How to make breadcumbs grab taxonomy terms and use tokens
First of all if you have not tried Taxonomy Breadcrumb module yet, give it a try.

If you already have tried that module and had no success , you can write your own code. 

the simplest solution can be achieved by implementing hook_preprocess_page, then you should check if there is a node and node is of a certain type used to store school type and then you can change $variables['breadcrumb'] to modify breadcrumb as you wish.

in drupal 7 you also can use drupal_set_breadcrumb to change breadcrumb in a code and in drupal 8 you can implement hook_system_breadcrumb_alter to make it done.