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‘For many children multiplication seems to be only something you do with numbers in mathematics sessions in school: it is not connected with any confidence to the real world.’
Understanding Mathematics for Young Children 5-9, Derek Haylock
With regards to learning mathematics, possibly one of the few things everyone agrees on is the importance of knowing multiplication facts.
However, why we need them and how best to learn them remain hot topics debated over the ages.
Having multiplication facts and times tables knowledge as a key part of our internal ‘mathematical tool box’ is hugely important because we use them constantly in everyday life.
Sadly, for most children (and for generations before them) the main purpose of learning these ‘times tables’ is not focused upon this meaningful and motivational application, but instead upon learning each fact rote-style to pass a class test, and often at speed. Which isn’t fun.
Failing to memorise and have fluent recall of these facts is cited by many adults as evidence that they are failures in mathematics, leading to a mindset that turns them off learning this wonderful subject altogether.
So, let’s agree that multiplication facts are vital skills to being a successful mathematician.
However, instead of traditional approaches using rote learning, which cause children to perform at a significantly lower level than those taught with a focus upon the ‘big picture and connections’ (Boaler, Mathematical Mindsets), let’s look at how we can all engage children with meaning and application building strong mental connections and flexibility working with how we know the brain learns best.
Having ‘conceptual understanding’ means we know why and how something works and not just ‘how to do it’.
Traditionally, maths in the UK has focused upon procedures and memorisation meaning children fail to engage in the vital processes taking place leading to both poor performance and/or a lack of appreciation of the actual mathematics.
So, what do we mean by multiplication? The answer is not as simple as we might think. Two main structures are at work here:
Equal-sized groups are added repeatedly eg 3 + 3 + 3 = 3 x 3 (incidentally, many children will tell you 3 x 3 = 6 when they see this written down demonstrating a lack of conceptual understanding)
A value is scaled by a specific factor eg ‘The giant was 4 times as tall as me’ meaning ‘If I am 1 metre tall then 1m x 4 = 4m so the giant must be 4m tall’
Learning multiplication tables is the first of these two structures – ‘repeated addition’ – but, as we can see, it’s also the knowledge on which we draw to solve problems involving scaling.
The following tasks aim to help teachers and parents present the learning of multiplication tables as meaningful and engaging for all children. The human brain learns mathematical facts most successfully through connection and meaningful application.
Multiplication as grouping is a form of ‘quick’ counting. When we have three groups of three we could count the objects in ones – 1, 2, 3, 4 etc – but knowing that the first group has 3, that by the end of the second group we will have 6, and that by the end of the third group we’ll have 9, means we only have to say three numbers (instead of nine).
This ‘quick counting’ only makes sense when children can see and connect the processes taking place.
All children need extensive experience counting in ones, and in comparing this with what happens when we only say the total of the groups we’re adding (so, three groups of three in this example), so they see what multiplication tables are and how they speed up our ability to count and calculate.
Teaching children to reel off a list of numbers can sound like they have multiplication knowledge: ‘Let’s count in 2s – 2, 4, 6, 8, 10…’.
But when we examine this more closely they are often starting in the same place every time and have essentially just learned to recite the equivalent of a nursery rhyme. Using and applying this knowledge flexibly then becomes almost impossible.
Adult: “What’s the capital of France?”
Adult: “Wow, yes. That’s amazing you were so quick. Tell me about Paris and France.”
Child: “What’s ‘Paris and France’?”
The body you have is a fantastic starting point for understanding and practising our times tables!
Start this activity by asking a group of children to come and stand where everyone can see them. What do we notice about equal groups of body parts? What can we see and describe? There are multiple things.
What comes in 2s? Hands, feet, legs, eyes, ears?
Can we see any other equal groups?
“Fingers come in groups of 10s and 5s,” or do they? What about thumbs? Does that mean that fingers come in groups of 8 and 4?
How many ears do 2 children have? Or 5 children?
Learning mathematics successfully is about the process far more than the destination. In order to ensure we are praising children’s ability to explain and prove their understanding (rather than spot facts at high speed which they may know but don’t understand).
Use interlocking cubes to represent the equal ‘human body’ groups the children notice.
Legs come in groups of two so here are three groups of legs – 2, 4, 6!
As our counting system works in base 10, use groups of alternating coloured interlocking cubes to create a number line and reason what is happening to the totals as we as we add more equal groups.
Here we have three groups of two (representing our ‘legs’ story’) and we’re comparing them with groups of 10. We can see that three groups of two is four less than 10. We need two more groups to equal 10.
Now we can talk about what we’ve built, and capture this first as language, then as symbols, to represent our understanding.
So, I can see 2 + 2 + 2 = 6
Have we ever thought about using animals to help us learn our times tables? Just think about how many different numbers of legs animals have – 2, 4, 6, 8, 10 – and some have so many it would be very difficult to count (and they don’t often stand still with that many legs!).
Get children to carry out research into which animals have a particular number of legs and sort them accordingly.
Now create concrete representations.
Dogs have four legs so this could be four dogs! Or lions, or giraffes, or ant eaters! 4, 8, 12, 16…
Now I can compare four groups of four on my number line made of 10s. I can see that four groups of four are more than 10 and less than 20. There’s space for one more group to equal 20 so five fours must be 20.
Learning to count in groups of 9 is considerably more challenging than counting in groups of 10. So, how can we use our understanding of counting in groups of 10 to find a pattern for counting in 9s?
What about 9 x 5?
Well 10 x 5 = 50. But each group is only 9, not 10. Nine is 1 less than 10, so there must be 1 missing from each group of 10. That’s five ones, or 5 x 1, which is 5. There the answer must be 5 less than 50, which is 45!
Try this trick for working with 8s.
We know 8 is 2 less than 10, so build an 8 x 5 model and see what you notice when you make it up to 10 with different coloured cubes, just like we did with the 9.
What about other ways of multiplying by 9? 8 x 9.
Well I know that 8 x 10 = 80, and if I’ve only got 9 groups of 8 then it’s 8 less than 80 which is 72!
See if this works with 14 x 9.
Perhaps start with what you know about 14 x 10, build a model and make the model tell a story eg ‘I have 9 bags with 14 items in each. How many items altogether? The story helps us visualize the problem and see how imagining we had 10 bags instead of 9 would help us calculate.
Imagine I am wondering how many days there are in 6 weeks. That would be 7 days x 6, so I will need either my 7 or 6 times table (because 7 x 6 = 6 x 7).
Let’s build it out of cubes. We’ll use an array:
Here are 6 groups of 7. I don’t know how to count in 7s yet.
Use ‘Notice and Wonder’: what can I see in the arrays? Can I see any facts that I already know or are easier to calculate?
Ah, now I can see the 5 and 2 inside 7. So I could calculate 5 x 5 and then 2 x 5.
The equal grouping of multiplication facts is most easily seen and understood when we create arrays.
We’ve used arrays already in this article by creating equal rows and columns of cubes.
Arrays allow us to see multiplication facts easily, and to partition facts into parts so we can see that 6 x 4 can be calculated as ‘double 6 plus double 6’ or ‘5 x 9’ as ‘(5 x 10) – 5’
Arrays exist everywhere in real life when we begin to look around us. They are a fantastic starting point for noticing and describing multiplication (and therefore division) facts.
They can also prompt some much higher-level challenge questions such as: ‘If we know the area of one of these ceiling tiles, how could we use this to calculate the area of the whole ceiling and do we need to count every single tile?’
To find out more about Karen Wilding Education and its services head to karenwildingeducation.co.uk.
Main image credit: Elzbieta Sekowska / Shutterstock.com
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