Grouping and multiplication – is 2x3 two 3s or three 2s?
I had an interesting debate recently with a teacher from Finland about the way that multiplication should be represented and modeled to children, with my more ‘fluid’ view in contrast to her opinion that there is just one way that it should be taught. So how do you see 2x3? Is it two 3s or three 2s? More importantly, how do your class see it?
It really is all about how you interpret that x sign. If it is ‘groups of’ or ‘sets of’, then 2 groups of 3 is 3+3. If you see that sign as ‘multiplied by’, linking it to multiplication as scaling, then it is 2 multiplied by 3, or 2+2+2. This is obviously important at the early conceptual stage where a child is trying to visualize the maths going on and then also relating it to the language. Consistency with the language to match the action is important – are you emphasizing grouping and repeated addition or scaling and making an amount ‘three times larger’?
It is the way it is modeled and represented that can cause problems if it doesn’t match the language (or expected outcome).
In this example, answers for 5x3 and 4x6 were marked as incorrect because of the way the child had interpreted the sign. Amazingly, it was even marked as incorrect for the very accurate array.
It is this second question marked as incorrect that is baffling because the array is the best way to illustrate that, say, 3x4 = 4x3. The commutative property of multiplication that makes this true and an array models it perfectly. It does not matter which way an array is drawn, it still shows four 3s or three 4s.
The columns and rows can be grouped either way at this stage in a child’s understanding of the mathematics – it becomes important with matrices later in school, but this is a long way down the line.
In whichever way that it is interpreted, at some stage a child needs to see that, for example, 73 x 5 is better represented as five 73s rather than seventy-three 5s, so the more understanding that the child has of the commutative property of multiplication through the use of arrays, the better.
Repeated addition can be a limiting representation and using arrays is a better model, so it is always a good idea to link repeated addition to arrays when exploring multiplication. It is interesting to ask children what they see for 3x4 and how they would arrange cubes or counters in an array to show it. Ask them to talk it through and you’ll get a good idea of how they make sense of multiplication for themselves.
So what about our multiplication 'tables'? Is it the tables for 6x or is it the x6 tables?
The primary strategy recommended and encouraged the use of ‘multiplied by’ with the number of the table to learn first (the multiplicand) followed by the number it is being multiplied by (the multiplier) – so 6x1, 6x2, 6x3, 6x4 etc. However, we now have the Shanghai and Singapore influence which will have an impact in our schools, and their model is consistently ‘groups of’, with the multiplier first and then the multiplicand, so 1x6, 2x6, 3x6, 4x6 etc. Which do I prefer? Oddly I haven’t really a preference – at the stage that children are learning facts then they should understand commutativity and say them both ways round. Instead of using the words multiplier and multiplicand I think it is it is better to use the word ‘factor’ to show this.
It doesn’t help that ‘times’ is vague and could mean ‘groups of’ or ‘multiplied by’. As a list on the wall it is easy to see the ‘tables’ you are practicing if it comes first. However, when practicing them orally with a class, we tend to chant ‘one 3 is 3, two 3s are 6, three 3s are 9, four 3s are 12…’ so it may help children to read it in the same order, 1x3, 2x3, 3x3, 4x3…
Have a look around your school at how these tables facts are displayed and read out in each class – it could lead to an interesting discussion about the favoured way in your school for reading and writing the tables facts.
Author for Keen Kite Books
Paul Broadbent is an experienced teacher and a best-selling author of over 500 maths books. Paul has a worldwide reputation for inspiring teachers through his courses on teaching and learning mathematics and through the support given on his website www.broadbentmaths.com.