A mat by any other name is a carpet, a rug, flooring, a cover up or something to walk or wipe your feet on. In a Montessori classroom, the table and the mat defines a student’s 
workspace. The mat is respected and walked around. It is meticulously rolled out, and rolled up. It is picked up and carried with two hands. It is an ingrained exercise in grace and courtesy.
It is as much a work material as the lessons given on it. There are lessons on how to properly roll and unroll a mat. A 3-year old progresses from pushing it open to rolling it open. A 6-year old can roll a mat so tight it stands up.
A lesson is given on how to walk around a mat. Heel to toe. First you learn to walk around a single mat, then several- without touching them. It becomes an unspoken rule. We treat work with respect, so the work on the mat is treated respectfully.
An indirect aim of the work mat is muscular control of fingers and hands, independence, and concentration. A parent once said, “My child is rolling every rug in the house!”
Yes. That happens too!