David Fielker considers an element of logical thinking that children can find difficult to grasp. I remember my elder son at the age of two or so, sitting on a friend's knee, and looking up at her and saying, "Not 'no' is yes, isn't it?" It was one of those one-off rare gems that occurs with children, and which they unfortunately seem less capable of reproducing when they get older.
Negatives generally seem to worry children, and sometimes adults. When Mastermind was all the rage (the pegboard game, not the TV quiz), I used to watch teachers at conferences playing. You had to guess the arrangement of four coloured pegs that had been hidden by one's opponent. This you did by laying out a series of guesses in the forms of a row of pegs, and the opponent replied each time with black and white pegs, a white one for each peg that was the right colour but in the wrong position, and a black one for each peg that was the right colour and in the correct position. If you received no black or white pegs at all then you knew that the colours you had laid out were all wrong. This was a great advantage, especially if it negated four colours out of the six available, yet many people considered the absence of black and white pegs some sort of failure, as though you needed something actually to be there in order to read any information from it.
A similar situation occurs with children when playing the game Twenty Questions, which we often used to play with Logiblocks. "I am thinking of one of the blocks," I said to my current class of seven-year-olds. "You may ask questions about it, but I can only answer yes or no."
A typical sequence was: "Is it red?" "No."
"Is it yellow?
"Is it blue?" Some groaned and complained. But often the same waste of a question occurred again. Getting a "yes" was positive, reinforcing, encouraging, confirming. It had an ethos to it. The cold logic of knowing that a "no" gave you exactly the same information was far more sophisticated.
Yet the same class was later coping effectively with three-set and even four- set Venn diagrams (which they had created themselves round the Logiblocks), arguing that a square could not go in a certain region because it was "not red", and describing regions as "red and thick but not triangles and not large".
We can use negatives in order to make definitions. There are various ways of defining even numbers, some of which are better than others, but the odd numbers are invariably defined as those which are not even. Thus far, this seems on the face of it fairly sensible. However, it poses some problems when you consider even numbers in the wider context of sets of multiples. Even numbers are multiples of 2, and this is perhaps the most satisfactory definition. This leaves the odd numbers to be those which are not multiples of 2. Now consider the multiples of 3, perhaps as I described in a previous Maths Extra ("One after the other", October 6, 1995), by looking at the whole numbers as a row of coloured Unifix cubes in red, yellow, blue . . .
R Y B R Y B in which all the "blue" numbers were multiples of 3. Here the not multiples of 3 are coloured either red or yellow, and there is a difference between the two sets: the reds are 1 more than a multiple of 3 and the yellows are 1 less. Our usual concern with multiples rather than with not-multiples bypasses the mathematics that can be obtained from this situation, where you can note, say, that red + red = yellow or red + yellow = blue, and similar equations. These build up a little structure which (in the ancient days of "modern mathematics") we called arithmetic modulo 3, and which can involve some purposeful manipulative algebra in proving, for instance, that 1-more- than-a-multiple-of-3 added to 1-less-than-a-multiple-of-3 is a multiple of 3.
Prime numbers also have a negative quality about them. They are the numbers which are left when we remove the numbers which are multiples, except the numbers of which they are multiples. That last sentence illustrates that one of the difficulties about primes is precisely that of explaining sensibly what they are! It is easier to use the "sieve of Eratosthenes" and on a 100-square colour in all the multiples of 2, except 2; the multiples of 3, except 3; and so on, as in Figure 1.
These are the composite numbers; the primes are supposed to be those that are left. In other words, the primes are those which are not multiples of other members.
Except that they are not. All numbers are multiples of 1. Oh, but we do not consider those!
In any case, 1 is left on Eratosthenes' sieve, so is it prime?
Well no. The reason is sophisticated, and has something to do with the so-called higher theorem of arithmetic, which says that any whole number can be expressed uniquely as a product of primes (disregarding order). For instance, 24 = 2 x 2 x 2 x 3.
If we include 1 as a prime, then 24 is also equal to 2 x 2 x 2 x 3 x 1 or to 2 x 2 x 2 x 3 x 1 x 1 and so on, and this makes the theorem false. Young children are not really worried about the higher theorem of arithmetic. But 1 is the odd one out: it is neither prime nor composite.
Zero, incidentally, poses some more interesting problems. It is another negative, an absence of things to be counted. That in itself should not make it difficult, but some teachers of young children think it is a more difficult idea than the other numbers, so they leave it until later; then, of course, the children do find it difficult, because after all, if it was as easy as 1, 2, 3, they would have learned about it at the same time, so it must be more difficult!
The awkward thing is that many children regard it somehow as not a number at all, and when asked, say, how many Logiblocks are in this (empty) set, they are inclined to use the word "nothing", an absence of anything, rather than use "zero", which attaches a number to it.
Later, when they realise that any number multiplied by zero is zero, they can relate this to sets of multiples, and see that zero is a multiple of every number. It is certainly not prime, and it has more divisors than any other number. This last fact is surprising when you have become used to the idea that numbers with a lot of divisors, like 360, tend to be large.
(If we consider the integers in general, then a number such as -6 is even because it has 2 as a divisor Q or more strictly +2, which is not necessarily the same thing in this context. However, can we say that 7 (or +7) is a prime when its divisors are 1, -1, +7 and -7?)
Those brought up on the traditional definition of a prime as "a number which can only be divided by itself and 1" are quite sure that 1 is prime. It depends how you interpret the word "and" in that definition. I prefer to define a prime as "a number which has exactly two divisors". This excludes 1, which has only one divisor, and includes the relevant part of the traditional definition, because the two divisors must be the number itself and 1. Or is that also too sophisticated? It is not for my current class of 11-year-olds, who now readily dismiss 1 as a prime when anyone suggests it, explaining that it only has one divisor.
Many years ago I was given the traditional definition by another class of 11-year-olds, and when I asked if that included 1 they said yes, it did. I asked if they could give me a definition that did not include 1. There was silence for a few minutes, then George said: "Any number that can only be divided by itself and 1 . . . except 1!"
George obviously understood the idea of negatives.
David Fielker is a freelance lecturer and writer, and is teaching part time at the American Community School, Egham