There are lots of classic mistakes in maths that learners of all ages make. For example, the idea that multiplication makes bigger or division makes smaller. The rules are not absolute.
Then you've got the theory that when multiplying by 10, you simply add a nought on the end. Not so. Think about 10x10.5, for example. And what about the view that the bigger the denominator, the bigger the fraction?
These over-generalisations often represent embryonic ideas that need challenging. Pupils talking, rather than teachers simply explaining, can pave the way for arguments and learning conversations.
One way of promoting a rich mathematical dialogue is to present pupils with a rumour to chew over. You could say: "I heard a rumour that the longer the arms of an angle, the bigger the angle." The idea is to get pupils talking about maths, sustaining that conversation and helping learners focus on proof.
To agree or disagree doesn't go far enough. Encourage pupils to find evidence to support or refute the claim. This rumour lends itself to drawing the same angle with arms of different lengths for learners to investigate.
A good way to get children talking about maths is to use concept cartoons. These are drawings that present a variety of ideas about a maths concept using speech bubbles.
Typically, four characters are pictured having a conversation. One character might say: "I think multiplying two numbers together makes a larger number." The others might say: "Multiplying two fractions makes a smaller number" or "Multiplying by zero makes zero" and "Multiplying one and one makes the same number."
Setting up a debate like this promotes multi-dimensional thinking, because there might be more than one right answer.
Another approach is to hold an odd one out discussion. For example, which is the odd one out in 11, 16, 27, 33, 50? Could it be 33, the only number that's a palindrome? Could it be 33 because it is the only number made up of two odd primes? Could it be 33 because it has a digital root of 6? What about the other numbers? An odd one out debate gives children an opportunity to use each other as sounding boards and debate different ideas based on their knowledge and understanding of numbers. It also helps us to see what they know, what they don't know and what they partly know.
Challenging activities that promote discussion come in all shapes and sizes to fit most learners. Just a few examples include balloon debates, true or false statements, deliberate mistakes, posters, spider diagrams, mind maps, card sorting, thought experiments and graphic organisers.
Within maths exchanges, every child counts and so does every contribution. All ideas are used as stepping stones. Don't sweep ideas under the carpet. This allows children to articulate their ideas freely without fear of embarrassment. These strategies promote rich and open discussion and provide opportunities for pupils to compare, contrast and learn from each other.
John Dabell is a Year 6 teacher and maths co-ordinator at Forest Fields Primary and Nursery in Nottingham
Resource Pack for Assessment for Learning in Mathematics, edited by Doug French (2005) pound;14.99; www.m-a.org.uk
Children's Errors in Mathematics, edited by Alice Hansen (2005) pound;15; www.learningmatters.co.uk
Multiplication Makes Bigger and other Mathematical Myths, by Pete Griffin and Sue Madgwick (2005) pound;19.50; www.devon.gov.ukonline_shop
Concept Cartoons in Mathematics by John Dabell, Brenda Keogh and Stuart Naylor (2008)pound;40; www.millgatehouse.co.uk.