It would be very nice to add additional local notifications in the app.
For example, we can have a background process that for every course check if they are new grades, new events in the calendar, course contents or if your completion status criteria has changed since your last access/check.
We don't need external services for doing that, we can just do the request not using the cache and comparing the result with the cached ones. If the hash/content is different we can alert the user (we can even implement this as an interfice/hook in every course plugin so each plugin is responsible of checking this).
In the case of grades for example, we can display the grade that has change.