Mott insulator at contacts

On the drive to Denver from Aspen, my wife (nonphysicist) asked me to explain Mott insulator --- and she questions the standard explanation of insulating state caused by strong (on-site) repulsive interaction -- how could it work if the Mott insulator is allowed to have contact with electrodes, if and one electron at one end of the Mott insulator is pulled (by e-field) away through the contact, creating a vacancy for the next electron to hop, and so on and won't be insulating. I found I can't explain this well and I don't really understand Mott insulator! Need to think more about it...