## Mulling over the maths

There are some mathematical spanners we can throw in the works to get children thinking about multiplication. One interesting way of doing this is to present your class with some statements spoken by other children.

You can use speech bubbles populated with mathematical statements along with cartoon-style pictures of children saying them. These ‘concept cartoons’ are an excellent activity to promote peer discussion. They encourage cognitive conflict and learners discover what they know, partly know, and do not know about a specific idea or concept.

What concept cartoons do so effectively is fire up discussion. They bring maths ideas and thinking to the surface and encourage further learning conversations. They also act as excellent formative assessment talking tools.

**For example, you can ask your pupils to consider statements like:**

*“I think multiplying always makes a number bigger”*

*“If you multiply by zero the product is zero”*

*“Multiplying by a fraction makes a smaller number”*

*“Multiplying by a negative number makes a smaller number”*

Ask them to think about each statement without chatting to anyone else. You can take a few ideas from individual pupils at this stage without agreeing or disagreeing with them.

Then, divide the class into small groups so they can talk and share their thinking with each other. Tell your pupils that their job is to try and reach a shared understanding which involves questioning each other and listening to what their group mates are saying.

Explain that there are different ways of thinking and that investigating these statements will help them reach a better understanding. As a next step, get your pupils to complete a ‘Thinking Board’ where they investigate the statements and provide proof for each speech bubble statement.

Using a concept cartoon and a reasoning board together can be particularly effective at getting learners to think about their own ideas and how they might need to develop their learning.

Here are two completed boards with a range of responses from one of the classes I taught:

### Thinking Board 1

Statement |
We agree / disagree |
Evidence |

*“I think multiplying always makes a number bigger”* |
We agree with this statement |
We tried 8 x 6, 9 x 11, 5 x 4, 10 x 12, 25 x 5 and these all made bigger numbers |

*“If you multiply by zero the product is zero”* |
We disagree with this statement |
If you multiply by 0 then the number stays the same, e.g. 0 x 10 = 10 |

*“Multiplying by a fraction makes a smaller number”* |
We agree with this statement |
We did 4 x ^{1}⁄_{2} and this made 2 |

*“Multiplying by a negative number makes a smaller number”* |
We disagree with this statement |
We tried -2 x -9 and this made 18 |

### Thinking Board 2

Statement |
We agree / disagree |
Evidence |

*“I think multiplying always makes a number bigger”* |
We disagree with this statement |
We tried 1 x 1 and this made the same number. We tried 8 x -4 and this made -32 |

*“If you multiply by zero the product is zero”* |
We disagree with this statement |
Any number multiplied by zero has a product of zero |

*“Multiplying by a fraction makes a smaller number”* |
We think this statement depends |
8 x ^{1}⁄_{2} = 4, but 6 x ^{3}⁄_{2} = 9 |

*“Multiplying by a negative number makes a smaller number”* |
We think this statement depends |
We did 4 x -5 = -20. Then we tried -2 x -2 = 4 and ^{1}⁄_{2} x -8 = -4 |