There, I said it.
Why? Basically because it makes the same error as we have seen with teaching writing: it imagines that "thinking" is a kind of universal activity or general skill. And while there is a level of commonality to all human thought and perhaps even some common cognitive reasoning capacity that is shared among all academic endeavors, that commonality, like the commonality we see among writing activities, is quite basic. Instead, I would suggest (and I don't think I'm alone here) that cognition is a systemic activity (distributed even) and that the "thinking skills" (critical or otherwise) required of students in different disciplines (and later in their professions) are situated and are not easily practiced outside of those contexts. In short the teaching of "critical thinking" faces that same kinds of knowledge transfer problems as writing instruction.
When someone tells me they teach critical thinking, my first inclination is to ask them "as opposed to what? Your colleague in the next building who teaches uncriical thinking?" In the humanities and English Studies in particular, critical thinking generally means "close reading," and close reading is really a set of disciplinary-specific skills for the interpretation, analysis, and evaluation of texts. Even these are not universal inasmuch as the skills that are required for literary interpretation are quite different from the demands of close reading in rhetoric, philosophy, etc. Each discipline has its own close reading practices as it does its own rhetorical practices for writing. If there is a meta-cognitive element to the evaluation of texts (in any field) it probably is something akin to basic rhetorical principles: purpose, audience, modes of argument, evidence, structure/organization, and so on. That is, all academic texts have these things. Personally I don't believe that "everything is an argument," so I won't say all texts have arguments, but I think it is fair to say that academic, scholarly texts make claims/arguments.
Now you could look at what I just said and think that I am arguing that rhetoric=critical thinking, but that's not my point at all. Because when it comes down to it, knowing that every text has an audience doesn't get you very far. It tells you that you need to figure out who this text is addressing (and why) but until you know the audience and so on, you aren't in a position to evaluate the text. We see this problem commonly in composition classrooms where there is a greater tendency to read across disciplines than in many other classes. A composition instructor, trained in humanistic critical thinking, might be deeply suspicious of social scientific, quantitative research or (just as problematic) they might accept it at face value. The problem is that we don't know enough to evaluate the merits of the argument. That's ok, of course. We can't expect to be able to do that. That is, we can't expect that a knowledge of rhetoric would give us broad critical thinking powers. It just tells us the kinds of questions that we need to answer when evaluating communication.
And needless to say (I hope), critical thinking, if such a thing existed, would extend far beyond the evaluation of texts. Math comes to mind, for example.
So why do I make this observation? Because the focus on "critical thinking" in general education, and even more broadly in our public discourse on learning, is something of a boondoggle. Should we teach our students to be skeptical, to not immediately accept the things they are told? Yes. We shouldn't accept the conclusions of an argument without being offered some convincing evidence. Sometimes the evidence is "I am an expert and you have to trust me." Maybe, but then convince me of your expertise, I guess, and I am still going to be skeptical. Instead, it might be helpful to teach students how knowledge systems work in some general sense but also to help them investigate specific knowledge systems, which is what general education does in an introductory way… I would hope or at least it could.
But "critical thinking"? Not so much.