ok -we're not going to see eye to eye on this but no matter. I agree with your 'feel' statements. I agree that computers/calcuation devices can back up or extend this 'feel'. Yes, they can't establish the basic concepts but no, the use of them does not automatically imply a lack of conceptual understanding. I still say use of a calculator does not mean a lack of understanding.
Mental numeric manipulation MAY lead to other thought processes but not always. Indeed there are many that are proficient at mental number calculations but are not able to generalise or go to the next stage of mathematical development.
Now, about that comment of yours that manipulating numbers is 'less than abstract'
Oh and still don't fathom how political leanings, political science or any of the other stuff is related to this at all.