From: cameron@symcom.math.uiuc.edu (Cameron Smith)
Subject: Re: Memorizing Multiplication Tables
Organization: none
Date: Sun, 27 Sep 1992 06:00:15 GMT

In article <1992Sep27.033427.24710@uva386.schools.virginia.edu>
dpicard@uva386.schools.virginia.edu (David Picard) writes:
>Just one teachers opinion about the multiplication tables. By
>the way, I teach 4th grade, the seeming make-or-break year for
>memorizing the "x" tables.
>As I see it, there are really 55 multiplication facts. Teaching
>kids early on that 6x7 is the same as 7x6 eliminates the need
>to memorize the fact twice. [...]
>Once the associative property is understood, kids can be
>taught some easy tricks to reduce the list of facts rather
>quickly. [...]

Ah, yes, but unfortunately this is not the associative property.
It is the commutative property.

>[...many multiplication tricks omitted...]
>
>Fifth, The "nines tables" are easily unlocked once students are
>made aware of the predictable pattern. If any number (less than
>10) is multiplyed by nine, the first digit in the product will
>be the number which comes before it. (ie, multiply 7 times
>nine, and the digit in the tens column of your answer is 6,
>which is one less than 7) The number in the ones column will be
>a digit which, when added to the first digit, would give you
>the sum 9. This sounds a little bit complicated, but it is not
>once students try it a few time. There is even a method which
>involves using our human calculators (fingers) to multiply by
>9 which is based on the same pattern of digits. Once these are
>learned, we have knocked off 6 more facts. We are left with 15
>facts.     

I still recommend teaching the products first and only afterward
teaching the tricks.  IMHO the students are in a better position
to learn tricks and observe patterns (and think about *why* they
work) after they are completely confident that they can get the
right answers without the tricks.  A student who KNOWS all the
products of 9 with other digits is then in a position to write out
the list (for himself) and stare at it and perhaps DISCOVER a
pattern in the numbers.  While he's still learning the rule he
can check the results he obtains with it against the separately
memorized products to see that he hasn't remembered it incorrectly.
Naturally, once he has reliably internalized the rule, he can use
it to compute 9 x whatever and forget about the individual products
(or he can remember the products -- it may be easier than applying
the rule).  To the student whose teacher tries to teach him the trick
for the "9 times table" without first teaching him the individual
products, I can only say that he'd better hope that he doesn't mix
that rule up with the rules for 2 or 5 or 11 -- and of course, there's
no way he could hope to find it for himself.

If the students are not going to learn why the tricks work, but
merely memorize them as aids to computation, then they might as
well simply memorize the products directly -- it's a few more things
to remember, but even 100 isolated facts wouldn't be too much for
them to handle (after all, we expect them to remember hundreds and
hundreds of irregular, non-phonetic spellings, so 100 numbers should
by contrast be a cake walk).

>[...] multiplicands (is that right?, oh well, one of the numbers
>being multiplied) [...]

The terminology is antique, but since you mentioned it,
here it is:

Arithmetic      First         Second
operation       operand       operand     Result
---------       -------       -------     ------

Addition        Augend        Addend      Sum

Subtraction     Minuend       Subtrahend  Difference

Multiplication  Multiplicand  Multiplier  Product

Division        Dividend      Divisor     Quotient


The operands in a multiplication can also be called factors,
those in a sum, terms.  Many of the terms above come from Latin.
The "-and" and "-end" endings come from the future perfect verb
forms, which translate into English as saying that something
"is to be done".  Thus your "agenda" is your list of things that
are to be done, and "propaganda" is (dis?)information that you want
to propagate.  Similarly an "augend" is a number that is to be
augmented (i.e., increased), and an "addend" is the number that
is to be added to the augend in order to increase it.  The
"minuend" is the number that is to be diminished, and the
"subtrahend" is the number that is to be subtracted from the minuend.
And so on.

Note that when surplus money in a corporation is to be divided
among the stockholders, it would be more correct (grammatically),
and more consistent with the usage of arithmetic, to call the
total amount *before* dividing the "dividend", instead of referring
to each shareholder's portion as his "dividend".  The dividing up
has already happened by the time the shareholder sees it, so it's
no longer "to be" divided, so the term "dividend" is no longer
appropriate.

>If they can't recall a multiplication fact, they
>should see if they can recall a fact which uses one of the
>multiplicands [...] and another number which is one higher or
>one lower than the other. (ie. I don't remember the answer to
>the problem 7x6. I do, however, remember that 7x5=35. If 5
>groups of 7 = 35, 6 groups of 7 must be 35 = 7 more or 42.)

As long as we're having a Latin lesson...
You don't mean "i.e." here (which abbreviates two words and so
needs two periods), you mean "e.g.".  "i.e." abbreviates "id est"
and may literally be translated "that is"; it is used when you
wish to re-state what you've already said in other words.
"e.g." abbreviates "exempli gratia" and literally means "for the
sake of example" (think of the MGM motto "ars gratia artis",
"art for art's sake"); you use "e.g." when you wish to give
an illustrative example that is a specific instance of
what you said before.  If what follows is equivalent to and
as general as what went before, use "i.e."; if what follows
is a specific application of what went before, use "e.g.".
A comma is usually grammatically appropriate after either
abbreviation.

Also, your "which"es in that excerpt should be "that"s.
The clauses are restrictive.

(OK, OK, end of grammar lesson.  No hard feelings I hope.
But the pairs "i.e."/"e.g." and "that"/"which" are so commonly
mixed up, and the rules for determining when to use which are
so easy to learn, that I couldn't help mentioning it -- especially
to a teacher, who I hope will pass on the correct usage to his
students.)

>I would agree that the teacher and his/her attitude toward the
>subject have a lot to do with how students do with their math
>facts. Teachers who continue to grow professionally and expose
>themselves to materials like Mathematics, A Way of Thinking by
>Robert Lorton develop confidence teaching math. Children pick
>up on this confidence (or lack therof) and emulate it.

And how!  But unfortunately more of the students I've dealt
with have picked up their schoolteachers' *lack* of confidence
in (and very frequently, their lack of *respect* for) mathematics.
They have quite successfully adopted their teachers' attitudes
that math is a hard, icky subject and that it's perfectly acceptable
not to be good at math and not to *want* to be good at math.

Many are even *proud* of being bad at math, as if their scornful
ignorance of and ineptitude with mathematics were proof that they
instead had some higher, purer, more ethereal right-brain sensitivities
and abilities that were unavailable to and even incomprehensible to the
numerically competent.

--Cameron Smith
  cameron@symcom.math.uiuc.edu