Letís look at the basic operations on whole numbers.
Adding whole numbers has a very natural interpretation.
ìIf I have this many over here, and that many over there, how many altogether?î Well,
just join the groups and count them up.
We use words like add, plus, more, join, together, increase.
The simplest strategy for adding is just to count on your fingers!
Of course, more than ten and you need toes as well, and then youíll run out of those
too, so you might need some counters or other physical objects.
Working in base ten, we can also use some special kinds of objects and shapes to help
us to subitise (which is a fancy word for counting quickly by grouping in easily-recognisable
ways).
Often weíll use a row of ten, a square of a hundred, a 3D box for a thousand.
For simple questions and small numbers, use mental strategies: basically, learn by rote
some basic addition facts.
Does this sound a bit old-fashioned?
Actually, learning the basic facts by rote means you have more brain-space available
later on for the hard stuff.
Iím a big advocate of learning and memorising basic number facts.
Of course, you can also use other strategies mentally if you need to: imagine a number
line, or visualise yourself writing the numbers and doing the sum on an imaginary piece of
paper.
The number line is a very useful tool for addition.
You can just count along it, or you can move in jumps for tens and then units, teaching
grouping and make-up strategies as studentsí skills improve.
There are many standard written algorithms for addition that work for adding two (or
more) whole numbers ó and they can be easily extended to decimal numbers later on.
The Australian Curriculum doesnít specify any particular algorithm, because the emphasis
is on teaching students a range of computation methods and strategies.
Students are to learn to be able to ìcalculate answers efficientlyî and ìchoose appropriate
methodsî.
In fact, the Australian Curriculum doesnít say that students need to be taught a standard
algorithm at all!
What we really need to do as Maths teachers is guide our students towards more efficient
strategies (which may include the standard algorithms) and help them learn how to choose
which methods to use when.
And of course there are calculators and computers for when the numbers get larger and the situations
more complex.
It is important that students learn what addition is and means first.
We donít want them just treating their calculator as a magic black box.
That wonít help them at all later on to know when and why to add (or subtract, or multiply
or divide).
Subtraction is the opposite of addition.
ìIf I have this many, then I take some away, how many have I got left?î Subtraction is
also the underlying operation for comparing two numbers, or for equalising two groups.
(ìHow many would I have to add to this group to make it have the same number as that one?î)
Words indicating subtraction include take away, reduce, minus, subtract, decrease, without,
difference.
Also questions like ìHow many more or less?î ìHow much bigger or smaller?î
The strategies for subtraction are the same as for addition.
Use fingers or counters or other objects.
Move backwards along a numbers line instead of forwards.
Notice the natural way that walking forwards and backwards for addition and subtraction
can correspond nicely with a well-established convention that a number line increases from
left to right.
This is why it is so important to set and to reinforce simple conventions like this.
Later on, when we start adding and subtracting negative numbers, we can build on this same
strategy and convention to help students to understand and work with more difficult concepts.
Multiplication is just repeated addition.
Itís indicated by words like times, product, of, by.
Be careful though: many of these same words are used for division as well.
The standard symbol for multiplication is the cross.
When we get to algebra, though, we often use just a single central dot, or even omit it
entirely.
Please be careful when typing mathematics (including on PowerPoint presentations) that
your times sign does not look like a letter X; make sure it doesnít have serifs.
(And in the same way, when doing algebra we need to make sure that our ìXîs donít look
like times signs.)
We represent multiplication by going back to its meaning of repeated addition.
For example, to multiply 3 by 4, you could: * Make three groups of four objects each.
* Or, put down three sticks, each of length 4.
* Or, make an array, with three rows of four (or equally, four columns of three).
See how the meaning of repeated addition helps us to arrive at the correct answer.
Reinforce that meaning all the time.
Again, when the problems become harder, like with fractions or algebra, the meaning of
multiplication being repeated addition will help our students to understand whatís going
on, even when they can no longer physically manipulate the objects theyíre multiplying.
(Indeed, the idea of multiplying with arrays is fundamental to the explanation of the distributive
law later on in algebraóbut thatís a story for another day.)
Personally, I feel it is importantóno, vitalófor our students to know their times tables off
by heart.
Why?
Well, itís about the way our brains work.
Your short-term memory can only hold about 5-7 things at a time.
Thatís a hard physical limit; itís just the way God made us.
The thing is, later on the Maths problems get more complex, and itís hard to keep everything
in your head.
If you donít know your times tables, then every time you have to do an operation as
part of solving a complex problem, you have to think carefully through the multiplication
or whateveróand then youíve forgotten where you were up to in the problem.
It really does make it very difficult to solve anything hard.
Having a calculator doesnít help you with this, eitheróby the time you find it, turn
it on, type in the numbers, oops made a mistake try again, Ö and then youíve forgotten where
you were up to again.
You really must, must, must know your times tables.
And of course, as a teacher itís even more important for you to know them.
You need to have a good ìnumber senseî, so you can have a good idea right away whether
a studentís answer to a question is about right.
This goes along with the skill of estimation.
It helps you to know what kind of answer to expect and whether the answer youíve got
makes sense.
Finally, division is the opposite of multiplication.
ìHow many times does this go into that?î ìHow many groups of Ö can I split this up
into?î
Division is indicated by words like times, groups, share, into, between, of, Ö and also
by more specific words like quotient, factor, remainder.
Notice how many of these words can also mean multiply.
There are three standard symbols for division.
The first, that short horizontal line with two dots, is actually really just a small
fraction sign.
Can you see the dots for the top and bottom of the fraction?
In fact, the line in the middle of a fraction just means to divide.
The slash is becoming more common nowadays as people use computers to type their Maths.
Personally I prefer the real division symbol, but most keyboards donít have a key for it
unfortunately.
And then when we write out the division algorithm by hand, thereís the square-corner symbol
that gets used to separate the numbers.
Now dividing really is just the opposite of multiplying.
If you know your times tables, thatís pretty easy.
If you want to represent it physically, try the same ideas:
* ìCan I break this set of twelve things into three equal groups?
How many will be in each group?î * Or, ìCan put these twelve things into a
grid with three objects in each row?
How many rows will I get?î
But the really interesting questions come along when it doesnít split equally into
groups.
If I have thirteen coins, and I want to divide them up equally among three people, Ö what
do I do with the one left over?
What do I do with the remainder?
Do I keep it myself?
Does one person get more than the others?
In the old money system I could split a shilling three ways, by giving each person four pence,
but in our metric system it just canít be done.
What do you think?
What do I do with that last coin?
Would your answer be different if they were lollies or pizza or something else you can
break up into smaller pieces?
(Or dare I say, fractions?)
You see, it really depends on the context.
And it can be quite tricky if you can only use whole numbers to give the answer.