Saturday, 2 October 2010

What is a general statement?

When we finally get rid of the crumbling 1999 National Curriculum for Key Stage 2, we will be able to bid farewell to a really duff example, which occurs in the Using and Applying Number section, under the heading 'Reasoning'.

Statement j is that pupils should be taught to understand and investigate general statements. Fair enough. But then an example is given that makes me wonder if the authors themselves understand general statements. This is it: 'there are four prime numbers less than 10'.

This is a true statement, but it is not a general statement. A general statement (generalization) is an assertion that something is true in general, in other words, in all the cases in a particular set.

An example of a generalization would be: 'there are four prime numbers in every decade'. As it happens this generalization is invalid. Assuming that the decades are 0–9, 10–19, 20–29, and so on, there are four prime number in the first decade (2, 3, 5, 7), four in the second decade (11, 13, 17, 19), but only two in the next (23, 29). So the decade 20–29 provides a counterexample showing that the generalization is false.

But the statement is nevertheless a generalization, because it is an assertion that something is true in a number of cases. General statements in mathematics usually use words such as: all, every, any, each, always, whenever.

Here's a generalization for you to investigate: in every third decade starting with 10–19 there are at least three prime numbers. [Every third decade would be: 10–19, 40–49, 70–79, 101–109, and so on.]

Is this a valid generalisation? Or can you find a counterexample?


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