[Numeracy 82] Re: Teaching math facts with MMR.

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Maureen Carro mcarro at lmi.net
Wed Jan 27 14:06:52 EST 2010

Carol, you make such a relevant point! What is missing here for LOTS
of students is "symbol imaging". Memorizing multiplication facts does
nothing about teaching the concept of multiplication... nor does it
help us "picture" what the symbols in math actually mean. Before
embarking on "teaching" multiplication facts... ie, helping students
commit them to memory and retrieve them...... they must have a concept
of what it means. "two times three". or " 2 X 3" can be meaningless
words or symbols for some students. All of us have difficulty
memorizing meaningless information..... so why wouldn't students
have trouble memorizing stuff that means nothing! Most adults know
how to count things, but many do not relate the symbolic numerals in
math as representing quantities that are counted. I start by asking
"What the heck do we need numbers for anyway?" Most of the time,
someone will come up with "to count things.... so we know how much"...
etc. Exactly! The counting numbers, that we use to show how many
things we'v e counted are called whole numbers... they show how many
whole things. It is worth reviewing this concept! Students are so
used to "throwing numbers and words around". Not being sure of what
is going on produces anxiety! As some people have mentioned, playing
cards, and dominoes are great for building this symbol imaging. The 6
playing card has six hearts, diamonds, spades, or clubs on it! the 9
has nine! Children quickly learn to match dots on dominoes, calling
them by the numbers: "double-sixes", etc. They learn the
"configuration" of dots that represents the numbers.

Back to multiplication: I often start out by documenting what each
"fact" means.... I might point to my own face and say I have 1 face
and two eyes.......... I document this on the board 1(face) X 2
( eyes)= 2 eyes. I call on another student and now we have 2 faces X
2 eyes each , so we have 4 eyes. I continue this , placing each fact
on a grid line, through 9. I call this the "Two's house" since the
common factor is 2 ... we're looking for eyes!
2x1=2 2X4=8 2x7=14
2x2=4 2x5=10 2x8=16
2x3=6 2x6=12 2x9=18

I take the time to go through each one and what it means. I might
develop "sixes" in the same way with having them look at a window in
the room that has six panes. Start over developing the concept: How
many panes in 2 windows, 3, 4, ....9....? and do the same thing on the
grid. Of course I do one "fact house" at a time until memorized.
They MUST have a concept of what it means before the memory drills
start!!!!! I cannot emphasize this enough! You can use hands for
fives.... "one hand times five fingers".... Find objects around the
room to represent 3's , 4's... or ask for suggestions. what can we
use that has "4 as a common factor"?

Next: I have the class or individual student recite aloud "two times
one equals two", two times two is four" ( you can vary the language
equals/ is / is the same as) through the whole grid up to nine.
I place a check after each recitation for 5 checks. then I erase
one...... and have them continue to recite with one missing, for five
times. Then erase another one..... recite for five more checks....
and another, until the entire grid has only blank lines...... recite 5
times with blanks.
Then: Just say the answers ( not the "two times" part). five times
in order ( this is skip counting really).
then skip around: who lives in this room? then another..... in
random fashion, pointing at the blank lines.
Next: where does "two times three live"? or "who lives here? and I
point to a blank line.... in random order.
Students are building a visual map for the auditory facts they are
saying and hearing.

It is more or less this progression with each group of facts.... After
all the fact houses are mastered, a student just needs to put a grid
of nine blank lines on top of the math worksheet, and they will be
able to get the answer to any fact they have trouble retrieving.
FACT GROUPS. Some will catch on sooner than others and go on to
learn on their own.

Another thing I do is use the "area model " of multiplication. I have
students draw out rectangles on graph paper representing 2 by 1
grid.... then a 2 by 2 grid, then 2 by 3, etc. they can see the
"area" grow.... i also tell them that each math fact is an equation
that represents a rectangle, and that each set of facts has one that
is a "square". We then draw a square around the math fact on each 9-
line grid, that is "the square" in the family. I tell them we are
already doing algebra here! I tell them that "algebra" is just a way
of thinking in a different way about lots of things that we already

This "area model" activity leads directly to the commutative property
of multiplication.... and you can go ahead and introduce it..... a 2
by 3 rectangle will have the same area as a 3 by 2 rectangle!
Students can cut them out and match them up if necessary. this is
what we call commutative..... they need to now make the word part of
the math vocabulary, and they can, because it will have meaning

I really suggest that anyone having the opportunity to take Making
Math Real 9-lines Intensive... do so. It works! And what I have
related here is just the tip of it to give an idea!

Maureen Carro, MS, ET
Academic Learning Solutions
Alamo, CA
mcarro at lmi.net

On Jan 26, 2010, at 3:03 PM, Carol King wrote:

> I could rearrange. If you say “3 x 4 =” I literally see in my head:

> “three times four is what?

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