Is that actually true? The entire literature on cultural cognition, and the myside bias, and all the other failures of rationality seems to point to it being pretty common for people to disagree irrationally. The advice given here is rather one-sided: one must assume the other person disagrees rationally. That's a huge assumption that is sometimes, maybe often, not justified, yet it's not mentioned. Furthermore, the second part of your comment about articulating their own argument is not related to the first, it's independent of the rationality and quality of their argument.
Often people are not as irrational as you'd think. It's just they only connect the dots one or two at a time. So, much of the knowledge they already have is not questioned. More often than not people appear to be irrational because they have learned something that is either false or perhaps true but very different to a false belief you hold. They appear irrational when assessing the same situation as you but arriving at different conclusions, because they are holding to a very different set of assumptions. People (my self included) are not very good at being aware of our underlying assumptions.
I guess that is being irrational given that you are not taking all the assumptions into account, but it's the level of rationality that most people operate at.
However if your talking about an emotionally charged situation, yeah rationality be damned emotions are running that show.
They may disagree irrationally, but they still disagree with causation. This is basically saying; figure out what causes them to believe what they believe, acknowledge the validity of that causal chain, and then introduce new information.
The new information could be about being less wrong (updating the rules of their causality) or it could be a new situation on which to apply their existing ways of thinking.
