You haven't mentioned what release of edubuntu as there would be major differences in how the default profile can be adjusted since unity first showed to now.
What shows in the Default launcher is determined by the key in
If opened in a text editor (root to edit) you'll see the key/schema -
<schema path="/com/canonical/unity/launcher/" id="com.canonical.Unity.Launcher" gettext-domain="unity">
Listed below between the <default> .... <default> are the default launcher icons.
You can carefully edit that list though a couple I wouldn't remove like 'unity://expo-icon'
You'd need to carefully look at how the key is written when adding new .desktops,(if you are), follow exactly & I'd advise backing up the orig. file 1st.
When done you'd run this command to set the schema
To return to the orig if needed - edit the file back using the backup as a guide or just replace your edited file with the backup.
sudo glib-compile-schemas /usr/share/glib-2.0/schemas
Then run the above command again to set.