Herman Rubin wrote (heavy snipping, since this is getting out of hand in length <<< However, it is quite possible for a person to know how, and be aware of this. When someone carries out routine calculations, or spells words, or uses mathematical concepts, while the use may be "automatic", this does not mean that it is not also conscious. Even if the person cannot describe it, it can be very conscious. >>>
I don't understand, then, what you mean by 'conscious'. <<< One should learn the concepts; details can come later. This is the approach opposite to what we are not taking. Learning the details first interferes with learning concepts, and even good mathematicians have trouble with this. >>> I can agree with this, mostly, but when should the details come? e.g., the concept of 'derivative', is, in one sense, very very easy: The slope of a curve. I think students could learn this idea very fast. I agree that it is important to get this idea across before teaching methods of differentiation. But WHEN should the details come? After learning the idea of derivative on day 1, should students learn integration on day 2, partial derivatives on day 3, and multiple integration on day 4, and THEN go back to learn details? And when should they learn about proofs? After all, concepts don't need to be proven; theorems need to be proven. <<< The approach is not automatically bad; it is rarely that good, and it certainly interferes with the student moving ahead in the subject. For those who think the current amount taught is what should be done, this might not matter, but it is a driving force in the dumbing down of our youth, and of dulling or even debilitating their brains. When one understands a concept, move on. Anyone who keeps a child back should have to pay heavily for the attempt. >>> I agree that this is bad; I disagree that it is new. It's been done for at LEAST 35 years (since I was in elementary school that long ago); anectodal reports are that it's been around much much longer. I had a 4th grade math teacher who refused to teach division to anyone who could not do a sheet of multiplication problems without error. And she was an old woman when she taught me; I assume she had been using similar methods earlier in her career. <<< Again, one can do things "naturally" and be quite conscious about what one is doing. This is the case even if the act occurs too fast for the consciousness to be in control. Those who consider statistics as a collection of algorithms might not be able to see this, but in that case, they have been very badly mistaught. >>> Again, I am confused as to just what you mean by 'conscious'. To my way of thinking, making something conscious is part of the nature of true understanding. If I read a theorem which I find intuitively obvious, I understand it at one level. But if I have to formally prove it, or even if I follow a formal proof in a book, I understand at another level. I think the main difference here is whether I am conscious of WHY the theorem is true, and how I KNOW that it is true. Of course, it is hard to define 'conscious' in a precise way, and this may not lead anywhere (but it is interesting, nonetheless) Peter . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
