Path: utzoo!utgpu!water!watmath!clyde!att-cb!att-ih!homxb!homxc!mchin
From: mchin@homxc.UUCP (M.CHIN)
Newsgroups: comp.edu
Subject: Re: Calculators in exams, was: Becoming CAI literate
Summary: abstraction
Message-ID: <1455@homxc.UUCP>
Date: 2 Mar 88 20:30:45 GMT
References: <2032@ukecc.engr.uky.edu> <3900008@nucsrl.UUCP> <7301@tut.cis.ohio-state.edu>
Distribution: na
Organization: AT&T Bell Laboratories, Holmdel
Lines: 51

> I also calculate things in my head while people are still fumbling with
> calculators. And I had quick arithmetic drills in school. The problem is
> that the two are totally unrelated. I failed every single speed drill I
> ever took; they regularly reduced me to tears. What gives me my speed is
> the non-traditional but simple calculating techniques that I learned
> from  my mathematician father - you know, the ones they used to fail me
> on math tests for using? All those math drills never taught me a thing,
> and my husband, who had the drills but not the mathematician at home,
> is utterly lousy at "routine mathematical calculations", even though
> he likes numbers and I don't. (I stopped letting him calculate tips long
> ago, having discovered that he was more often wrong than right. Now I
> let him do it again, having dicovered that the reason that he screwed
> up was that he was doing cross-multiplication, just like the books have
> you do. That gets good scores on tests, because it looks right. I
> was moving the decimal place to get 10% and then multiplying that,
> which math teachers fail because the intermediate steps aren't "right"
> but it sure works.)			
> 
> >	Richard Harter, SMDS  Inc.
> 
> Elizabeth D. Zwicky


Let's hear it for calculating things in our heads.  I do this too, in many
instances, especially when it concerns shopping.  Even now, I still do some
of the easier multiplication and division problems on paper because its
faster than finding a calculator or calling one up on the computer.  In the
instance of tipping, if you live in a state where tax is 5 %, 15 % tip comes
really easily.  What I often end up doing is dividing by 10 (move the
decimal) divide that by 2 (very easy). and add the two numbers.  What can I
say, its easier than multiplying by 3 and dividing a memorized number by 2.

Just to add my two cents worth, I think what is needed is a good sense of
abstraction.  This means that after having done the drills, you are able to
do the arithmetic.  This doesn't mean you know the shortcuts.  Word problems
teach you the shortcuts.  Once you know several different ways to attack a
problem, you are able to take the shortest step.  If this means reaching for
a calculator to multiply 3 four-digit numbers, so be it.  But, until you are
familiar with every process, you won't know the quickest method.  Now, word
problems, by not stating the method or the formula, make you think more
abstractly.  This draws up, in general, a larger picture of the problem in
your head.  Therefore, you are able to view it from several different angles
as opposed to through a microscope.  Given the extra viewpoints, you are
able to choose the easiest/quickest path to the solution.  

At least that's me view.  Unfortunately, this always posed a problem when
they started throwing in extraneous variables that had nothing whatsoever to
do with the problem :-)

Michael Chin
ihnp4!homxc!mchin