So, what are they? Concept cartoons are cognitivedrawings or visual disagreements that use acartoon-style design to present mathematicalconversations inside speech bubbles. The viewpointsportrayed are all different and it is this differencethat acts as a catalyst for further conversations, aslearners talk together to discuss their thinking.They make learners ideas about mathematicsexplicit and they make mathematics interactive anddiscussion-based. Why has mathematics been soslow to catch on to their potential? As formativeassessment tools they have real virtue because theyare a highly effective way of probing learnersmathematical conceptions and, crucially, theirmisconceptions. Presenting alternative ideasencourages everyone to think about things from a different point of view.

Debate and disagreementWhat concept cartoons do so well is set up adebate, invite cognitive conflict, inspire dialogueand promote participation. They encourage learnersto compare and contrast, to look for evidence andjustify their own reasoning in response to thecharacters in the cartoons. Everyone in the class isexpected to have an answer and contribute to thelearning conversation and all answers, whetherright or wrong, are used as stepping stones toachieve a fuller understanding. They are thereforeused as talking tools, to help learners build on eachothers ideas, use each other as sounding boardsand work creatively together to make sense of a keyidea. Talk is not an optional extra in a concept

cartoon, it is an essential ingredient; it is expected.A disagreement helps set the learning agenda

for what happens next in a lesson or series oflessons. A failure to agree is actually a success forlearning.

Organisational issuesConcept cartoons are more effective when discussedin a mixed-ability group of learners. This results ina greater degree of exchange and allows differentideas to surface, which can then be debated. Theyare a particularly valuable tool to employ withlearners with special educational needs and learnerswho require confidence in putting forward theirviews in mathematical contexts.

There are no hard and fast rules about how topresent concept cartoons and it is worth experi-menting with different ideas. Traditionally whenused in science, one speech bubble contains acorrect idea and other speech bubbles presentdifferent ways of thinking, ideas that are commonlycalled misconceptions. It is important to point outhere that a misconception is not faulty thinking buta concept in embryo or local generalisation, asMalcolm Swan points out in his outstanding bookCollaborative Learning in Mathematics. In my ownresearch, I have played with the original conceptcartoon formula and adapted the speech bubblesfor different concepts and different age groups. Forexample, one correct speech bubble and threeincorrect, two correct speech bubbles and twoincorrect or all speech bubbles correct.

In this example there are two possible correctanswers. The party hats shown are clearly cones buthow many learners will recognise them as pyramidswith circular cross-sections? You can see why some

MATHEMATICS TEACHING INCORPORATING MICROMATH 209 / JULY 200834

USINGCONCEPT

CARTOONSJohn Dabell describes how heuses visual disagreements to

advance his learnersunderstanding of mathematics.

A

ndy

Robb

(Bee

hive

Illus

trat

ion)

In the last few years,John Dabell has toured

Britain deliveringConcept cartoons in

mathematics inservicetraining, spoken at

conferences and workedwith a focus group ofteachers from the Cityof Edinburgh, writing

20 concept cartoonposters and 120

concept cartoons for abook and interactive

CD. These are available from:

www.millgatehouse.co.uk

children might call the party hats triangles.Naming them prisms shows us that the propertiesof 3D shapes are not well understood and are ripefor development. A rich follow-up to this conceptcartoon might involve an exploration using actualparty hats, getting learners to make their own,examining solid cones and thinking about real-world cones outside of the classroom.

A case studyIn my own research into using concept cartoons Ihave been impressed with the way children haveresponded to them. I have used a variety ofcartoons to address a range of conceptual obstaclesand alternative frameworks including the idea ofmultiplication makes bigger, which is generallyconsidered a classic mistake. I recently tried themultiplication makes bigger concept cartoon in aY6 class.

There are various ways of bringing this conceptcartoon to life. Initially I present the conceptcartoon on the interactive whiteboard and askchildren to think about each statement privatelybefore discussing it with anyone else. I might takeone or two ideas from children at this stage, or doa quick survey with the whole class. This mighttake the shape of me reading out each statementfor children to then draw a smiley face on theirwhiteboards to indicate whether they agree or not.This tells me a lot about their understanding andchildren dont have to worry about being right orwrong as I dont give them any answers.

I then divide children into small groups ofthree to talk together to share ideas and askwhether we can make sense of the concept cartoontogether. I make it very clear to children that the

focus of the task is on understanding and thatunderstanding is a team effort involving differentways of thinking. Children are encouraged tocomplete a group table (or thinking board) wherethey try to justify their groups thinking and provideproof for each speech bubble statement. You canimagine the range of responses you might get:

When presented with conflicting evidence likethis it is time to seize the moment and milk themaths in order to progress childrens learning.What happens next is crucial. A technique thatworks well here is to use envoys to push childrensthinking further. This involves one person fromeach group visiting another group to discuss theirtake on things. I normally do this for about fiveminutes, after which time the envoys return totheir group to share ideas they have picked up andthen revise their thinking boards if necessary, afterfurther discussion. The act of writing also helpspupils refine their arguments and summarise theirthinking. This is also the time to open up the debateto the whole class in order to showcase particularexamples, such as multiplying by negative numbers,proper fractions or improper fractions.

On top of this, I play devils advocate, orprovide more background information whereappropriate, by encouraging particular groups to

Statement (conjecture) We agree /disagree Proof

Multiplying always makes We disagree with We tried 1 1, and thisa number bigger this statement made the same number.

We tried 2 2 and this made 4

Multiplying by zero means We agree with Any number multipliedthe product is zero this statement by zero has a product of

zero

Multiplying by a fraction We think this 4 2 but makes a smaller number statement depends 4 6

MATHEMATICS TEACHING INCORPORATING MICROMATH 209 / JULY 2008 35

A

ndy

Robb

(Bee

hive

Illus

trat

ion)

Statement (conjecture) We agree /disagree Proof

Multiplying always makes We agree with We tried 4 4, 5 6, a number bigger this statement 7 3, 8 2, 9 9 and

these all made bigger numbers

Multiplying by zero means We disagree with If you multiply by 0 thenthe product is zero this statement it gets ten times bigger

because you put a zeroon the end

Multiplying by a fraction We agree with We did 2 and this makes a smaller number this statement made 1

2

think along different lines if they arent doing soalready. I try not to reveal any answers andactually make a point of saying: Im going to askyou not to trust what I have to say. You might tellme something and I might agree with you and goalong with it or I might disagree and give yousomething else to think about.

The most obvious next step is for children toshare each others ideas and focus on proof, tryingto convince each other whether they think a state-ment is conceptually sound or not. This particularcartoon throws up some delightful thinking. Beforereaching an end-point, looking through a variety ofmathematical dictionaries (book and online) is aworthwhile step so that more evidence can begathered.

Finally I encourage the children to try and RIP(refine, improve and polish) the statements in thecartoons. For example, the statement Multiplyingalways makes a number bigger can easily be disproved.One group thought that it would be more accurateto say Multiplying two numbers together sometimesmakes a bigger number. The statement, Multiplyingby a fraction makes a smaller number, was also foundto be too vague to be helpful and another groupgenerated their own response, Multiplying by animproper fraction makes a number bigger. Theyprovided the example, 3/2 4 6. This waschallenged by another group who argued that animproper fraction was not always a top-heavyfraction. They researched some website dictionariesto find a definition of an improper fraction as afraction where the numerator is larger than or equal tothe denominator. They said that 7/7 was an improperfraction which, when multiplied by a whole number,would result in the same product.

From one small statement a whole dominoeffect is generated, whereby children respond toeach other creatively and help build layers ofunderstanding. Leaving the classroom, I amconvinced that children have a much strongerunderstanding of multiplication and that multipli-cation doesnt always make bigger. As one Y6learner told me, Concept cartoons help you to focuson what you know and on what you have to say.

Some conclusionsIn all the examples above, it is clear that there is alot to talk about. Concept cartoons pitch one ideaagainst another. Some of the speech bubbles seemplausible, some are ambiguous and others are wayoff the mark. Whatever way, they promote thoughtfriction, spark ideas and kick-start conversation.Each concept cartoon also requires classroom

follow-up in order to investigate the ideas further,explore understandings and practise skills.

One frequently-asked question is which agegroup concept cartoons are suitable for. Thisdepends. Clearly some of the cartoon charactersmight not appeal to older learners, but the actualcontent of the cartoons straddles key stages. Somecartoons have been used successfully with learnersin key stages 1, 2, 3 and 4, depending on theirprior knowledge. I think concept cartoons arerather like a swimming pool of learners they holdso many abilities. Some learners are doing lengths,some widths, others struggle to do doggy paddleand some are too afraid to get in! Learners access aconcept cartoon at a level that suits them and weare there to supervise them splashing their ideasaround and its our job to help them make waves intheir understanding without actually doing theswimming for them.

Using a concept cartoon follows a cognitiveconflict model of teaching. Here is one suggestionof a teaching sequence to follow: present learners with a cartoon that focuses on

a conceptual obstacle allow learners to reflect individually on the

challenge encourage small-group discussion provide opportunities for feedback across the

class about the alternatives try to reach agreement discuss a possible line of investigation to find

out more hold small-group enquiries share outcomes of enquiries whole-class discussion pull ideas together consider how learners views might have changed consolidate and practice.

Concept cartoons can be used as lesson starters,develop into whole-class lessons, as plenary activi-ties or as summative assessments. Why not trythem in a maths club? Or display some aroundschool to promote discussions amongst parents,carers and visitors? When learners are familiar withconcept cartoons, encourage them to create theirown or hold a competition. The scope is endless.Give concept cartoons a try and judge for yourselfthe impact they can have on the rate, depth andbreadth of your pupils learning. Write and let usknow how you get on.

John Dabell is a mathematics coordinator at LawnPrimary School in Derby.john@johndabell.co.uk

MATHEMATICS TEACHING INCORPORATING MICROMATH 209 / JULY 200836

References Keogh, B. and Naylor, S.(2000) Concept Cartoons inScience Education, MillgateHouseSwan, M. (2006)Collaborative Learning inMathematics: a challenge toour beliefs and practices.National Research andDevelopment Centre forAdult Literacy andNumeracy (NRDC) andthe National Institute ofAdult ContinuingEducation (NIACE)

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.