I also don't have a lot of respect for EM and other reform/constructivist methods. They're all supposed to teach mathematical concepts, but I think they fail. I understand that many teachers and textbooks don't do a good job of explaining why a standard algorithm like long division works, which puts kids into a position of having to memorize the procedure without understanding it. This is not good. But the solution is to
explain why the standard methods work, not just jettison them in favor of other stuff.
From what I've read today, the Bridges program is in this group.
This page reviews it briefly and mentions some of the things people criticize about EM, such as lack of emphasis on standard algorithms, group work, etc. The US Department of Education says that
none of the studies of Bridges to Mathematics are scientifically sound. This means that the Dep't. of Ed. couldn't come to any conclusions about the effectiveness of the program. IMO, this is a huge red flag, especially because this was 0/18 studies, plus 5 more that didn't even meet their standards for examining them.
yells at me that he's "not allowed to do it that way" when I tell him to just do the problem the old way if that works better for him.
I think you have two problems here. If I'm right, teasing them apart might help you solve one or both of them.
The first problem is a bad curriculum. The second one is your son's reaction to it.
I think that one of my parental duties is to teach my kids that sometimes stuff is wrong, even stuff in school. I took this approach with EM with my eldest, and I do the same with my youngest when she gets bogus homework questions. Unless you leave the school or negotiate independent study, this is a way around the bad curriculum.
If my kids were to complain that teacher says I have to do it this way, it's time for me to have a talk with teacher. A good teacher will accept that any correct route to the answer is reasonable. If the teacher won't budge, I might talk to the principal or find a way to explain to my kids that sometimes some ideas are wrong and/or that if the standard methods make the most sense to them, they should use them. I have very strong opinions about the failings of reform/constructivist mathematics, though, so bear this in mind when I say that I wouldn't let a badly designed approach to math damage my kids.