From: bdsm@fermat.WPI.EDU (Billy Don McConnell)
Newsgroups: k12.ed.math
Subject: Re: Memorizing Multiplication
Date: 11 Oct 1992 06:23:14 GMT
Organization: Worcester Polytechnic Institute
Distribution: world
NNTP-Posting-Host: fermat.wpi.edu
Originator: bdsm@fermat.WPI.EDU


In article <7355.2AD6B884@puddle.fidonet.org>, Grover.Partee@f58.n343.z1.fidonet.org (Grover Partee) writes:
|> RT>One type of error which I hear my students say is "...times two by three...".
|> RT>The meaning of times is:  'multiplied by'. Hence, the above phrase would be
|> RT>interpreted : "two multiplied by by three", which does not sound
|> RT>grammatically correct.
|> 
|> It doesn't sound correct because it isn't! :)  But then neither is a
|> direct substitution; to wit: "multipled by two by three."  Neither is
|> "Multiplied two by three..." quite what we had in mind.  The problem
|> here is, perhaps, the use of "times" at all.
|> 
|> ...
|>
|> Without the second "by," we could say "times two by three" means
|> "multiplied by two, three..." There is one possible advantage to this
|> formulation.  "Multiplied by two, three" can perhaps lead more easily
|> to "multiply (two, three);" thence to the commutative aspect of
|> multiplication, that "multiply (two, three)" is equivalent to
|> "multiply (three, two);" thence to examination of other functions,
|> add(x,y), divide(x,y), subtract(x,y), etc., to see whether they are
|> commutative; thence to associativity; and so on into more complex
|> algebras.
|> 
|> IMHO, however, sacrificing language skills even to gain better math
|> skills is not acceptable.  Sacrificing language skills with no gain in
|> or possibly even a loss of math skills is reprehensible.
   
   I agree, but am at a loss to elaborate.
|> 
|> 
|> --  
|> uucp: uunet!m2xenix!puddle!343!58!Grover.Partee
|> Internet: Grover.Partee@f58.n343.z1.fidonet.org


I have some thoughts on this "times" terminology.  It really isn't BAD, it's
just misused.  A hypen would make things clearer, but it may obscure meaning
in other ways.  Let me just get to it...

It seems that we as a society have advanced beyond understanding what 
"two times three" means.  We seem to make the word "times" and operator,
hence the complaints quoted above.  That's because we just don't think
the way the folks who coined the phrase thought.  To them, "two times three"
_meant_ "twice three"...that is "taking three TWO TIMES".  The two is what I call
a repetition number, while the three is what I call an object number.  Similarly,
"three times 10" means "thrice 10", or 10 taken three times.  We have lost that
part of society that appreciates the real meaning of "x times y".

To counter this, I agree that maybe we need entirely new notation, like 
"multiply (x,y)". Or we could just alter the way we speak of the multiplication
operation...not "two times three" but rather "three, two times".  Along the 
lines of the second option, we could use a hyphen to make an adjective out of
the repetition number..."two-times three".  Here "two-times" attempts to capture
the quality of the word "twice";  "three-times" would signify "thrice";  "four-
times" and "N-times" make the logical leap to larger repetition numbers.

There is an argument here that either re-phrasing ("three, two times" or 
"two-times three") conceals the commutativity of multiplication.  I actually 
think this concealment is a good thing.  First of all, it forces teachers to 
make the specific point of commutativity early on:  x-times y  HAPPENS TO BE THE 
SAME AS  y-times x.  Secondly, it implies that there are situations that may be
Non-commutative, and therefore it lays the seed for studying more abstract
algebra.  Multiplication of real numbers is introduced as being exceptionally
convenient, as with addition;  student might ask if there are things that aren't
quite as convenient.  Just a thought.


Also, I believe that people today don't have the proper understanding of the
fraction terminology.  I don't just mean "numerator" and "denominator", either:
some students just don't understand the meaning of "three fifths" as a quantity.
I'll get back to numerator and denominator in a minute, but I want to concentrate
on "three fifths" first.

"Three fifths", the spoken "name" of the fraction 3/5, is like "three times five" 
in that it has a repetition number, three, and an object number, a fifth.  The 
problem is that students don't recognize that the fraction name breaks down like
this.  All they know is that to name a fraction, you say the top number and then
the bottom number-with-a-"ths" on it (although they are used to halves and 
thirds and quarters as special cases).  Translations I have seen of ancient Greek 
works use fifth-parts to capture the "part-of-the-whole" quality.  Our "ths" is
supposed to capture that same quality...and it does, but we aren't attuned to the
nuances of language enough to see that sometimes.  I think that students (and
teachers) need to be reminded that the terminology here has a subtle charm.  Then
"three fifths" being interpreted as "three fifth-parts" might make multiplication
by fractions less complex:  "three fifth-parts of 20" says outright that we 
should probably find out what _a_ fifth-part of 20 is and then duplicate it 3
times.

As for numerator and denominator, I'll just comment that these words, along with
commutative, associative, distributive (to a lesser degree), exponent, radical,
and a host of other words (including even multiply), are just that....WORDS...
to many, many students.

Sure, we explain how commutative means you can exchange order, and how
associative affects grouping, and so forth.  Students know that the numerator of
a fraction is the top, while the denominator is the bottom.  The exponent (or 
power) is the little number up and to the right, while the radical is the check-
mark box thing.  It's sad, because students are taught what the words mean TO
MATH, instead of what the words mean TO THE WORLD.  

Although I am sure that most teachers make the appropriate "commute, commuter"
and "associate, sociable" and "distribution" analogies to get across the true,
underlying meaning of Commutativity, Associativity, and Distributivity,  I must
say that I don't recall ANYBODY in the teaching profession informing me that 
"numerator" (like "enumerate") tells "how many", while "denominator" (like 
"denomination") tells "what kind";  I came upon this bit of knowledge through
my studies of ancient languages.  That numerator/denominator insight greatly 
aids the understanding of fractions, as I suggested above.  To too many students,
numerator and denominator may as well be dkeirjs and irdkcqii...they have as 
much significance.

There are two ways to go with my realization.  The first is to make sure we 
teach students the real, underlying meaning of the words we use in mathematics,
so that they can understand why we mathematicians have applied them to particular
situations.  The problem is, we have to explain the commutative property very 
early on, to kids who really can't appreciate the true meanings of the words.

The second solution is to rename all of our favourite terms.  Give things names
that even kids can understand.  Not the "commutative property" or the
"associative property" but maybe the "switching property" and the "grouping 
property".  (I think "distributive" works pretty well as it is.)  We can save
the "Highbrow" words for the Mathematicians who will study not only "commutative"
concepts but "non-commutative" ones as well.  (You know, I'm not even sure we
need to specify both the "switching" and "grouping" properties.  At the 
elementary level, all students need to do is compute, which involves knowing only 
"add 'em up in any order you find convenient".  "Commutativity" as a concept is 
important here, but "Associativity" is doesn't need to be singled out very early 
on.)

I'm afraid that I sound as though I'm advocating "talking down" to students.  I
suppose in a way I am.  But I would much rather have a student say "switching
property" with full conceptual knowledge of the phrase than recite "commutative
property" with less understanding of the term.  I LIKE the terminology we have.
As I mentioned, a lot of it has a certain charm...a charm that gives deeper
meaning to the concepts involved.  However, the terminology usually requires quite
a bit a maturity to be appreciated.  Consequently, the charm of our terminology
is mostly or completely lost on our students.  If all students were missing out
on were knowing smiles when they use the term "numerator", then I'd say "Fine";
but since students are missing out on concepts because they find themselves 
attaching seemingly meaningless words to their ideas, I say "Something's got to 
be done."


(I know of a guy in Austin, Texas, who's writing a book on Math Term Etymology.
As I understand it, he intends to explain WHY we use the terms we use, WHERE they
came from, and all that.  I'm not sure if I have his plan straight, but if I do,
then every teacher should get a copy of the book and study it cover to cover;
then they can begin to enlighten students about the real meanings of the terms
they use.  Unfortunately, I don't remember the guy's name.  The book is still a 
work-in-progress, though, so don't look for it yet.  If you find it, be sure to 
let me know.)

This post is getting way too long, and I'm afraid that once again I am being
too inarticulate to make my point effectively.  I think in the future I will try
to compose my thoughts elsewhere before typing them into the computer.  

My apologies for taking up so much time.  But I would like to know the thoughts
of others on the "renaming" terms idea.  I have another idea about "multiplication
and division signs" (similar to "positive and negative signs") that I have
intended to post for review, but I haven't worked it all out yet....


Thanks,

Billy Don Sterling Mc Connell
   (just call me "Don")