Any curriculum worth its salt…

When asked for my opinion about the MTC, I can’t help but think that it’s an easy win for subject leaders and schools in general.

Quite simply, it appears to me that there’s very little reason the vast majority of children should leave Year 4 unable to recall multiplication facts in their entirety, particularly in a world where Times Tables Rock Stars exists. This is our third year using it and not only have we found there to be a correlation between the MTC trial scores and pupil usage but we’ve also seen a correlation between performance in standardised testing and usage, as well.

In some cases, success in the realm of TTRS has been the trigger for success in the subject as a whole, presumably as the child twigs, “Hold on a second, maybe I’m not as bad at this maths carry on as I though,” and doubles down on effort in maths lessons – yes, in my mind, they all sound like Jim McDonald.

As I see it, recall of multiplication facts and an understanding of multiplicative reasoning are related but not as closely as we might think. It’s entirely possible that a child be able to recall the multiplication facts without actually understanding the concept of multiplication. They’re related, for sure, and it’s hugely beneficial to know them when exploring how multiplication works but they aren’t completely interdependent.

Which is why it’s a quick win in terms of accountability, to attain well at the end of KS2 you can only get so far by smashing the arithmetic paper, some level of understanding will be required. For the MTC, I don’t believe this to be the case.

I can say this without fear of retribution, however, because any curriculum worth it’s salt will indeed contain a thorough exploration of the fundamental laws of arithmetic, the field axioms if you will.

Our focus will lie in this area for the minute, with the laws that govern much of what can and should be utilised across the primary phase.

In their 1994 paper, Gray and Tall are resoundingly convincing in positing that pupils who see mathematical symbols as both procedure and concept expend less effort than their peers who see them as separate entities. I’ve paraphrased above but look at this example…

On the one hand we have a pupil making 17 individual marks and laboriously totting them up to find the sum. Then, when the 7ness of 7 is utilised, we find the calculation all the more palatable and executed with fluency you might say. As I understand it, much of the fluency we talk about now comes down to seeing these relationships and symbols as procepts. Multiplicative reasoning is but a mere extension of this line of thinking.

Key for us today but far from an exclusive list – even the first diagram above isn’t complete – are the commutative, associative and distributive properties of multiplication.

As early as Year 1, the idea of an array is essential and, hopefully shown here, I can’t help but think what is an array if not the embodiment of the commutative law?

Similarly with associativity and distributivity it’s apparent how the child thinking in terms of procepts is a child who understands multiplicative reasoning. But they won’t develop this understanding by osmosis, the ideas should be central to everything that takes place in the mathematics classroom.

Just take a quick glance at the end of KS2 papers from 2017 and 2018 and you’ll see how this looks in action. (I’m using SATs papers because they’re freely accessible don’t read too much into it. It should be clear from the examples that the axioms are essential across the primary phase)

There are a number of ways to efficiently navigate each problem but only if the underlying structures, the laws are securely understood and internalised. If you’d like to discuss any of these in greater detail or I just haven’t been clear, hit me up on Twitter @Kieran_M_Ed

I’ve recorded the times on one image because the idea that mental processing is always more efficient doesn’t hold true. For question 20 it was much more efficient to use the formal written algorithm because I had to hold far too much in mind to be productive and was way beyond a minute when I tested this on myself. For one digit by 3 digits it was half the time, for two digits by 3 digits it was double and rising.

In the main, however, the ability to think about the structure of multiplication made manipulation of these broadest sense problems much more manageable and provided the ability to efficiently navigate with minimal effort. As always, I would describe the laws to pupils, make it clear that these are the terms we should use and offer opportunity to experience them to the fullest. This, you see, is the key to unlocking multiplication to those who just can’t magically see it.

Some children might come to this understanding seemingly naturally. For the majority of us, however, we’re relying on thorough exploration and instruction as part of our daily mathematical diet. We can all think in procepts and we can all get our heads around the field axioms. If, and it’s a big if, someone more expert teaches us about them. Fancy that.

@Kieran_M_Ed

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