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12 Feb 2006 - 16:43
the cognitive unconsciousEd and I were just going over Old Grouch's drawing & Anne Dwyer's notes....whoa nelly. They're right. The figure can't be 'solved' as drawn. what did your C.U. know and when did it know it? Anne also had this to say: Yes, yes, yes!!! I knew there was something totally bothering me about this problem. If you use the marked numbers on the test of 4cm for the height and 13cm for the base, the hypotenuse of the triangle on the left is the square root of 185 which is between 13 and 14. But the base of the triangle on the left marked 10cm cannot be correct. If you draw a line parallel to this 10 cm base with its start at the right hand corner where the circle is and drop it to the bottom, you get a right triangle with a hypotenuse of 14 and sides 3 and 10. This is not possible since 32 + 102 does not equal 142. Also, since the horizontal line marked 6 cm is parallel to the horzonal line marked 9 cm, the two vertical sides of the resulting parallogram have to be parallel and the same length. But, as I've pointed out, the length cannot be 14 cm because the sides are 3 and 10. So the student who said he couldn't do this problem was absolutely right. You can't do this problem if you try to get all the numbers right because they don't come out right. The funny thing is, when I put in my original post about this being a problem for high school geometry students, I believe it was because my intuitive brain recognized the problems, but didn't articulate the words to my verbal brain. I think Anne is right. I've pulled out my copy of Arthur Reber's Implicit Learning and Tacit Knowledge: An Essay on the Cognitive Unconscious (Oxford Psychology Series, No 19, and will try to get around to posting some of the key passages from the book shortly. In the meantime, I'm sure Anne is right. As I understand it — and I'm still learning this area — the cognitive unconscious does most of the heavy lifting. This is one reason why I feel it's wrong to 'privilege' the 'verbal mind,' as constructivist math curricula do. The verbal mind, as I understand it is the conscious mind, and the conscious mind is the last to know anything about anything. I think. (Irony intended) Here's an abstract of a 1987 article in Science: Contemporary research in cognitive psychology reveals the impact of nonconscious mental structures and processes on the individual's conscious experience, thought, and action. Research on perceptual-cognitive and motoric skills indicates that they are automatized through experience, and thus rendered unconscious. In addition, research on subliminal perception, implicit memory, and hypnosis indicates that events can affect mental functions even though they cannot be consciously perceived or remembered. These findings suggest a tripartite division of the cognitive unconscious into truly unconscious mental processes operating on knowledge structures that may themselves be preconscious or subconscious. source: the death of implicit memory? As I say, I'm still working my way through this field. The next article I'll read will be Willingham on The Death of Implicit Memory: Abstract I suspect that Willingham's & Preuss's article doesn't contradict Reber's book, but I don't know. Here's Willingham in a passage from Malcolm Gladwell's article on The Art of Failure 'The Art of Failure': "Choking" sounds like a vague and all-encompassing term, yet it describes a very specific kind of failure. For example, psychologists often use a primitive video game to test motor skills. They'll sit you in front of a computer with a screen that shows four boxes in a row, and a keyboard that has four corresponding buttons in a row. One at a time, x's start to appear in the boxes on the screen, and you are told that every time this happens you are to push the key corresponding to the box. According to Daniel Willingham, a psychologist at the University of Virginia, if you're told ahead of time about the pattern in which those x's will appear, your reaction time in hitting the right key will improve dramatically. You'll play the game very carefully for a few rounds, until you've learned the sequence, and then you'll get faster and faster. Willingham calls this "explicit learning." But suppose you're not told that the x's appear in a regular sequence, and even after playing the game for a while you're not aware that there is a pattern. You'll still get faster: you'll learn the sequence unconsciously. Willingham calls that "implicit learning"--learning that takes place outside of awareness. These two learning systems are quite separate, based in different parts of the brain. Willingham says that when you are first taught something--say, how to hit a backhand or an overhead forehand--you think it through in a very deliberate, mechanical manner. But as you get better the implicit system takes over: you start to hit a backhand fluidly, without thinking. The basal ganglia, where implicit learning partially resides, are concerned with force and timing, and when that system kicks in you begin to develop touch and accuracy, the ability to hit a drop shot or place a serve at a hundred miles per hour. "This is something that is going to happen gradually," Willingham says. "You hit several thousand forehands, after a while you may still be attending to it. But not very much. In the end, you don't really notice what your hand is doing at all." Under conditions of stress, however, the explicit system sometimes takes over. That's what it means to choke. When Jana Novotna faltered at Wimbledon, it was because she began thinking about her shots again. She lost her fluidity, her touch... I'm still confused about the relationship of the conscious, verbal mind to 'knowing,' 'understanding,' and 'expertise." However, I'm not confused about the fact that expertise means you've reached automaticity — and automaticity means you can do something without thinking about it. Again, this is why I'm uncomfortable — very uncomfortable — with the constant demand that math students explain their answers in words. On the one hand, it's true that I discover gaps in my knowledge when I force myself to put things in words. So....having to explain an answer, I think, functions as formative assessment. On the other hand, I don't see where 'explain your answer' necessarily has to mean 'explain your answer in words' as opposed to 'show your work.' At this point, I think Feynman's aphorism — unless you can show how to arrive at an answer through 5 different paths, you don't know it — (NOT FACT-CHECKED) describes true conceptual understanding for me. (Personally, I'm not shooting for 5. Two will do. Two or perhaps 3.) I have the strongest feeling that, in forcing children who are just learning math to 'back up' into words every time they learn how to solve a particular kind of problem, constructivists are pulling these kids back out of real mathematical learning and comprehension — back out of implicit memory into explicit memory. -- CatherineJohnson - 12 Feb 2006 Back to main page. CommentsAfter entering a comment, users can login anonymously as KtmGuest (password: guest) when prompted.Please consider registering as a regular user. Look here for syntax help. Sounds like the centipede: A centipede was happy, quite, Until an ant, in fun, Said "Pray, which leg comes after which?" Which raised his doubts to such a pitch, He fell befuddled in the ditch, Not knowing how to run.(When I was Googling around for a source, I found this story "The Centipede Who Went to School" which isn't quite the same, but seems like it'd fit in around here ;-) -- OldGrouch - 13 Feb 2006 I have days when I feel just like that centipede. Seriously. -- CarolynJohnston - 13 Feb 2006 I love it!!! That's it!!! You have to be on auto-pilot if you're a centipede, that's for sure! -- CatherineJohnson - 13 Feb 2006
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