In Carraher (1985) the authors write about a series of case studies where young children selling products on the streets were able to quickly manipulate numbers in their head to arrive at the correct answers to multiplication questions. When presented with the arithmetic more formally, they failed.
Their first example features a twelve-year old, asked how much four coconuts would cost if each were Cr$ 35; they quickly calculate the results with aplomb. The same child is then asked was 35 x 4 would come to, and they fail.
How on Earth could the same child be successful in the first case, and not in the second! It’s all the more baffling when you realise the child presented the correct answer only moments before, and doesn’t appear to have realised their own incongruence.
Elizabeth Green’s book, Building a Better Teacher, tells a story that hints at the refrain we’ve become too familiar with: The problem is that the children didn’t understand the formal process. If only they understood why it worked, they would be better able to use the algorithm.
There is a little more to Green’s narrative, and she certainly doesn’t make this case quite so explicitly herself, but it is there. Carraher, interestingly, concludes not by railing against standard algorithms taught in school, but by asking where the proper pedagogical departure might be from using informal mental processes to more formal written procedures.
I think both have missed something much more important.
We don’t need a formal written algorithm to compute 35 x 4, provided we do have a reasonably well-developed sense of number.
This is part of a bigger problem that I see cropping up time and again in curricular. I currently suspect it might be this, more than anything else, that leads to that syndrome we’ve all witnessed, of mathematics perceived as detached from every day experience, exemplified in the case study above. In that instance, the child perceived ’35 x 4’ as something related to school, and so brought out the ‘school method’ they had been taught, and learnt poorly, resulting in failure. The report mentions other cases where the questions are framed as word problems, but with contexts not directly relevant to the children’s lives, resulting in similar failure.
Another example might be introducing ‘solving by rearranging’ with the equation
We instruct the pupil to subtract 8 from both sides, only to realise what was obvious to everyone from the start, that it’s 2 which we add to 8, to make 10. Why all that formal procedural nonsense?
It wasn’t a lack for understanding that meant the street children couldn’t perform the written multiplication methods, it was simple incompetence. It’s entirely possible to ‘understand how’ the method works, and ‘understand that’ the method works, and perform it expertly, without ‘understanding why’ it works. Equally, it is entirely possible to understand why algorithms work, yet struggle to actually execute them, or retain the ‘why’ for only a moment, before forgetting. Herein lies the false assumption, that knowing why automatically leads to knowing, and forever remembering, how.
It’s clear the children in the study aren’t connecting the numbers they work with every day with the numbers they experienced in school. To them, the two different kinds of problem involve two different kinds of mathematics, even when the numbers used in both contexts are identical. Also, the inability to check the first answer against the second will, again, be all too familiar to teachers of mathematics; it’s the inability to ‘sense check.’ But then, how often do we demand sense checking from our pupils?
Interestingly, in none of the cases presented in Carraher was the arithmetic involved much more complex than 35 x 4. I wonder, how would the same kids have done with 37.8 x 4. Personally, I can just about do that one in my head, but it’s a struggle, as it uses up quite a bit of working memory. Here’s where the written process can become a useful tool.
Consider instead a mathematics curriculum that focussed entirely on number sense and mental arithmetic, even up to and including arithmetic such as 35 x 4. When it was noticed that pupils were struggling to complete the arithmetic in their heads, perhaps then the written procedure could be introduced; it becomes ‘the aspirin to the headache,’ as Dan Meyer puts it.
Here comes the clever bit: say pupils were struggling even with 37 x 4, or maybe it’s 37 x 7… Before teaching them the written method for exact calculation, they are first taught to estimate. Maybe it’s 35 x 5, maybe it’s 40 x 5, or even 40 x 10, but all the rich thought that can go into careful estimates is explored (40 x 5 should produce a better estimate than 40 x 10, for example.) When they are then introduced to the formal written method, they are asked to do what they’ve been doing all along, to estimate their answer first. As a part of the written method they are taught to sense check their calculated result against the estimate.
Suddenly, there are connections between their mental manipulations of quantities and their written methods. Without these connections, should we really wonder that the child in the study was oblivious to his faux pas? Now instead, the child is free to to calculate in their head when they can – as most of us do – but has the written method available when they need it, rather than when they are told to use it. If they were told to use it for something like 35 x 4, my bet is that they would now automatically check the result against their mental calculation, and realise if they made a mistake.
Understanding is not the problem because ‘understanding,’ as I hear people discuss it, is not a thing; finite and concrete, a pale shadow of the multiplicity of interconnections between ideas that we could bring forth, if we’re smart.
Kris Boulton is deputy head of maths at King Solomon Academy.