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are they motivated by pieces of coloured cloth or by having passed a test of competency?
let's make it simple, let's say there are two students attending the same class, they started at much the same time, they have passed the same grades, but one, let's call him jobo, doesn't wear his grade round his waist. whilst the other one, , let's call him JR does, what benefit does Jr get from wearing a belt, that jobo misses out on ?
I can't answer for how nor why others are motivated by belts. I've heard a lot of different answers from a lot of different people. Remember... just because something doesn't make sense to you doesn't mean it doesn't make sense to someone or anyone else.
The physical belt doesn't motivate me. Improvement motivates me. When I get a new shiny belt, it's definitely because I improved. Otherwise I either wouldn't get a new one or I would've skipped the previous one. So if that means I'm still motivated by new belts, then so be it. If there wasn't a belt but the process was exactly the same, it wouldn't change my motivation one bit. Saying "I'm a brown belt" is quicker and easier than saying "I'm a level X student." It's easier for a teacher I'm unfamiliar with to see my belt than it is to ask me what level student I am and have me explain what I know and don't know. So belts make my life easier.
If your hypothetical "Jobo" didn't wear his belt, but hypothetical "JR" wore his...
In the hypothetical small dojo the hypothetical "JR" and "Jobo" train at together, nothing. Once they step out of that hypothetical dojo and into an affiliated one, hypothetical "Jobo" would be given a white belt, be made to practice solely with the white belts, and would only work on white belt syllabus material.
In the real dojo I train at, if you decided you didn't want to wear the belt my teacher gave you (and approved gi), you would be shown the door. The rules are the rules, if you can't follow them for whatever reason, leave. No hard feelings, it's just not the right fit for you. We've got a way of doing things, and you'd be told those ways before you joined. If you can follow the rules, welcome aboard; if not, there's other places that have different rules and will gladly welcome you. They'll hopefully welcome you, anyway.
People from different organizations and styles have inquired about joining our dojo and asked if they can wear a belt from their previous/other school. The answer is always no. A guest would be allowed to wear whatever they want, a formal student isn't.