NetNews Usenet Archive 1992 #16/NN_1992_16.iso/spool/sci/math/9635

Internet Message Format | 1992-07-29 | 2.9 KB

Path: sparky!uunet!stanford.edu!rutgers!dziuxsolim.rutgers.edu!no.rutgers.edu!bumby
From: [e-mail] (Richard Bumby)
Newsgroups: sci.math
Subject: Re: Computational aids. Was: Re: An interesting limit problem.
Message-ID: <[e-mail]>
Date: 29 Jul 92 17:47:47 GMT
References: <[e-mail]> <[e-mail]> <[e-mail]> <[e-mail]>
Organization: Rutgers Univ., New Brunswick, N.J.
Lines: 46

[e-mail] (Herman Rubin) writes:

>In article <[e-mail]> [e-mail]
>(USENET News System) writes:

...[introduction and other quotes deleted]...

>>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 :-)).

>I must concur.  There is no evidence whatever that learning to compute
>gives any understanding of what the computation does, and why it is done.

This reminds me of the Applied Linear Algebra course (beginning
graduate level) that I am teaching right now.  While doing homework
exercises designed to aid understanding of concepts just introduced,
the students frequently make arithmetic errors leading to
qualitatively different results than they have been told to expect.
Frequently this discrepancy is clear at an early stage of the work,
yet calculation continues.  I still have the rest of this week to try
to teach them the right answer to the question:  "Which do you trust
more: the theoretical principle we proved two days ago or your
ability to perform the arithmetic you learned in grade school?"

>A six-year-old can understand the Peano postulates with little difficulty;
>they do not mention anything except counting.  Then the understanding of
>what the arithmetic operations mean can be taught.  Those who have
>manipulative ability will learn the time-saving mechanics, and those
>who do not are far better served by using calculators.

>.... [remainder of article deleted]

I know that my own school training suffered from an emphasis on
manipulation over understanding, and things seem to be getting worse.
We have ample evidence of the major pitfalls in our approach to the
basic operations on numbers, and have successfully taught tiny chips
of silicon to avoid them.  However, the schools persist in advancing
students whose future learning will suffer because of a lack of
understanding of these concepts.
--
R. T. Bumby **  Rutgers Math ||   Amer. Math. Monthly Problems Editor
[e-mail]       || P.O. Box 10971 New Brunswick, NJ08906-0971
[e-mail]     || Phone: [USA] 908 932 0277 * FAX [phone]