From: dpicard@uva386.schools.virginia.edu (David Picard)
Subject: Re: Memorizing Multiplication Tables
Organization: Virginia's Public Education Network (Charlottesville)
Date: Sun, 27 Sep 92 17:20:11 GMT

cameron@symcom.math.uiuc.edu  writes:
> 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.

Sorry, I've been out of practice. Thanks for the correction.

> >[...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.
> 
I would agree with all of this if memorizing the products first
was quick and/or painless. Unfortunately, it isn't. If students
are taught to recognize patterns with color and shape
first, they don't need much prodding to identify a pattern when
they look at the products when multiplying by 5. They can use
their knowledge of the pattern as an aid to memorization rather
than memorizing the products first and then saying "Hey, what
do you know!"

As for mixing up the rules, students must be taught the concept
of number and estimation so that if they come up with 63 as the
product of 2x7, bells will go off indicating that the solution
is not very realistic.

> 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

Ah, there it is again, *simply*. From my limited exposure to
research on human memory, I've learned that students are most
likely to retain information that is meaningful and enjoyable.
Student's also have different learning styles which often means
presenting the same material in different formats (e.g., Oral
recitation may aid the aural learner, while the visual learner
needs to see the fact written out. The tactile learner may need
to actually build their answer with objects before s/he
understands it. For these reasons, I would never expect my
students to just "memorize the products directly" without
giving them some strategies. That is why I teach the *tricks*
along with lots of manipulative practice to demonstrate what
multiplication is and why it works as it does.

> -- 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).
> 
This is true, but we never make students feel guilty for using
the dictionary to make sure. Math students aren't allowed such
support.

> >[...] 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
> 
> 
Thanks!
           


> 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.

Like I said, Thanks.
> 
> >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.
> 
See above illustrative example. How did I do?
 
> 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.
> 
I was just reading an article recently about how Americans view
math ability as genetic! Many parents say "Oh, well, I was bad
at math so I guess they get it from me." Ours is one of the
only countries on this planet where success in math is seen as
the result of good breeding rather than hard worUnbelievable.

> 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.

Having been one of these students myself, I can understand
their point. Unfortuntely, I regularly had teachers who did not
expose me to the relevance of what I was learning. Instead,
they focused on neat notebooks, assignments without messy
erasing, percentages on daily quizzes and other such trivia. It
was not until I began teaching math from the "bottom-up" that I
began to appreciate what it was all about.

David Picard

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