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sums will set you free

how to teach your child numbers arithmetic mathematics

minus and zero,
dealing with no-thing and less than nothing!

‘This nothing is a very important something, since it is that out of which god created everything’. [1]

• zero
• minus and zero, dealing with no-thing and less than nothing!
• borrowing and lending, and paying back

zero

Zero (nought, naught, nil, null, nothing, nowt) usually written as 0, looks so innocent; but if taught sloppily or incorrectly, zero becomes the foundation for much difficulty and confusion.

Zero (0) has two main uses:

• As a placeholder, as in a thermometer; and
• to indicate nothing.

But there ain’t no such thing as nothing (see Dividing by 0, and also see Counting and addition). You can have an empty basket, or a basket without any eggs in it, or with no fairies in it, or even without a bath full of water in the basket. Of course, there is probably air in and around the basket, and there might be some dust, but in all circumstances the basket remains, whether empty or full of Easter eggs.

In the number 1017, the zero is a placeholder and it tells you that there are no one hundreds in this number. It is a placeholder in the sense that if you wrote 1 17, you may well misread the number as 1 and 17, or even as 117. Of course, you could put other placeholders, such as 1X17 or 1☺17. And it was by such devices that numbers did develop at times, until the zero became the standard placeholder for an ‘empty’ column.

Thus in standard numbers, zero is both a placeholder and an indication that there is nothing in the basket or column.

Now here, in the photo of a thermometer, you see a zero that is only being used as a placeholder at nought degrees Centigrade (0°C). You may well realise that at zero degrees Centigrade, there is still a lot of heat remaining in the air, or whatever else is being checked for temperature. in fact, there are another more than 270 degrees.

Now, obviously, a young child learning arithmetic will not grasp all this and more complexity (see below). But you should have some awareness in order that you do not introduce confusions to the arithmetic which will cause problems later.

You can have no eggs in your basket, or no socks in your washing machine, or even no clouds in the sky over your house, but you cannot have nothing at all at all.

minus and zero, dealing with no-thing and less than nothing!

Zero can also act as a reversal point. A reversal point is another form of placemarking, the place of reversal does not indicate that there is nothing there. Jack and Jill went up the hill to fetch a pail of water. Well, they went up the hill, step by step, getting higher and higher (no, they weren’t taking drugs).

Count as you are going upstairs - one, two, three, four etc., then as you are coming down - ten, nine, eight, seven, six. It matters where you start. Start counting from half way up - one, two, three, four. Now start counting as you go down - three, two, one. Now the next stair is zero, and as you continue, minus one, minus two.

Now you can number any stair zero and count forward as you go up, and count downwards as you go down. Or, if you feel like being awkward, you can count down when you’re going up, and count up when you’re going down. You can decide to call any one of the stairs zero, your starting point. You can even decide that your starting point is seven [7] or one thousand three hundred and twelve [1312]!

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In Britain, and other places the ground floor is, effectively, zero. Show the child the buttons in a lift, if you have the chance. In the United States, the ground floor is commonly referred to as the first floor. If there are floors below the ground, you will see buttons with -1, -2 etc.

Get the child used to this, walk forward counting steps, then turn round and walk back. Or walk backwards and take the chance that you will trip over something or somebody. Use the square tiles in the precinct.

Pick a square and see if it ends up on the same number on the way back. In free walking, do you end up from where you started, how regular is your step length?

Tiling of this type is useful to teach a child about distance and estimating distance. In the picture above, you will see a tape measure in the inserted picture, in this case each tile measuring 40 cm. Thus five tiles is two metres, and, therefore, fifty times five tiles [250 tiles] is one hundred metres [100 m]. The person can guess at how far twenty metres or one hundred metres would be, and then check by counting the tiles.

There are many other ways that you can learn to estimate distance, for example, by checking the milometer on a car, or the distance of running or other race tracks.

As you will see the Tour de France cyclists ended the day at a lower height than that at which they started. Notice that the height numbers start at 1213 and, eventually, the cyclists finish at 425 metres. The numbers are relativised to a notional sea level (remember that the tide comes in and the tide goes out). Continue walking downhill as you reach the sea edge and you could easily become wet, or you could turn round and reverse your steps to go back up hill.

Zero is often chosen for convenience, forward and you will become wet, while the ground floor is often the best way to leave a building!

In the next diagram, the start point has been relabelled as zero, and the rest of the heights matched to that level.

borrowing and lending, and paying back

If you lend money to another person, you have less money in your pocket or bank account, and they have more. Assuming you expect to the money to be repaid, when you add up your total money, you may also add in the money you expect to be repaid. Thus it is that banks call the money that they have lent out assets, despite the fact that they now have less money in their vaults. Likewise, the borrower is wise to include his debts as a negative (debits) when adding up his money. This is all a bit strange, and not quite an intuitive way of using numbers.

cartoon: banker

The problem is to keep books balanced, and to be very careful to know what you mean by the numbers that you are using. After all, the borrower actually has the money and can use it for a while when running a business enterprise. Meanwhile, the lender, who cannot use that money at the moment, commonly charges interest (more on this at this linked sums page).

Health warning: Remember the borrower may not always pay you back - and think what that will do to our sums and your bank account. That is what a lot of the recent banking crisis [2008] was about, the banks couldn't get their money back. (For much more detail, visit abelard.org's economics and money zone.)

A central idea of mathematics is balance, used in keeping accounts, and a myriad of other matters.

Equals is the point of (approximate) balance.

Well, it wasn’t quite in balance - if you look carefully,
you will see half a handful of gravel I placed on the left-hand seat.

Of course, no five year old will be expected to follow this with the mind of an experienced adult, but the more you know and the more background you have, the more easily you will be able to understand the concepts that the child is learning. The important idea here is to grow the experience, which in this case, is an understanding of the idea of balance.

Five equals five? 5=5?

Five eggs is the same as five eggs (equals), or are they? Are the eggs in the left basket the same as the eggs in the right basket?

Five equals five? 5=5?

Or does one chicken egg equal five quail eggs?

What you count depends on what you choose to count, and how you choose to count it. There is no holy scripture that makes five equal five (5=5). Even the two numbers 5 (five) are not in the same place on your screen. When you write 1=1 (one equals one) on a piece of paper, the two ‘ones’ are written at different times, on different paper, with different ink, with different movements of the hand. It is important that the child gains an understanding that counting involves continuous decisions and choices.

Keeping score

pic of money

end notes

1. Fridugisus, 9th century. From
   ‘Carolingian debates over Nihil and Tenebrae: A Study in Theological Method’,
   Speculum, 59, (1984), pp. 757-795.
   
   
2. 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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