Recently, I’ve been thinking about the driving lessons I took when I was 17. By the time I first stepped into his car, my instructor had been teaching people to drive for decades. Every sentence he said felt reassuringly rehearsed. This was especially true of a maxim that he repeated whenever I was struggling with an aspect of driving:
“First, we think. Then we practise until we don’t have to.”
Sometimes he’d justify this little maxim by describing hypothetical situations:
“When you see a football rolling into the road in front of you, I want you thinking about the kid that might be chasing after it. You can’t do that if you’re thinking about how to change gear or how to use the brakes. How can you think about the important stuff if you’re still thinking about the little stuff?”
In other words, my driving instructor recognised the importance of chunking to the development of expertise.
Chunking is the process through which individual pieces of information are consolidated into larger meaningful units. For example, in my first few driving lessons, I had to pay conscious attention to operating the clutch, moving the gear stick and assessing whether my speed required a gear change. Through conscious thought and practice, these separate actions each became automatic and were eventually chunked into a single unit: changing gear:
As I developed as a driver, changing gear, using the steering wheel and working the brakes were then chunked into a larger single unit: manoeuvring the car:
Eventually, these aspects were chunked into an even larger single unit: driving competently.
I am able to drive to a legal standard precisely because various skills and bits of knowledge were consolidated into larger and larger chunks through conscious effort and practice. (The visual model above is borrowed from the brilliant Efrat Furst. You can find more about this here: https://sites.google.com/view/efratfurst/pyramidmodel)
This chunking of bits of knowledge and skills into larger and larger meaningful units so that we can do more and more complex things is a pretty powerful way to think about the learning process.
And this brings me onto the year 4 multiplication table check. In relation to the check, an education journalist recently posted the following tweet:
At first glance, we might agree that this is a remarkable state of affairs. After all, don’t we want pupils to grasp that 8 x 7 can be reimagined as (8 x 5) + (8 x 2) or as (8 x 10) – (8 x 3)? Isn’t this reasoning about the distributive property exactly the sort of thing we want pupils to become familiar with? It certainly is. Nevertheless, it seems that the government doesn’t want pupils to use such reasoning to work out basic multiplication facts forever. They want pupils to have chunked these reasoning steps into a single multiplication fact in each case by the end of year 4, as evidenced by the need for rapid recall to pass the multiplication table check. Why might the government see this as necessary?
If we look at the components of the national curriculum that are commonly taught in year 5, it is clear that pupils need to put their knowledge of basic multiplication facts to a lot of use. And if pupils are reasoning their way to basic multiplication facts like 8×7 (i.e. reaching the answer of 56 through multiple steps), then we are adding extra steps to any chain of reasoning that they undertake.
Let’s consider an example of what this means. Imagine that we want to teach pupils that we can work out 8 x 69 using the distributive property. We might consider this as one particular chain of reasoning:
8 x 7 = 56 → 8 x 70 = 560 → 8 x 69 = (8 x 70) – (8 x 1) = 552
This is tricky stuff. It takes time and exactly the sort of understanding of the distributive property that pupils will have been introduced to when they were initially learning about multiplication. Every pupil that has to go through multiple steps to find the answer to 8×7 is forced to add extra steps to an already complicated chain of reasoning. In contrast, those that can fluently recall 8×7 can focus on this more advanced application of the distributive property.
Let’s consider another example. Imagine that we want pupils to simplify 42/48. If they have to use multiple steps to divide each of these numbers by 6, they are less likely to focus on the underlying mathematics that allows this process to work. (One way of thinking about this process is to recognise the equivalence between 1 and 6/6 and to know that dividing by 1 leaves a value unchanged → 42/48 ÷ 6/6 = 7/8)
In other words, pupils should absolutely be taught to reason their way to basic multiplication facts. But this is part of the learning process, not the end goal. At some point, pupils should be encouraged through practice to recall each basic multiplication fact without having to work them out. (Michael Pershan has a cracking blog on this subject: http://notepad.michaelpershan.com/what-people-get-wrong-about-memorizing-math-facts/)
I have met many people over the years who have stated that they coped fine with mathematics without being able to recall multiplication facts. This isn’t a surprise. I have no doubt that some are capable of overcoming almost any impediment in almost any situation. The issue really is that not all will overcome these impediments. Having spent a decent chunk of my career working specifically with those who have struggled academically, I am certain that it is these pupils who are most impeded by a lack of foundational knowledge on which to rely. The learning of number bonds and multiplication facts to fluency has often been the catalyst that has led to positive changes in what children deem themselves capable of in mathematics. I make no apology for advocating the fluent recall of multiplication facts as an aim for the vast majority of pupils.
Of course, there are other questions to consider in relation to the multiplication table check:
- Are the expectations of the national curriculum in year 5 reasonable?
- Do accountability measures such as the year 4 multiplication table check achieve what they aim to?
- Will teachers prioritise the learning of multiplication facts with children for whom other aspects of maths might be more urgent (i.e. number bonds inside 20)?
These are interesting questions, and this blog is not a defence of the check itself. Instead, this blog is simply a reaction to the belief that fluent retrieval of multiplication facts (or other foundational knowledge) is somehow at odds with mathematical reasoning. It isn’t. Regardless of one’s views on the multiplication table check, it is perfectly sensible to want pupils to fluently recall basic multiplication facts if we also want them to apply these facts as elements of more advanced reasoning.
In short, chunking knowledge and skills into larger and larger single units is essential to learning, and the development of arithmetic is no exception to this. Pupils find it much harder to reason with basic multiplication facts if they are still reasoning their way to basic multiplication facts. Aiming for eventual rapid recall of basic multiplication facts is a perfectly sensible aim within any primary mathematics curriculum.