From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Pedagogy
Date: 29 Jul 92 19:39:28 GMT
Distribution: na
Organization: Purdue University Statistics Department

In article <25018@dog.ee.lbl.gov> sichase@csa2.lbl.gov writes:
>In article <1992Jul29.000223.27339@massey.ac.nz>, news@massey.ac.nz (USENET News System) writes...

>>I believe that it is far more important to understand the meaning of
>>the operations of arithmetic than to be able to work through the
>>algorithms like trained monkeys without having a clue why they work.
>>(I know some bright spark will ask how many monkeys I've trained to
>>do long division - well, no actual monkeys, but a lot of great apes :-)).

>This is certainly the crucial point.  As a physics instructor I am convinced
>that unlike in botany, one cannot learn the concepts of math or physics without
>*doing* the work.  And the only acceptable proof that a student deeply 
>understands a concept is to *apply* it to solving a problem.  So I think that
>the distinction you make between understanding and being able to work through
>the algorithms is completely falacious.

I agree partly with this.  One cannot demonstrate any understanding of a
concept except by applying it.  But the application does not require that
the problem be solved; it only requires that the application of the 
concept reduces the problem.

>I remember having a specific dialog once (as a physics TA) with a premed 
>student who came to my office on the day they learned Gauss's law in EM.
>She was accustomed to memorizing fact and considering the matter done.

So what does that have to do with concepts?  Memorizing the definition 
of an integral has nothing to do with understanding what an integral is.
And if one thinks of an integral as an antiderivative, one is even
farther from understanding the concept of integral.

>ME:  Did you follow today's lecture?
>HER: Sure.
>ME:  You understand Gauss's Law?
>HER: Sure.
>ME:  OK.  Please calculate the field of an infinite uniformly charged
>cylinder using Gauss's Law.
>HER: But I understand Gauss's Law.
>ME: OK, then just do the problem.
>HER: I said I understand it!
>ME: No problem.  Then just solve the problem.

At this point, I would have asked her to SET UP the problem in symbols.
This is the major mistake made in teaching mathematics.  If the problem
is set up properly, a computer can solve it.

			.........................

The idea that applications must be previously shown pervades the elementary
and secondary education system.  It has permeated the university level as well.
There have been excellent students who were never taught to use symbols to
formulate word problems; I know of a case who only needed to be told about it.
The high school geometry course involving proofs is no longer available at
many schools; instead, they are taught geometric theorems, and even given
the "advanced" material of analytic geometry.

BTW, those who learn botany that way do not understand botany, either.
Or history, or language.

-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)