Re: Another Route to Grade Inflation.

In article <[EMAIL PROTECTED]>,
Alberto Moreira <[EMAIL PROTECTED]> wrote:
>Said  [EMAIL PROTECTED] (Herman Rubin) :

>>The Sumerians counted to 12 on one hand; they used the
>>segments of the fingers.  Is the thumb a digit?  They also
>>came up with the use of base 60, and base 60 calculations
>>done by them and the Babylonians were more advanced than
>>any Western base 10 calculations until after Greek times.
>>Our use of minutes, seconds, etc., for fractions comes from
>>that.

>As far as mathematics go, we're neither Sumerian nor Babylonian. We're
>kind of Greek, but their mathematics, considered from our modern
>viewpoint, were a bit incipient. As far as mathematics go, our true
>origins are Arabic, we inherited the decimal system from them and
>that's what made the transition from antiquity to modernity possible.
>And you don't have to look twice to see that our hour/minute/second
>system of measuring time is a baroque anachronism. 

We did not get the idea of using a decimal system to 
calculate from the Indians through the Arabs; all that
we got from them in this direction was the use of the
same characters in all places, with a "0" character to
indicate no term with that power of 10.  Otherwise,
there is no difference between arithmetic using Egyptian,
Greek, or Roman numerals.  In fact, it might be best
from a pedagogic standpoint to start with Egyptian, which
(at least as I read it in the encyclopedia) just used a
symbol for each power of 10, repeated the appropriate
number of times.

>>See the above.  The oldest arithmetic records we have were
>>base 60, not base 10.  More than 4000 years ago, they had
>>the arithmetic operations, a symbol for 0 in a place, division
>>with remainder, and sexagesimal fractions to many places.

>Yet that's not where our mathematics comes from. We're deep rooted in
>the Arabic tradition. 

We are accustomed to doing it that way; it is important that
we recognize that it is no more than that.  The Greeks used
9 symbols for 1-9, another 9 for 10-90, and another 9 for
100-900, and indicated multiplying by 1000 by a bar over the
symbol.  Other than having more symbols, their arithmetic
was no different from ours.

And this is arithmetic, not mathematics.

>>Agreed; I still suggest that anyone teaching arithmetic be
>>conversant with a good development from the Peano Postulates,
>>and if a person cannot read the first part of Landau's book,
>>that person cannot understand arithmetic.

>That may be true to math majors - it isn't true to the rest of us. The
>Peano Postulates aren't but a formalization of our counting intuition,
>and our own intuitive machinery is well more suited to learn
>elementary arithmetic than Peano's formalization. I do not find the
>need for any kind of formalization before 5th grade or so !

VERY wrong!  The need for formalization is before too much
is learned essentially by rote.  Variables, as used in
mathematics and its applications, but not in computers, are
formal extensions of language, and are needed for concise
precise communication.  With variables, word problems
become much more trivial.

It is a major problem to teach students formal concepts
after they have gotten poor intuitive ideas, or just
learned methods.  It is the opposite of the way
educationists have it; one needs certain abilities to
replace miscellaneous garbled intuitions with structure or
formalism; one does need to keep not only the formalism,
but the concepts, in mind as well.  We had much less of a
problem of teaching good formal mathematics in high school
a half century ago than we have in teaching it to college
students who have had more facts and computational
procedures today.

                Time will
>come when the formalization can be introduced naturally and as a
>consequence of the need to look at things with a more critical eye -
>but that's probably not the most efficient way to teach the stuff to
>young children. 

Formalization is natural for children; it is after they
learn in a sloppy manner that it becomes difficult.
-- 
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, Department of Statistics, Purdue University
[EMAIL PROTECTED]         Phone: (765)494-6054   FAX: (765)494-0558