I agree that teachers should encourage critical thinking skills and how to solve problems without just memorizing facts and formulas but rather learn to interpret and solve problems with their own reasoning and like you said "learning to think".
I love your comment 'We should shed the notion of the "one-size-fits-all" approach to teaching because students are not clones of each other: What works well for one student may not work well for another student. Maybe some of those turned off by traditional curricula might like math better because they have more options that now appeal to them or simply because the traditional curricula was taught to them in these bad ways."
I recently watched a video clip of a math teacher name Dan Meyer. He made some very eye opening comments about the way we teach math these days. It is not a "one-size-fits-all" curricula. Why would it be? We clearly know that students learn differently yet we are still forced to teach in ways that doesn't recognize this. In his video, Dan suggests that as teachers we should be less helpful to our students. Stop giving them all of the information and encourage them to plug it into a formula to come up with an answer but instead include them in the formulation of the problem. We should give them the "bare essentials" and let them go from there. Allow the students to make mistakes and learn from them, to come up with their own formulas and reasoning.
Students may start to see math in a different way when we as teachers stop doing the "textbook" way of teaching math and start bringing it to life! As Dan put it in his presentation, math makes sense of our world, it is the vocabulary for our intuition"