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By John Dabellteacher, writer and Ofsted-trained inspector

How big are your gaps? After watching the recent Channel 4 Dispatches programme about the state of maths in our primary classrooms, you might be wondering whether the Strategy booklet ‘Moving on in mathematics: Narrowing the Gaps’ is aimed at teachers or children. The programme highlighted that some teachers are less mathematically confident than the children they are teaching. How long will it be before we introduce attainment trackers for teachers?

Maths teacher

But before we press the fit-for-purpose eject button let’s press pause and remind ourselves that not even subject specialists know everything there is to know. Primary polymaths have no chance of attaining even a reasonable level of expertise in one subject during their training unless they are direct descendants of the da Vinci family. Maths surgeons aren’t trained overnight. Expertise takes time to build and so does confidence. What we need is to share and discuss generic techniques that can be adapted across a range of maths contexts.

My CPD work as an inset provider and fellow teacher has taught me that colleagues value opportunities to talk about their maths uncertainties in a supportive, informal atmosphere where they aren’t judged or tested. But you need a context. What work remarkably well as tools for discussion are visual disagreements or concept cartoons. Most teachers know about these through their science teaching but they are less well known in other subject areas.

Maths concept cartoons play a valuable role in the professional development of teacher subject knowledge and understanding. Colleagues find them helpful tools to review and develop personal subject knowledge, by asking questions that they had never thought of asking themselves. So, as well as helping to recognise the kinds of misconstructions and uncertainties that children have, teachers can use them as an instrument for pinpointing their own.

Here is an example of a concept cartoon that I have used with children and teachers:

Maths concept cartoon

Unfortunately some children hold the idea that when multiplying a number by ten you just ‘add a zero on the end’. This maths myth rears its ugly head from time to time and has been innocently mistaught somewhere along the line. Discussion could centre on examples where the ‘adding a zero’ rule does not apply. What about 25.8×10, 0.36×10 or 3/4×10? This concept cartoon also provides fertile ground to talk about another maths myth – the idea of the decimal point moving. Presenting alternative viewpoints like this invites learners to make public their own ideas in a safe context. If they are wrong then they can blame a concept character instead!

Maths concept cartoons can be used in early years right through to mature adult learners and are excellent tools for developing maths conversations and argumentation.

See over 120 more examples of maths concept cartoons by John Dabell (with Stuart Naylor and Brenda Keogh) including interactive versions on CD: _Concept Cartoons in Mathematics Education at

In order to upgrade knowledge there are plenty of other talking tools you can use to unpick maths locks and open the door to understanding. Examples include agony aunts, balloon debates, card sorts, deliberate mistakes, graphic organisers, KWHL grids, matching exercises, odd one out, posters, rumours, spidergrams, true-false statements, word definitions and games such as Who wants to be a Mathionaire. These are all active ways of assessing what children know, what they partly know and what they don’t know about a specific idea or concept.

One novel way of assessing thinking and the level of sophistication learners are working at is to try the ‘Dear Doctor’ strategy. The basic premise of Dear Doctor is born out of the idea of a newspaper or magazine problem page and follows a similar format to that of an agony aunt. Children write a maths question that is troubling them to a fictitious maths expert who then provides a ‘personal’ answer. The expert in this case are groups of children who think, talk and write a mathematical response. These corporate efforts can be compared and merged into creating a class response. How would you respond to the following example?

Dear Doctor Portion,
A group of friends at school couldn’t agree on a little problem our teacher set and we were wondering if you could help? The question we’re trying to work out is whether 4/5 is bigger or smaller than 3/4? Some of us think that 4/5 is bigger because that has the biggest number under the line whereas some of our other friends say that the smallest number means it is bigger. We’re still not sure!
Yours, in pieces,
Balroop, Lialle, Mohammed and Sofia

Children are highly tuned when it comes to confidence deficiency in their maths teachers. They can spot the dog-eared answer book discretely opened on a busy desk. They can read the nervous body language when an explanation falters or a question can’t be answered. They can also sense when their teacher is out of their subject comfort zone and follows a bullet-pointed sat-nav scheme of work. But we are all learners and our knowledge and understanding grows with every lesson. Perhaps that’s something we should share with children more often.

If you are a subject leader then why not hold a maths misconceptions ‘amnesty’ in the style of a maths audit questionnaire to staff. This could form the basis of a whole school inset day around common mistakes and misconceptions children and teachers make. It could be the sort of ‘hand on heart’ day you need to identify weaknesses; something that will make everyone stronger in the long-term by closing some gaps.

For other ideas of ways to promote discussion about maths, see Active Assessment: Thinking, Learning and Assessment in Mathematics by Brenda Keogh, John Dabell and Stuart Naylor available from


  1. ab fab
    on 18 March 2010

    Concept Cartoons... brilliant!

    Thanks so much for your article. As someone who always “got” maths and who sometimes realises she’s rushing over the head of her class, it was interesting reading in its own right. But your pointer to Concept Cartoons was fantastic. What a great resource! I’ll use them often.

    4out of 5


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