Before children focus on learning multiplication facts, make sure they have spent time exploring the structures within a times table. If children understand the structure of a times table, and can link related facts, it will not only increase the accuracy of their recall but will also impact on the speed of recall and their confidence.
How does understanding structure helps children remember times tables facts?
If you go down the traditional route of teaching and learning times tables through chanting and regular repetition of facts you are relying purely on children remembering linguistic and sound patterns. When asked the prompt question “3 twos are?” we supply the answer six because our auditory memory kicks in and tells us that “six” is the missing sound in the sequence; in other words, “six” sounds right. If we answer the prompt question instantly, we don’t answer “six” because we have swiftly conjured up a mental image of three groups of two and worked out it is six. We are literally just supplying the missing sound at the end of a familiar soundbite in the same we know what to say after the prompt: “Twinkle, twinkle little ?”
The problem with relying solely on linguistic and sound patterns is that they are subject to pattern interference. Professor Keith Devlin wrote this in The Maths Gene (2000):
“The reason we have such trouble [with multiplication tables] is we remember the table linguistically, and as a result many of the different entries interfere with one another.”
“9 sevens are 64” sounds plausible. We know 64 is a times table answer. Intuitively, we know the answer is about the right magnitude, and in this 7-syllable soundbite, only the last syllable is wrong. In other words, it sounds nearly right. Without recourse to an easy way to check this answer, we might settle for it.
When linguistic or sound memory fail completely, or when the answer proffered by the auditory memory doesn’t sound quite right, that’s where knowledge of structure can come to the rescue.
So what bits of knowledge about multiplication structures could help?
Obviously, there is the commutative law. This is widely taught and it is often used by teachers and students to check a fact. If your linguistic memory fails you on “9 sevens”, reverse the factors and try “7 nines”.
But what else could and should we be teaching our students about the structure of times tables to help support accurate recall of multiplication facts and develop their fluency in the broadest possible sense?
Odd and Even patterns
It is worth exploring the patterns of odd/even products in a times table. There are basically only two patterns. For any even times table, all the products are even. For any odd times table, the pattern is alternating odd-even-odd-even …
It is worth exploring why this is the case, and this investigation with the FunKey Times Tables cards is a good way to go about it. Children will discover that to get an odd times table answer, you need both factors to be odd. When considering the answer “9 sevens are 64” we want children to feel, intuitively, that this can’t be right. Both factors are odd, so the product must also be odd.
We also want children to be able to swiftly use facts they are 100% confident about to derive or cross-check other facts where they are not so confident.
Children can use known facts to derive or cross-check others using the distributive law.
If for example a child knows that 10 x 7 = 70, then it is easy to derive 9 x 7, which must be 7 less than 70. Or, they could work out 9 x 7 as the sum of 5 x 7 and 4 x 7, or 8 x 7 plus 7 etc. There are lots of ways to do it, and it is a useful exercise when practising the distributive law to show the multitude of different ways to derive one fact from other facts.
Doubles within a times table
Consider the first twelve multiples of any times table. How many multiples are double another multiple? Few teachers are aware that there are six multiples which are double another multiple. So, few students are taught this fact, which is a shame because doubling and halving known times tables facts is a great way to derive other facts.
Imagine the question is 8 x 4 and the linguistic memory is just not supplying an answer. What to do? We could use any of the following doubling or halving approaches.
- If we know 4 x 4 = 16, let’s double 16 to get 32.
- If we know 8 x 2 = 16, let’s double that to get 32.
- If we know 8 x 8 = 64, let’s halve that to get 32.
What we are in effect doing here is factorizing the factors in the times table question to our advantage.
In the first example, instead of seeing the question as 8 x 4 we are converting it to 2 x (4 x 4)
In the second example, we are converting 8 x 4 to (8 x 2) x 2
In the third example, we are converting 8 x 4 to (8 x 8) ÷ 2
The technique of factorizing will stand children in good stead in their primary maths journey but it also lays solid foundations for their work in secondary school.
We need to teach children about the underlying structures within a times table for a number of reasons, not just to help them with accurate recall of times table facts. There are four techniques listed above, all of which develop key skills, but only one of which is widely used in primary schools. I highly recommend exploring and practising all four techniques; children will benefit for years to come.