#3843. math facts -- a method that works

Mathematics, level: all
Posted Sun Oct 29 22:05:38 PST 2006 by victoria (advance.tutors@sympatico.ca).
advance tutors, Montreal, Quebec, Canada
Materials Required: poster board, markers; other concrete counters such as abacus, base ten blocks, or pennies
Activity Time: five to ten minutes daily all year
Concepts Taught: memorization, math facts, addition, multiplication

Post: math facts -- a method that works -- bringing to top
Posted by victoria on 10/29/06

Well, this seemed to be an important topic and I answered
it seriously and at length, so I'm copying and reposting it
before it falls off the board. It would be nice to know the
OP saw it, please and thanks.

Re: math facts -- a method that works
Posted by victoria on 10/28/06

On 10/28/06, KelleeF wrote:
> Math facts memorization seems to be a constant struggle
> for everyone I talk to. So, I'm curious what everyone's
> expectations are. What is mastery? How many facts do you
> expect kids to complete and how much time do you allow?
If
> your students are expected to complete a facts worksheet,
> do you expect them to go in order or is it ok to skip
> around? Does anyone know of any studies out there on
> memorization of facts?
> Thanks

I work on this ALL the time.

There are tons and tons of studies out there about learning
and memorization. They tend to be listed under psychology
rather than education so teachers rarely pay attention to
them, sigh.
You could search ERIC or google or for that matter
Scientific
American and Discover and find enough to fill a room.

Almost all of these studies back up Mrs. Ross, my Grade 3
teacher (who was on top of things, believe me!) and my
grandmother and mother (who also really knew what they were
doing).

The very best way to learn a fact permanently is to do a
combination of three things:
(1) CONNECT it to other knowledge, ie work on
mutliplication
by repeated adding; and CONNECT to other senses, ie
practice
with concrete objects and/or visual representations.
(2) PRACTICE it *ORALLY*
(3) PEPEAT, repeat, repeat -- *but* in MULTIPLE CONTEXTS.
Use
it in as many different contexts as possible, ie do not
just
do table practice and then leave it, but rather do all
sorts
of applied problems where these facts will be needed.

Please note that filling out worksheets is nowhere on this
list. Filling out paper copying from one place to another
is
of very little benefit in putting things into memory.

Yes, I do use worksheets and workbooks, but I use them in
an
interactive fashion, discussing the meaning of the problems
as we go along; or, I teach the material orally, make sure
it
is understood, and then give the paperwork as a backup for
reinforcement.

I teach addition and multiplication facts to students who
are
in deep deep trouble academicaly. Since I work as a private
tutor and people have to pay me cash out of pocket, I only
get called in after everything else has failed. The
techniques I am suggesting work with any student who is
teachable at all.
I have aso used this approach in a classroom, and it works
even better with students who don't have problems!

The following method is presented as you would use it in
Grade 3 to learn the facts from the ground up. The SAME
approach, streamlined a bit faster for review, works right
up
through high school.

Method:
(1) Present the logic and reasoning. For addition, three
red
dots and two blue dots; count the red dots (three) count
the
blue dots (two) count the total (five); SAY out loud "three
plus two is five" -- the WHOLE thing, question and answer;
your goal is to connect question and answer so separating
them is counterproductive. For multiplication, three rows
of
four dots; how many rows? (three) how many dots in each
row?
(four) how many altogether? (twelve); SAY *out loud* "three
times four is twelve" -- the WHOLE thing, and make
sure "times" is clear, NOT "and" which means plus.

Once you have worked on four or five facts in the three
timeses, then start to build the table. One row of three
dots, one times three is three. Add a second row, two times
three is six. Add a third row, three times three is nine.
Keep giving concrete and visual models and having the
students actually count the dots. Then note that you are
adding three each time you add a new row, and build the
facts
up by adding. **As many different *logical* routes to the
answer as possible**. Work on these facts every possible
way
you can think of and get books to give more ideas.

(2) Give paper and pencil work as a backup to each stage. I
have old books that have things grouped in three -- three
children to a table, three books to a stack, whatever --
and
all sorts of questions where students have to count from
given pictures, then make their OWN pictures/dots and
count --
this making their own models is a huge step, one that is
too
often ommitted and that leads to much trouble later.

(3) Once the concept of what this thing *means* is down
solidly, *then* we go into memorization. Recite the table
ORALLY at a good steady rhythm. Three times one is three,
three times two is six, three times three is nine, and so
on.
Takes a minute or two to go through the whole table.

Now, here is the big, big trick -- it is called "spaced
practice". You recite through the whole table **as a class
group** once or twice every morning and again every evening
for a week or two.
This takes less than five minutes at a time and can be done
as a transition activity or a "sponge" activity while
standing in line. Kids like oral recitation and it is fun
to
do. Make sure that they are all saying things in a good
steady rhythm, NO mumbling and NO racing to the end.

(4), Now, more paper and pencil work. At THIS stage you can
give the fill-out-the-table worksheets, and the mixed
practice of one times up to three times in random order.
NOW
these exercises mean something and actually connect to
something in the brain so they lead to real learning.
However I recommend very strongly against speed drills and
flash card drills at this stage -- you are teaching
accuracy
FIRST, speed later. There is NO benefit in a fact mistake.

(5) Most students, after a week or two of spaced oral
drills,
will simply say "three times five is fifteen" -- they have
ingrained the pattern into their memory. The weaker ones
however will need more work. Don't panic!
(a) teach them that when they forget a fact they can count
up
the table to it, from three times one on up. (b) Send them
home with instructions to spend five minutes morning and
night (no more!) reciting the table with parents. (c) work
with them again on adding three more each time. If,
however,
you find a fundamental weakness in addition and basic
numeber
sense (I often do) then you may find it useful to leave
multiplication to the side for a bit and teach the addition
facts in the same way -- that student can come back around
to
multiplication in a month or two. (d) some weak students
benefit from counting pennies or beads on an abacus or base
ten blocks. For you it may feel like a very long time, but
usually it is only a few months (your centuries) total
while
they get a grip on number sense and can start to move ahead.

(6) Once the three times are set, then the fours, the same
exact way. One little bit at a time. A week or two per
table,
and in twenty weeks or just over half the school year, the
class KNOWS multiplication and will never forget it.

(7) When appropriate -- I use good books and follow along
with them for consistency -- you show the students that
three
rows of four can be turned on its side to make four rows of
three, so three times four and four times three are the
same.
This message is repeated **concretely** with
pictures/arrays
a few hundred times over the course of the program so the
students understand solidly how it works.

(8) You are doing the table recitation at spare times while
standing in line or while transitioning into math class,
five
minutes morning and afternoon. Students who are not
catching
it fast are also doing it for homework morning and evening -
-
and strees to parents no more than five minutes at a time.
This is vital, but it certainly is NOT your whole math
class.
Now during class you are doing all sorts of applied
problems -
- the ones where there are three children at each table and
seven tables, so that makes how many children?; the ones
where you have four rows of three flags and three rows of
four flags, and is that the same or different and why or
why
not? ; and also the review/maintenance ones where Mary has
read 25 books and John has read 18 and how many all
together,
and how many more has Mary done. As you build up you get
into
new types of problems, areas and money and so on.
You also spend a reasonable amount of time -- I would say
one
day in four or five -- on math fluency practice, just
number
skills.
Each of these, table knowledge, concrete application, and
fluency practice, feeds back to the others.

(9) when multiplication and its meaning and the first half
of
the facts are all quite quite well understood (my book
starts
after the four times table) you introduce division, again
*concretely* as undoing multiplication, and do all sorts of
interesting problems with that.

(10) My books also start introducing two-digit
multiplication
without carrying quite early, with carefully chosen
problems
using the known table facts AND with concrete models to
show
that two tens = 20. This is empowering to students who get
to
understand how the whole system works while spacing out the
memory load.
If you try to swallow the memorization of all 100 or 121
facts in one lump before you actually get to do anything
with
it, you will choke.

At the end of the year, your students know all the facts
really well and automatically, and more importantly they
can
USE this knowledge to answer real-world questions,
including
many that are meaninful to them.

They have also learned a powerful technique for
memorization
of any facts they need to know in the future, from second
language verb conjugations to medical school anatomy.