My colleague EHK told me today about a difficulty she experienced in her first encounter with arithmetic, aged 6. She could easily solve “put a number in the box” problems of the type
buy counting how many 1’s she had to add to 7 in order to get 12
but struggled with
because she did not know where to start. Worse, she felt that she could not communicate her difficulty to adults.
A brief look at Peano axioms for formal arithmetic provides some insight in EHK’s difficulties. I quote Wikipedia, with slight changes:
The [...] axioms define the properties of the natural numbers. The constant 1 is assumed to be a natural number, and the naturals are assumed to be closed under a “successor” function S.
- 1 is a natural number.
- For every natural number n, S(n) is a natural number.
Axioms 1 and 2 define a unary representation of the natural numbers: the number 2 is S(1), and, in general, any natural number n is Sn-1(1). The next two axioms define the properties of this representation.
- For every natural number n other than 1, S(n) ≠ 1. That is, there is no natural number whose successor is 1.
- For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.
The final axiom, sometimes called the axiom of induction, is a method of reasoning about all natural numbers; it is the only second order axiom.
- If K is a set such that:
- 1 is in K, and
- for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.
Thus, Peano arithmetic is a formalisation of that very counting by one that little EHK did, and addition is defined in a precisely the same way as EHK learned to do it: by a recursion
Commutativity of addition is a non-trivial (although still accessible to a beginner) theorem. Try to prove it — you will be forced to feel some sympathy to poor little EHK. If it is a trivial task for you, write a recursive rule for multiplication and prove, from Peano axioms, commutativity of multiplication. Then write a recursion for exponentiation and try to explain, why this time commutativity fails, even if the recursive scheme appears to be the same.
In an ideal world (I emphasise, in an IDEAL world), primary school teachers should be taught Peano arithmetic — of course not because they have to teach it to their pupils, but because they have to appreciate intellectual challenges that their pupils have to overcome.
It is a strange concept of British model of teachers training that teachers need to know only the stuff that they pass to pupils. For a successful teaching, at least in mathematics, a teacher has to know much, much, more.
I have to emphasise: I do not propose to introduce Peano arithmetic into teacher training courses. I talk about an IDEAL world.