How to allow users to edit their submissions and allow them to add information?

Can the students accessing the forms somehow store what they start and add to it during a month?

You may allow the user's to edit their submissions alter on, this way they can add information if needed, this guide will help you doing so: https://www.jotform.com/help/40-How-to-Let-Users-Update-Their-Form-Submissions-at-a-Later-Date 

If you have questions, let us know.