Tuesday, December 27, 2011

Brian Rude, kcab and ChemProf on The virtues of explaining your answers


Brian Rude said...

I think it was in about fifth grade when one day the teacher told us, "If you can't say it, you don't know it." That always stayed with me, not because I believed it, but because I questioned it. I wondered if it were really true. It's totally understandable that a teacher would say that, but that doesn't mean it's really true. Over my lifetime I have given the idea some thought and concluded that it's not a simple matter. There are times when explicitly and precisely verbalizing what we are learning is very important. There are other times when verbalizing what we are learning is practically impossible and certainly not worth the time trying. Language is obviously a very powerful tool that we ought to develop as much as we can. But it is definitely not true that thought is only linguistic. With a little reflection one can come up with plenty of examples of nonverbal thought.

I have a few years experience teaching college math. Many times a student would come to my office for help. A common pattern was repeated many times. After getting oriented to the students difficulty (not always a quick or easy task) and selecting an appropriate problem to work on, I would be struggling to find just the right words to explain something, when the student would suddenly say, "Oh, I get it!". That would be my cue to shut up. Language is a wonderful tool, but a lot of thought consists of assembling ideas together in certain ways. When helping students I use language identify the mathematical ideas needed and assemble them in a way that will apply to the problem at hand. But the actual assembling of ideas itself is not linguistic so much as conceptual.
Most of us are quite adept at using a computer word processing program for writing. That involves a lot of learning. Were we forced to verbalize each step along the way when we learned? Can you verbalize everything you know about writing with a computer? Would it be beneficial to try to verbalize all that? What about driving. Have you ever tried to verbalize everything you know about driving? Would that be beneficial?
All this is not to say that language is not important. Obviously it is. But you can over do a good thing. I have been aware of the "explain your answer" fad in math education, but have always considered it just that, a fad. In a math class I think "explain your answer" should be translated into "show your work". And writing in a math class should simply mean "show your work".

kcab said...

I think the very best math students are able to explain their answers as well as work the problems. In reading a couple of books recently, "Count Down" and "Perfect Rigor", it seemed to me that the systems described for developing top math contest competitors had a lot of emphasis on teaching kids (not necessarily neurotypical) to explain their thought processes. My own mathy child is extremely good at explaining work, particularly aloud to others (the process of writing it seems tedious to him at times).

I think there is something to the idea that being able to explain one's work requires understanding a problem in a different way. I wonder though, whether anything is lost in the translation to words - is it necessary sometimes to hold off on the verbalization for a bit so that the problem can be completely seen? Wasn't there something about verbal descriptions of a memory altering and weakening the memory itself? I wonder if other non-verbal information is similarly affected.

ChemProf said...

In General Chemistry, I teach basic quantum mechanics. I occasionally get a very verbal student who wants me to explain why atoms act the way they do at a more fundamental level. But at that fundamental level, you are really talking about mathematics (or a description of mathematics -- saying "a superposition of basis sets" to a student who is in Calc I is not too meaningful). A few have been insistent that I must be able to explain in words, since "if you can't say it, you don't know it," and I've had to shut them down with a little bra-ket notation! So, there are a few things that aren't easily explained in words. And the people I know who really internalize quantum (I am not one of them) seem to do so in some non-verbal way.

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