I'm really surprised that the teachers have given up - he so obviously has SOME kind of disorder, and (no slam on overworked public school teachers) I wouldn't expect private school teachers to give up so easily and pack him off to a "special school" without at least trying to get a diagnosis so you know what methods work with such kids.
Again, it might be worth talking to his past teachers at the Montessori school to see what his behavior was there and, if he did have these problems, how they dealt with them. If he knows the material and is doing the problems correctly in his head but just won't write them down, I would hate to see him shunted aside just because the school didn't have the time, flexibility, or inclination to figure out how to work with his particular idiosyncracies. To start with, try the computer option - see if you can just get him to type in the answers, or tell them to you so you can write them down. I know that won't work as a long-term solution, but it'll help you see if he's getting the answers right so you can focus your attention on what the real problem is: is it his ability to do the work, or is it his ability to write things down? One you are trained to deal with, the other needs to be treated by someone who knows what the underlying problem is.
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"If ten million people believe a foolish thing, it is still a foolish thing."
- Anatole France
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