[comp.edu] Mental arithmetic, was: Calculators in exams, was: Becoming CAI literate

levy@ttrdc.UUCP (Daniel R. Levy) (02/28/88)

In article <24954@cca.CCA.COM>, g-rh@cca.CCA.COM (Richard Harter) writes:
> In article <1988Feb24.224849.928@jarvis.csri.toronto.edu> tjhorton@ai.toronto.edu ("Timothy J. Horton") writes:
> 	If you have a reasonable aptitude for arithmetic, it is faster to do
> the odd multiplication in your head than reaching for a calculator -- if you
> have been trained for quick mental arithmetic.  I amuse myself and startle 
> people by giving the answer to routine calculations while they are still 
> fumbling with the calculator.
> 	I am inclined to think that quick mental arithmetic ought to be
> taught in the schools.

As my own $.02, I found in elementary school that I had the worst problem
with the high-number multiplications (where one or more of the numbers was
greater than 5).  The smaller numbers I could handle by adding in my head.
But the fact that "8 x 7 = 56" for example kept slipping from my mind.

Then from a book of puzzles and games I learned the "finger multiplication"
trick for numbers 6 <= n <=10.  You hold both your hands with the palms
facing you and the thumbs uppermost.  Then you assign the numbers 6 through 10
to the "pinky" through the thumb of each hand, respectively.  To perform
a multiplication between two numbers in this range, touch the tips of
the corresponding fingers together.  Take the sum of the number of fingers
below and including the touching fingers, and multiply that by ten.  Then
add the product of the number of fingers on one side above the touching
fingers and the number of such fingers on the other side.  The result is
the desired multiplication.  For most combinations, the product "above" the
touching fingers will be less than ten, so it's a matter of the tens digit
below, and the units digit above.  The exceptions are 6 x 6 = "twenty-sixteen"
and 6 x 7 = "thirty-twelve" which took some mental addition afterward,
but it still beat rote memorization for me.  Once I knew that trick, I fairly
flew through my arithmetic lessons.  I showed it to a number of my teachers,
none of whom had known it before.

Surely this isn't the only trick around but it sure sounds like a useful one
to teach the munchkins in elementary school who can use all the help they can
get in gettin' down those ole debbil multiplication tables....
-- 
|------------Dan Levy------------|  Path: ..!{akgua,homxb,ihnp4,ltuxa,mvuxa,
|         an Engihacker @        |  	<most AT&T machines>}!ttrdc!ttrda!levy
| AT&T Computer Systems Division |  Disclaimer?  Huh?  What disclaimer???
|--------Skokie, Illinois--------|

tjhorton@ai.toronto.edu ("Timothy J. Horton") (02/29/88)

In article <2208@ttrdc.UUCP> levy@ttrdc.UUCP (Daniel R. Levy) writes:
>In article <24954@cca.CCA.COM>, g-rh@cca.CCA.COM (Richard Harter) writes:
>> In article ... tjhorton@ai.toronto.edu ("Timothy J. Horton") writes:

I did not write the following.  In fact, I disagree entirely with the position.
Richard Harter wrote it -- credit where it's due.

>> 	If you have a reasonable aptitude for arithmetic, it is faster to do
>> the odd multiplication in your head than reaching for a calculator -- if you
>> have been trained for quick mental arithmetic.  I amuse myself and startle 
>> people by giving the answer to routine calculations while they are still 
>> fumbling with the calculator.
>> 	I am inclined to think that quick mental arithmetic ought to be
>> taught in the schools.

I don't think mental arithmetic is very important for strong mathematical
facility.  'Amusing oneself and startling others' may be important for TV
game shows, but how much else?  If the 5 functions on your basic calculator
were the core of mathematical prowess, every student in the world capable
of pressing buttons would surpass the accomplishments of Einstein, given
enough 9 volt batteries.  Understanding arithmetic is important, but I
question the value of the hours that would be needed to teach QUICK, MENTAL
arithmetic in schools, when some much else of value is 'out there' in math
to be learned.

spf@whuts.UUCP (FRYSINGER) (03/01/88)

In article <1988Feb28.224421.6922@jarvis.csri.toronto.edu>, tjhorton@ai.toronto.edu.UUCP writes:
> I don't think mental arithmetic is very important for strong mathematical
> facility.  'Amusing oneself and startling others' may be important for TV
> game shows, but how much else?
> ... every student in the world capable
> of pressing buttons would surpass the accomplishments of Einstein, given
> enough 9 volt batteries.  Understanding arithmetic is important, but I
> question the value of the hours that would be needed to teach QUICK, MENTAL
> arithmetic in schools, when some much else of value is 'out there' in math
> to be learned.

Well, I think you've hit the nail on the head.  Yes, in most
situations calculators improve performance (speed & accuracy).
Unless you don't have one handy.  Or it's broken.

The ability (and willingness) to do mental arithmetic is certainly an
advantage when no calculator is around, as happens with me often.  I
even carry a calculator in my "go to work" shirt pocket, and still
frequently find myself in arithmetic situations without one.
Furthermore, if one loses the ability (or confidence, or willingness)
to DO mental arithmetic, how will one recognize (A) when your
calculator or your fingers have made a mistake, or (B) when someone
else or their calculator (e.g. a cashier) has goofed?

Since proper training to do mental arithmetic is in fact a very minor
effort (mostly done by parents, not teachers), it's well worth doing.

Steve Frysinger

P.S. By the way, the same goes for doing arithmetic long hand.  When's
the last time you balanced your checkbook by hand?

tlh@cs.purdue.EDU (Thomas L. Hausmann) (03/02/88)

In article <1988Feb28.224421.6922@jarvis.csri.toronto.edu>, tjhorton@ai.toronto.edu ("Timothy J. Horton") writes:
> 
> I don't think mental arithmetic is very important for strong mathematical
> facility.  

  What do *you* mean by strong mathematical facility, I am not agreeing or
disagreeing with what you say here, I am just asking.  Because (as I stated
earlier in a discussion with elg@killer) doing mental calculations quickly
lets me concentrate more on my eventual computational goal.

>         'Amusing oneself and startling others' may be important for TV
> game shows, but how much else?  If the 5 functions on your basic calculator
> were the core of mathematical prowess, every student in the world capable
> of pressing buttons would surpass the accomplishments of Einstein, given
> enough 9 volt batteries.  Understanding arithmetic is important, but I
> question the value of the hours that would be needed to teach QUICK, MENTAL
> arithmetic in schools, when some much else of value is 'out there' in math
> to be learned.

Here I agree with you, heuristics that work for one person may not work for the
next. We all have our own mental tricks for doing quick computations and hence
we have preferences as to which to apply in a given instance.  Although, there
are some heuristics that can be taught (e.g. grouping additions of single digits
into pairs of treys s.t. intermediate sums are multiples of ten) others are
more obscure (15% of x [for tips] is .10*x + .5*(.10*x) because moving decimals
and halving are easy mental operations.)

-Tom
-------------------------------------------------------------------------------
Tom Hausmann       Dept. of Computer Sciences     Purdue University
tlh@mordred.cs.purdue.edu    | My ideas?  There has never been an original
...!purdue!tlh               | thought since Plato.

roberta@eleazar.Dartmouth.EDU (Roberta Millstein) (03/04/88)

In article <3849@whuts.UUCP> spf@whuts.UUCP (FRYSINGER) writes:

>Well, I think you've hit the nail on the head.  Yes, in most
>situations calculators improve performance (speed & accuracy).
>Unless you don't have one handy.  Or it's broken.
>
>The ability (and willingness) to do mental arithmetic is certainly an
>advantage when no calculator is around, as happens with me often.  I
>even carry a calculator in my "go to work" shirt pocket, and still
>frequently find myself in arithmetic situations without one.
>Furthermore, if one loses the ability (or confidence, or willingness)
>to DO mental arithmetic, how will one recognize (A) when your
>calculator or your fingers have made a mistake, or (B) when someone
>else or their calculator (e.g. a cashier) has goofed?
>
>Since proper training to do mental arithmetic is in fact a very minor
>effort (mostly done by parents, not teachers), it's well worth doing.
>

Is anyone else here *unable* to do mental arithmetic?  Yes, I had all the
drills, etc, etc, and have absolutely no problem with a pencil and paper, but as
soon as I try to keep more than two numbers in my head, I'm lost.  I can
manage the bit about doing the decimal point, and can keep a running total of
groceries if I round to the nearest dollar--but even the latter takes some
concentration.  Forget multiplying in my head, I can't remember where I've
been.  Is this just me or what?


>Steve Frysinger
>
>P.S. By the way, the same goes for doing arithmetic long hand.  When's
>the last time you balanced your checkbook by hand?

Actually, believe it or not, I almost always do it by hand.  It's too annoying
to have to worry about whether or not I've pressed the right keys.  I can
do it much faster and more accurately by hand.  With a pen.  ;-)





-- 
###############################################################################
Roberta Millstein                                 roberta@eleazar.dartmouth.edu
                                                  ...dartvax!eleazar!roberta
"A friend of the devil is a friend of mine....."  --The Grateful Dead