AirDrop.
Click here to read more : How to copy the content between two iPads?
Please click here to refer to Apple's End User Licence Agreement - EULA