Austin K. A. (2000)

Comment on...

"The rebirth of proof in school mathematics in the United States" by Eric Knuth
(published in the Proof Newsletter, May/June 2000)
© A. K. Austin

The text of Eric Knuth is available on-line at...

The author reports that the majority teachers feel proof is not  appropriate for all students. I would suggest a possible explanation for this : teachers believe proof is hard because they have seen only hard proofs.
  An example of an easy proof; this would be appropriate for ages 5 to 7 (Key Stage 1 in the UK).

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
THEOREM

If we take an even number between 1 and 13 and add 6 to it then the sum is even.

PROOF

We list all the possibilities and check each one in turn.

Number

2
4
6
8
10
12

Number + 6

8
10
12
14
16
18

Is Number + 6 even ?

Yes
Yes
Yes
Yes
Yes
Yes

So the theorem is true.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Some people may dismiss this proof because it is not like the well-known proofs, say, in geometry. However that would be like dismissing shallow learner swimming pools because they are not like olympic pools. In fact this proof above shares the same pattern as the harder well-known proofs, namely,

Pattern: ...

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The Theorem states the following.
the LIST (or SET) the theorem is about;

here it is - 2, 4, 6, 8, 10, 12.

the CONDITION each member of the list does or does not satisfy, which the theorem is about;

here it is - adding 6 gives an even number.

The THEOREM makes a STATEMENT about which members of the list satisfy the condition;

here it is - all of them.

In the PROOF :- we have to CHECK EACH MEMBER OF THE LIST TO SEE WHETHER OR NOT IT SATISFIES THE CONDITION

then

SEE THE STATEMENT IN THE THEOREM IS TRUE.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The difference between the easy and the hard proofs is that in the latter the list/set is too large for us to write down all the elements and so we have to use the techniques of proof - language, symbols and logic.
  Thus the easy proofs provide a good starting point for pupils and teachers by giving a cocrete context for proof - a context which helps to explain the origins and rationale of the techniques of proof.