Part 1,        Contents, Introduction, Starting Up: meaning of number, counting, naming,
                relative value, place value, games

Part 2, p.7     Addition and Subtraction

Part 3, p.19   Multiplication and Division

Part 4, p.28   Other Thoughts: two figure addition and subtraction, fractions, least
                common denominator, prime factors, 111 day, Casting Out Nines.

Part 5         A Flexible Worksheet, Fact Sheets and Number Charts to copy.

    This booklet is just intended to be a framework for thinking about the teaching of
early math, or arithmetic, whichever you choose to call it.

    Mainly it is about adding, subtracting, multiplying and dividing.

    It is not written in technical language, and it is not completely correct as to the use
of widely accepted terms.
    It is not a text to follow and it will not tell you how to teach.

    You will find many spaces to fill with your own knowledge because there will be
holes in it.
    You will have to decide when and how to use what you find here.

    The purpose of this little book is to help you provide children with their own simple
framework for building number knowledge on, using words they are familiar with, no
matter what text or program you may have to use, or wish to use.

    I intend to speak often in the first person to constantly remind you that these are
simply my opinions, based on experience.

    I spent 35 years in Elementary education, some with Grades 4 to 7, most with
children from Kindergarten to Grade 3, ages 4 to 9. One year was taken off to earn an
M.Ed in Early Childhood Education.

The Game:

    Children like to think in terms of what they know. Most know and love games.

    I believe I did best when I taught math as a game-like subject.
    In many respects it is just like a game. There are definitions to learn and rules to follow. How well you do depends on knowing definitions, rules, and the many
facts that make one quicker at the game. Like many games, speed is important,
but second to accuracy.

    I don't intend to say how a young child's teacher is going to get this game of
numbers started; just that the child, and all the other teachers the child will ever have
for number work, are depending on this being done well and thoroughly.

September, 1997

What is a number, in common usage, and what is number?

    Well, most of the time it is a symbol for a certain quantity.
    Sometimes it is the name of a number that is a symbol for a certain quantity.
    Sometimes it is just a name.
    Sometimes it is a position in an ordered group.
    Sometimes the symbol is the same, but the name is different (3, three, third).

    We could talk about number and numeral and ordinal and names. The point is that
the question is not as simple as it seems, and common usage flits from one thing to
another.

    Young children have to be taught this from the beginning and reminded from there
on. And basic to the main usage of number is the concept of one, or the unit.

    They have to know what game the speaker is playing with number.

    If they aren't sure of all this, then is it any wonder their minds can go astray,
especially if they aren't brilliant and mathematically gifted?

    Consider these questions:
    What's on the back of that player's shirt?       How many cookies do you want?
    How many is 5?       What do you say when you see this shape? 5
    Who is fifth in line?       Is this the third time you've used that excuse?
    Are you number 8 in line?       Could you count to 20 for me?
    What time is it?       How long is it until lunch?       When is my birthday?

All are about number in the most general sense.

    Much of this learning may be started before a child gets to school.

    Most teachers do not see these early beginnings, no matter when they start, as
having the importance or status of calculus, for example.

  &nbspBut, if the many uses of number are not clear, and tied to the main concept of
number, then a child can be expected to have difficulty learning the basics of
arithmetic or math.

    Before going on, please consider introducing the word 'equal' very early, and
keep it up as you go on with number work.

    Equality is a very important concept.

    It can be a little song sung with meaningless sounds.
    It can put things or people in order.
    It can be a way of finding the amount of things or people in a group.
    It can skip numbers if you count by equal groups rather than by ones.
    It has a direction and can be done backwards and forwards.

    Counting should have a purpose, but it means more to the young child.

    It can grow into a way of developing a sense of the relative size of numbers when
the real quantities cannot be seen and felt.
    A lot of counting should be done.

    At first the main concern may be to focus on the joining of number name and object
(two, and two cookies in the hand), number name and numeral (spoken two, and
symbol 2 ), or the one-to-one relationship of object-number-numeral.
    Later this becomes more abstract as larger numbers are dealt with.

    Early on, the system of naming numbers should be brought in and one should not
underestimate the difficulty the names between 10 and 20 can cause. My
kindergarten classes enjoyed calling these the "terrible teens", even though they
weren't all "teens".
    From eleven to nineteen, the English naming system does not have the simplicity
found from 20 on, and it does not parallel the simplicity of the written numbers.
    Still, it provides a useful contrast that can help children see how numbers are
named from 20 on.
    Again, my Kindergarten children enjoyed thinking about what counting would have
sounded like if we had a completely consistent system: "tenty", "tenty-one (11), tenty-
two, tenty-three, ...", or better still, "onety(10), onety-one, onety-two, onety-three, ..."

    As children learn to count beyond 100, they must continue to develop the idea of a
naming system, or reading system.
    It is important that children see how easy it is to read large numbers. Once they get
past 19 there is relatively little that is new to learn, if they can see how we keep using
old names in new ways, with very few new names.
    In particular, it is useful for them to see that when we learn to read 387, we can also
read 387,387,387 with only two new words needed. Children enjoy doing such things,
and doing so gives them a very important feeling of confidence.

    It is essential to emphasize this simplicity as often as possible. Many children need
to be helped to see how easy numbers really are to work with. They have been told
often that Math is hard and their mother, or father, or brother, or sister had trouble with
it, or they are just naturally inclined to worry about what is to come.
    There will be plenty of times to challenge the more able, without discouraging those
unsure of their ability.

    In terms of relative position, and size, one may later start to emphasize questions
such as:
    What comes before 76?    What comes after 76?    What is between 47 and 49?

    Those adults not in on the early stages of counting should note that there are some
big steps for children to make in answering: What comes after 49?  What is before 60?

    In my classes we were counting every day. This was the most regular part of it:

    There was a solid paper strip from a cash register roll stapled high up on the wall
along the front and one side of the room. On it there were all the numbers from 0 to
203... . Every day we moved a stiff paper marker along this strip to show how many days
we had been in school.
    Of course we used this strip for other counting as well. We could easily count by
tens as every 'tens' number was underlined.

    As experience was gained we also used it for varied skip counting of the type you
need to count your change. That is, we may have counted the days by tens, and then
ones; or by tens, then fives, then ones. ( Five and ten places were marked with color.)

    Then there was a section of the chalkboard which had three vertical rows of the
numerals 0 to 9 printed on it, counting upwards. We had a steel chalkboard and used
markers on magnets to indicate the numeral for the day in each of the hundreds, tens,
and ones places.
    This way the children could sense the relative value of each place, how the ones
climbed from 0 up to 9 before the tens could move up one place, and the long wait for
the hundreds place to move up one.

    To show how the numerals are used over and over, we also had a small chart with
three hooks, one for each place value position, and on each hook hung a set of cards
from 0 to 9. We changed the cards daily to indicate the days we had been in school.
(We also kept a calendar with hooks for the numbers to be hung on daily.)

What is place value?

    Place value came in with counting, but it also needs to be looked at by itself.
    We normally use a Base Ten Place Value Number System. We represent numbers
by using our numerals 0,1,2,3,4,5,6,7,8, and 9 over and over again in a very special
way. Put simply, it means that every time we get a group of ten, we can call it
a new kind of 1.

  &nbspThis contrasts with what could be called an Additive-Subtractive System as in the
Roman Numeral System where the numerals "I, V, X,L,C,D,M" are used in an entirely
different way.

    With Roman Numerals there is no need for a 0. With our system the zero symbol is
very important, and is the only symbol with two functions. First it is used to show the
amount is zero, and second it is used to occupy a place and show that place is there,
but empty. In 30, the 3 means three, but that's only part of it. The 0 to the right of it sets
the 3 in a special place and for us that place has meaning. A 3 sitting there is read as
three tens. In 300, the 3 means three, but the two 0's push it into a new place where
we read the three as three hundreds.
   This is a good time to reinforce the idea that numbers can have different jobs to do.

    Children need to understand this thoroughly. They need to see the real difference
between 30 and 03, 57 and 75, 103 and 130 or 301 or... Real objects help here.

    Roman numerals can provide a useful contrast to draw their attention to the
workings of place value in our number system, and of course grouping and counting
real objects is very valuable.
Note: This need to understand place value comes at a time when many children are
prone to reversals, reading and writing 'on' for 'no'.

    When we kept track of the days at school each morning in my classes, the number
line that ran around the room gave little place value practise, but the two ways we
used the numerals 0 to 9 to record the number of days definitely did emphasize place
value.
    I made up a simple little set of "number makers" to practise place value, but they
also had other uses. These were made of stiff durable paper (Bristol Board), about 3
inches by 4 inches. I used a sharp knife to cut slits in the paper, through which I ran
two strips of the same material, about .5 in. by 6 in. On these strips I printed the
numerals 0 to 9 in a vertical line. The slits were placed so that the strips could be
threaded in from the back and out through the back again, leaving just enough of each
strip exposed to show one numeral. Another plain strip was stapled to each end of the
strips to give more strength, and keep them from being pulled through.
    This allowed the children to practise making numbers from 0 to 99, while forcing
them to concentrate on place value. We also had little games where each had a partner,
made a number secretly, and then had to decide who had made the larger or smaller
one, depending on what I called out after their number was made.
  &nbspIt can take quite a bit to convince some children that 40 means more than 39.

Note:
    Some teachers like to give emphasis to the ideas of place value by making up
counting activities using a different base, eg., forming a group every time you reach 5
and calling that group a new kind of 1. This would be using base 5.
    This can be a good way to force adults to see what young children have to learn,
but I found it could be confusing for many children and the time spent on it was better
used on added attention to place value, base ten.

Games:
    Here I'm going to mention Gordon's Games, a set of games I developed which is
so named, not for ego's sake, but to try to get away from problem of choosing a name
someone else was using and liable to object to my using it too. I had called
them, "Two Up", a more appropriate name, but it had many conflicts with others.

    Most of the games in this booklet use a simple set of 20 cards having the ten
numerals, 0 to 9, twice each, and are usually played with two children together.
    These games deal with simple matching, counting, number order, comparison of
value, place value, addition, subtraction, multiplication, division, multiples, factors, and
even fractions and decimals if you want to carry them that far.
    They are useful in practising skills just covered and keeping up old ones.
    For example, my classes enjoyed the games where they picked up two cards each
and tried to make the larger number, or smaller number if that was the object of the
game. Any misunderstanding of place value showed up quickly here and was soon
corrected. As well, the left - right orientation problems of some were helped.

   (Some class games and language related games are at the end of the booklet.)
    One of the class games is designed to give practise in addition, subtraction,
multiplication, and division facts.
As well it requires children to form their own questions to ask.
    These questions start out to be simple, but grow to be to be more complex, in their
eyes. That is, 3 + 4 = __ can progress to __ + 4 = 7.

    I won't say any more about these games, but they are available from me at my
e-mail or snail mail address, and, for free, through my home page, all on p.1.

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