Editor note: This blog post was originally published on 25 July 2018
Multiplication doesn’t always make bigger numbers
One common maths fallacy that often crops up in the classroom is the idea that ‘multiplication makes bigger’. But, it doesn’t, not always.
Children initially learn that multiplication is repeated addition, so it would make sense that multiplying two values together makes a larger product than both of the multipliers. But this isn’t always true, and exposing this misconception isn’t always the most effective teaching strategy. Sometimes, it’s better to create a purposeful context and let children fall into a misconception trap so they can make their own discoveries, reach their own conclusions and upgrade their own understanding.
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|
Upgrade to first-class thinking
These statements will promote cognitive conflict and you can see within these responses that there are rich opportunities to unpack the statements and delve deeper to uncover ideas and see the thinking behind them.
It shows which groups are working at a greater level of sophistication and also highlights where we can help each other by using all responses as stepping stones towards better understanding. Your pupils can then work towards upgrading the statements and rephrasing them as their understanding develops.
For example, learners might associate an improper fraction as being ‘top-heavy’ but is this true? An improper fraction can be defined as ‘a fraction where the numerator is larger than or equal to the denominator’ so
Tackling misconceptions to better understand needs
So clearly multiplication doesn’t always make a number bigger in the same way that division doesn’t always make a number smaller. Presenting conflicting viewpoints and weaving them into a scaffolded discussion allows your pupils to help each other shine a light on erroneous thinking and help avoid misconceptions in the future. Learners and teachers can start to understand their learning needs and begin the process of restructuring their understanding.
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Dabell, J, Naylor, S and Keogh B (2008) Concept Cartoons in Mathematics Education. Sandbach: Cheshire: Millgate House