In article <33024F7D.firstname.lastname@example.org>, sara lehman <email@example.com> wrote: >Marc Whitaker wrote:
>> Nicole Budiselich wrote:
>> > gino crocetti wrote: >> > I agree with what gino crocetti had to say about the more advanced >> > students helping the other children in the classroom. All of the >> > students will be able get a better understanding of the topic that is >> > being covered in the class.
>> As a parent, I would resent my daughter taking time away from her >> studies to help those who couldn't or wouldn't keep up. Rather than >> spending half her class time assisting others, she could be progresing >> at a faster rate, and should.
>> I mean the word proof not in the sense of the lawyers, who set >> two half proofs equal to a whole one, but in the sense of a >> mathematician, where half proof = 0, and it is demanded for >> proof that every doubt becomes impossible.
>> Karl Friedrich Gauss (1777-1855) >> In G. Simmons Calculus Gems, New York: McGraw Hill inc., 1992.
>A true test of students' learning is to ask them to explain what they >have learned to others.
This is very definitely not the case. There are things which can be explained, and things which cannot. Also, it is quite possible to understand to some extent, or even fully, and not to be able to explain.
The real test if understanding is the ability to use the material in new situations. This cannot be explained, but merely presented, and hopefully evoked. Ask a good writer how thoughts are put into words, and a useful answer will not result. Teachers fool themselves when they claim to be teaching writing; mechanics and some criticism can be taught, but that does not help getting the thoughts onto the paper.
A mathematician can prove theorems, but cannot explain how the proofs are found. When an artist tries to just daub, it still comes out looking artistic.
At the time of Gauss, one could come close to explaining what a full proof was, but not quite. A century later, it could be taught in the schools. This can be very helpful in learning how to prove theorems, as it does allow mechanical checking of a formal proof, rarely done. When my son learned how to find proofs himself, he could not have explained even the checking process to anyone; he just knew it.
Also, being able to explain something to someone might depend on the learner being receptive. Often this reduces to the problem of being able to unlearn; it is better not to be in this position. It seems that about half of the children taught to read by the whole word method never could learn to decode words not taught in this manner.
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 firstname.lastname@example.org Phone: (765)494-6054 FAX: (765)494-0558