If a child understands additive reasoning and the relationship between the whole and its parts, it is a fairly straightforward conceptual step to understand multiplicative reasoning. Multiplicative reasoning should be modelled as repeated addition in the first instance. Adding multiple equal parts (for example 5) might look like this:
5 + 5 + 5+ 5 is equal to twenty. Children need to understand that multiplication allows for efficient repeated addition. You have your thing to be multiplied (5) and the multiplier (4): 5 +5 + 5 + 5 = 5 x 4. Creating arrays and deliberately connecting repeated addition with multiplication makes for sound understanding.
How children work out the whole should not be taken for granted. At first, children might count each item in the array. Counting in multiples can be achieved by first skip counting. Children might whisper the numbers while counting except for the last in each row, which is said out loud. Then replace the whispering with counting in their heads and then simply saying the multiples. Over time, given sufficient practice, children will internalise these times tables.
Commutativity is important here – the array used above shows 5 x 4 but rotated it shows 4 x 5. Times tables taught systematically and with such conceptual support should be straightforward for children to learn comfortably before the end of year 4, especially when we consider it like this:
Of course, children need time to practise well and multiple representations help children to make connections. Graham Fletcher’s blog post describes the use of pictorial representations on flash cards – an approach that is a great form of low stakes testing to support the learning of times tables.
This image supports the understanding of having a ‘thing to be multiplied’, a multiplier and a whole. With practice, children will be able to subitise from glancing at the flash card, becoming fluent and accurate with times tables recall.
Some children will grasp all this quickly and can work at a greater depth while children that need more practice with the basics get it. Still using the array, children can easily begin to think about distributivity simply by splitting the array into parts:
The part above the line is 5 x 2 and the part below the line is 5 x 2:
5 x 2 + 5 x 2 = 5 x 4.
There is lots of scope for systematic thinking about equivalence with a task like this.
Arrays are perhaps not the most efficient of representation so a progression is to get children to be able to represent multiplication in bar models. First though, Numicon to work on the language of size of each part, number of equal parts and the whole:
Numicon is a great manipulative to represent multiple parts because of its clarity of the ‘size of each part’. Multi-link cubes could work too, but children would need to organise the parts into different colours to differentiate between them:
Building worded statements using a manipulative will ensure children practise the language needed to internalise the concept of multiplicative reasoning. Dropping in some of the inverse relationship between multiplication and division could be useful here too. Doing it systematically can also help keep times tables knowledge conceptual and not shallow:
Commutativity could be brought in again – showing that 3 groups of 4 is the same as 4 groups of 3 using manipulatives arranged with intent. Alongside this, comparing the similarities and differences with the worded statements will get children to think with clarity about equivalence between two multiplicative expressions.
Bar models are a versatile representation that can be used to solve a wide range of problems later on, so getting children to sketch out multiplication and division statements using bars enables them to practice a versatile skill. We should expect great accuracy in their drawings – they should be representing equal parts. If children also represent the same expression on a number line beneath the bar model, we can encourage links between representations and lay the foundations for trickier calculations and problem solving as they progress through school.
Update: The NCETM recently published this account of teaching the six times table, with some great ideas for depth.