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how to teach your child numbers arithmetic mathematics

multiplication and still more counting

The approach to teaching mathematics and other languages is profoundly unsound, but the habits are deeply ingrained.

I have already provided documents explaining where the problems, are on a technical and adult level. To dig into this, start at why aristotelian logic does not work and laying the foundations for sound education.

This sub-section of abelard.org is designed to lay out a rational and logical base for teaching arithmetic and mathematics from basics. I shall not always justify the methods in this section as I go along, but the methods are very relevant and purposefully structured. Throughout this section, many of the yellow links take you to a more advanced, or technical, explanation.

It is vital to understand that there is no fundamental or logical difference between the symbolism of teaching English and teaching arithmetic/mathematics. This congruence becomes part of the learner’s understanding. It is a deep and dangerous pedagogical error to allow the learner to imbibe the erroneous concept that mathematics and English are different “subjects”.

On these pages, you will be given basic methodology and necessary examples. You will not be provided with hundreds of examples, those you make up as you work with the learner, adjusting those examples according to the person’s problems. Some examples should be interspersed which are easy for the learner, in order to reinforce and to give experience of success, while others should be aimed at specific difficulties.

multiplication, or repeated addition

Multiplication can be regarded as repeated addition, or constant addition. But often, instead of adding just one each time, a collection or set of items is added again and again.

To illustrate, we are using fairly uniform objects: Cuisinaire rods, in general the smallest (one unit) rods.

Thus, for example, two-times multiplication can be regarded as
       two add two

add two add two and so on.

A human can scan, while counting, numbers of objects up to about seven or eight objects at a glance. The greater the number of objects, the more difficult this becomes.

After a while, working out a name for each of these new collected groups of objects could become very tiresome.

So to make things easier, we make collections, or sets, of the numbers.

A convenient number of objects in a set, that has become traditional around the world, is ten. Thus, each time we gather ten things, we tend to regard them as a set or unit. I shall return to this shortly.

collecting tens

In the picture above, you will see seven collections of ten (and a couple of ten blocks along side).

Here we have six sets of ten and three blocks left over, 6 lots of 10 and 3 more. We don’t want to mix up the tens with the singles, any more than we want to mix up the snails with the lettuces. Keeping everything neat and sparse, we therefore assign a column to the singles and another column to the left for the tens.

advertising disclaimer

Yes, I know you understand all this, it has become habitual to you, but you need to be in a position to recall the details in order to explain to the learner. You don’t need to make a meal of it, but you do need to explain to the child in words and concepts the child can follow. You don’t need to spell out every detail as I am doing here, but you should be organised enough to understand the complexity of what you are trying to teach. Remember that this is damn complicated if you are two, or three, or five, or eight, meeting this cascade of details for the first time, or if you have been taught by rote.

In the picture above, we have arrived at ten sets (collections) of ten, so we run out of the numbers one to nine. Thus, we have to form another column of the groups of ten tens, that is for the hundreds. Thus:

Don’t be over-eager for this to go in and stick immediately or at one time. Be content that it will seep in on repetition and as you move back and forth among the concepts/problems.

placeholders

Remember the cloth with no things on it.

The idea of nothing is causing people considerable confusion, and you don’t want that confusion to enter the mind of the learner.

Nothing, or no-thing, does not mean that there is nothing there, the cloth is there and the air around it. The just isn’t a block on the cloth at the moment. There ain’t no such thing as no thing. If there were, it wouldn’t be there...where???

What we do need zero for, is as a placeholder where there is no ones or tens or hundreds or whatever.

Using a number as a placeholder
In the number 201, the 2 represents 200, not 2, and the zero means that there are no tens (or completed sets of ten), and only the 1 indicates an untrammelled one!

cross tables

Below is a completed multiplication cross-table for the numbers 0 to 10.

Click to see blank multiplication cross-table for printing out. [Opens in new tab/window.]
Another page, prime numbers and factors, the sieve of Eratosthenes for developing an understanding of and a familiarity with tables.

Learning by rote suits some people and not others. You will see from the cross-table above that I have marked up the two-times, the five-times and the ten-times tables. These are extremely easy tables to learn. The diagonal on the squares of numbers is also marked. These tend to be fun to learn and memorise.

If you examine the cross-table, you will see not one of the remaining numbers (those that are white) is more than one step away from the marked-up (coloured) squares. For example, examining the cross-table above, nine 8s (eight 9s) is 64 plus one more 8, or 81 minus one 9, or 80 minus one 8; and similar adjustments can be made for any of the other numbers outside the easy times-tables and the square numbers.

Working these numbers out can accustom people to playing with these numbers, rather than learning by rote. To build solid foundations upon which to advance, the more that can be understood the better. Teaching mathematics through a lot of memorised formulae will eventually leave a person bored and not understanding what they are doing. Thus using the mathematics in new situations, in order to think clearly about real-world problems, will not be developed or enhanced.

If a child is finding this sort of thing fun, then they may like to build up cross-tables for 12, 16 (hexadecimal) or 20, or more if they become ambitious.

Remember the following from the addition page, teach the order of the numbers by pointing to one block and then moving to the next block and stopping, then moving to the next block and stopping. If necessary, hold the child’s pointing hand in your own and move from item to item. Your are teaching the child to separate objects, but never lose sight of the fact that the world is continuous.

In the pictures below, you will see the sort of arrangements of objects that can be used to teach the basics of multiplication. Now, instead of just counting along a line of objects, the child learns to count the objects in two different directions. This is followed by counting all the objects on the ‘square’. As you will see in the picture, the objects are separated a little to allow the child to point without becoming muddled. This distance can be increased or reduced according to the acuity or neurokinesthetic control (brain-body control). This control is another part of the child’s learning. To an extent, it proceeds at a natural rate, but this is no reason not to introduce practice.

Thus, as control, visual acuity, concentration and counting fluency develop, the items can be moved closer together and the overall number of objects increased. Still think learning basic arithmetic is so easy when you’re five years old? So...

concentration

Be patient and remember the brain is a physical object which actually changes during the child’s learning. That process involves nerves making new physical connections, and this process takes time. Thus, constantly repeated, fairly short learning sessions, followed by time to let the brain get on with its growing, are an integral part of learning.

In this context, it is also important that the child learns to concentrate. Without concentration, memories will tend not to be laid down permanently (physically); only a short-term retention will occur, such as when you memorise a phone number before transcribing it to paper or organiser (only to be forgotten by the morrow).

By the time the child is beginning to understand the basic tables, they are already doing simple multiplication. Three 3s or seven 8s is not just a parroted noise like nine or fifty-six, it is three plus three plus three (3+3+3), that is three times three (3x3).

Here we have a three by three block moved out of a ten by ten block (here, for the moment, the ten by ten block is merely context and, of course, you don’t have to do it this way). Note the use of both ‘one’ blocks and ‘three’ blocks. As you see, the blocks are shaped in proportion, thus one green block lays alongside three white blocks, and so on. Another simple multiplication is 8+8+8+8+8+8+8 or seven lots of eight, that is seven 8s or 7 x 8, which is, of course, 8 x 7.

Of course, we haven’t yet come to writing down multiplication soms, but as you should see, several of the basic nuts and bolts are now available to move onto that stage.

Below are various arrangements of three by four or four by three blocks in a block of one hundred.

abelard.org maths educational counter

[This counter functions with javascript, you need to ensure that javascript is enabled for the counter to work.]

The full version with more detailed instructions, go to the introduction page.

So, to practise multiplications, for example 3 x 4,

• Reset Counter Value to 0;
• Change Step to 3;
• Switch Direction (if necessary) to Increasing;
• Now click on the Manual Step button four times. The red number counts to 4.

The counter counts up (increasing): 0, 3, 6, 9, 12. Thus 3 x 4 = 12.

Now help the learner to try other multiplication sums. Each time, click the red Reset button to return Manual Steps (the red number) to zero.

Remember that trying much bigger sums, such as 134 x 572, would mean wearing your finger out trying to clicking the Manual Step button 572 times. So when you reach sums and numbers beyond clickability, it is time to start introducing a calculator. At first, work using the calculator with small number multiplications and match the results on the abelard.org counter with those on the calculator.

multiplying positive numbers and negative numbers

Now try multiplying

• positive numbers with negative numbers, for instance 3 x (-4),
  by putting Reset Counter Value to 0 and Change Step to -4.
  (Don’t forget to click the red Reset button)
  Click the Manual Step button three times.
  
  
• negative numbers by negative numbers, such as (-3) x (-4),
  by putting Reset Counter Value to 0, Change Step to -4, and Switch Direction to Decreasing.
  (Don’t forget to click the red Reset button).
  Now click the Manual Step button three times.

Why is the result to this second sum 12 (positive 12)?

Three lots of 4 (positive four) is 12, while three lots of -4 (negative four) is -12.
But three negative/minus lots (-3) of -4 is +12,
just as some interpretations of not (not a chicken) is deemed to be a chicken. Remember that what and how you count are matters of choice.
The use of minus at this level tends to mean ‘to reverse direction’.
So if plus four means going up four steps, then minus four means going down four steps.
And minus minus four means go upstairs because you have reversed twice. Minus minus minus four means go down stairs, and so on.

One lot of plus four is 4, that is 1x4 = 4
One lot of minus four is -4, that is 1x(-4) = -4
Four lots of minus one is -4, that is 4x(-1) = -4
Minus one lot of -4 is +4, that is (-1)x(-4) = 4

This is often taught as a ‘rule’ such as, ‘multiplying similar signs gives + (plus, positive) and opposite signs gives - (minus, negative)’.

For a learner in the early stages, this may be going too quickly, remember that the objective is the learner understands, not merely memorises rules.
I may return to this on a later page.

At this point, you may find it useful to review the section minus and zero, dealing with no-thing and less than nothing! or, if you wish to dive into the really deep waters, look at the asymmetry of ‘not’ and follow the links there.

end notes

1. Reversing an operation is often loosely referred to as ‘doing the opposite’. For example, the opposite of riding your bicycle from Oxford to London, in clement weather, when you’re fit and fresh, is riding back at night, in pouring rain, after a hard day’s partying.

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© abelard, 2009,16 june

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the web address for this page is https://www.abelard.org/sums/teaching_number_arithmetic_mathematics_multiplication.php

2650 words
prints as 10 A4 pages (on my printer and set-up)