Yes. So in that case, your menu is in an editable region. Normally that is not a good idea because it means that changes to the menu must be done individually to each page. The benefit of using a template for your pages is that you avoid that situation.
To remedy this situation, you would need to edit your template to move the menu code OUTSIDE of the editable region where it is now found. Once it is located in the non-editable part of the template, changes to the menu will automatically propagate to all child pages. Make sense?