This is a convenient dichotomy for that part of all of us that wishes to think ourselves “critical thinkers”, but not at the cost of looking carefully at the me

This is a convenient dichotomy for that part of all of us that wishes to think ourselves “critical thinkers”, but not at the cost of looking carefully at the meaning of words. Math and Science are the two fields whose understanding requires this. This is the first and most important of the requirements of Math and Science: know, as precisely and accurately as you can, what you mean. Simple to express, but difficult to achieve in practice. It takes unrelenting effort. I know that for myself, all my instincts conspire against it. It is much easier for me to think I am a “critical thinker” because I can criticize what some other people think, without the discomfort of examining my own preconceptions and assumptions, or even getting a clear idea of what they are. And it is all the more easy with legions of academic departments willing and even eager to train me and support me in the practice of trading verbal tokens with my colleagues, and affirming my own notions about the folly of the notions of others. I suspect that is much of what happens with the liberal arts. I have some experience with the teachings in math education. Here, the liberal art of teaching and the hard science of mathematics meet. In my experience, the hardness of hard science is always softened in the service of “education”; “education”, which is split into factions with zealous supporters, wins by force. Here's how this plays out: someone observes that, say, arithmetic with integers can be modeled with, say, tiles with different colors on each side. They create a lesson to help students get a conceptual understanding, using integers of small absolute value, to get avoid unnecessary details in the computations. Than an educator “realizes”: “hey, as long as they ‘get the idea’, well, why continue with practice in difficult computations when they already ‘get it’ with the tiles, and the tiles are so much FUN?” Training and practice in more involved computations is dropped, along with practice in a lot of the trickier details of this new kind of computation. They can use calculators, anyway. The learning becomes, at best, almost exclusively qualitative, and those students who complain that the class is become boring and unchallenging are labeled malcontents---err, “resistant to modern teaching methods”. Later, high school algebra students need calculators to calculate “-3 * 4”, and a high school algebra teacher must reteach arithmetic, perhaps doing some “conceptual” lessons of his own, and the progress of learning is retarded. Students abandon this “math” in frustration. The DOE demands more money to solve the education crisis in America. Rinse. Repeat.