A further thought on composite numbers definition

Going back to the erroneous 'definition' of composite numbers in the Year 5 Programme of Study for the Mathematics National Curriculum for England ... where we read that pupils should learn about 'composite (non-prime) numbers'.

In my previous post I pointed out that you cannot use 'non-prime' as a synonym for 'composite' because the integer 1 is neither prime nor composite.

On reflection, I realised that the error here is much more substantial than this. The concept of 'prime' applies only to positive integers. So, a prime number can be defined as an integer with precisely two factors (which will be 1 and itself).

This means that 'non-prime numbers' would include all numbers that are not positive integers. So, the identification of composite numbers with non-prime numbers would imply that numbers such as –3, 2.4, ⅚ and √2, for example, are all composite!

Of course, they are not! 'Composite' is also a concept that applies only to positive integers – those integers with two or more factors greater than 1.

That was a sloppy bit of work by whoever wrote that programme of study.

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Any reader looking for professional help in the area of primary mathematics education might like to take a look at the recently-launched website of my colleague, Ralph Manning: www.manningeducation.co.uk

Context-free calculations

So we now know that the national tests for children at the end of Key Stage 2 from 2016 onwards will include one paper (of the three) that will consist of context-free calculations.

That seems a strange idea to me. Mathematics – as far as I understand it as a humble mathematician –
does not generate calculations without a context. Calculations only ever occur in a context. Normally this would be a practical context in which mathematics is being applied, and where the numbers are likely to be attached to sets of items or units of measurement of some kind. Even in pure mathematics, on the rare occasions you might want to do a calculation it would be to investigate the relationships between two numbers that have some property – like finding the ratio of successive terms of the Fibonacci  sequence (1, 1, 2, 3, 5, 8, 13, 21, ...) – so there is a context which provides the need to do the calculation.

Yet our children will be given a test paper consisting entirely of context-free calculations, to be done by prescribed formal written methods, the argument being that these calculations are 'the fundamental processes' of mathematics.

NO THEY ARE NOT!

Mathematics national curriculum oddities

The statutory requirements for the new primary mathematics curriculum (to be taught from September 2014) have some odd aspects for which it is difficult to find a rational justification.

Here are two examples.

1) Under algebra, children (the curriculum prefers to refer to them as 'pupils') in Year 6 are required to learn how to

·    find pairs of numbers that satisfy an equation involving two unknowns

but nowhere are they required to find solutions to equations with one unknown!

2) In learning about geometric transformations, children will meet the concept of 'rotation' in Year 2, but then the word does not appear again in the primary curriculum. So, after Year 2 there's nothing at all about recognising and describing a transformation of a shape in terms of rotation and there's nothing anywhere in the curriculum about rotational symmetry.

"By George, she's got it!"

These are the words of Professor Higgins in My Fair Lady when Eliza shows significant progress in her learning! We use a lot of different idioms to indicate that we understand something or do not understand something. For example, for understanding, we might say things like:

• Oh, I see! 
• Now it’s clicked. 
• Everything is falling into place. 
• I think I've cottoned on! 
• I get the picture. 
• My eyes have been opened! 

And for not understanding, we have phrases like these:

• Sorry, I don’t get it. 
• I’m still in the dark. 
• I can’t see the sense in that! 
• That's as clear as mud! 
• You're talking double Dutch! 
• It just went over my head. 

What's your favourite way of saying 'I understand' or 'I do not understand'?

Expressions like those above reveal a number of insights into the nature of understanding. They show, for example, the importance to us of understanding things, of making sense of them, rather than just learning by rote. We can see within the expressions used a clear sense of closure, of things fitting into place, a sense of relief almost, when we 'get it'. And, by contrast, phrases that indicate not understanding reveal a sense of frustration. Understanding or not understanding is as much an emotional experience as a cognitive one!

Then we might notice that understanding brings clarity and light to the learner. Many of the phrases we use for understanding talk about 'seeing' something, having our eyes opened, or, by contrast, being in the dark.

Finally, these expressions for understanding support the idea that learning with understanding is all about making connections: connecting some new experience with other experiences or existing understandings. So, we talk about understanding as though it feels like things 'clicking' (like two Lego pieces fitting together), or 'falling into place' (like a piece in a jigsaw connecting with other pieces and suddenly making sense), or 'getting the picture'. And when we don't understand, the words 'go over our head', so we have a sense of not being able to connect them to anything and embed them in our minds. 

To learn that mathematics can be learnt with understanding in the ways indicated by these idioms is the most important thing for children to learn about this subject by the time they leave primary schools.