Rote Learning and Mathematics

Most people who care about what they do, in any profession, can suggest a couple of ways in which they feel that their field of expertise is being ill-served by its conventional wisdom. Sometimes, these will be cries to tear up the rulebook, new and progressive ideas about how everything could be moved forwards.  Just as common, however, are rather more reactionary ideas – a conception that their peers and colleagues have gone ‘too far’ in some particular direction. In my own case, this contradiction comes to the fore in the matter of mathematical rote learning.

When I taught at Bruern Abbey, a specialist school for pupils with learning difficulties, I often thought about a sign that hung on the wall of the staffroom. “If a pupil can’t learn the way you teach,” it read, “can you teach the way he learns?” This is splendid advice, and though it might be challenging to apply in a class of 40 pupils, it’s very important for smaller groups and virtually a moral requirement for one-to-one work. Those of us who were true believers in the Bruern Way talked a lot about ideas like kinaesthetic learning, and enjoyed thinking about how far ahead we were, or thought we were, of traditional, talk-and-chalk pedagogues.

The Bruern Abbey maths department put a great deal of emphasis on understanding the underlying logic of things like multiplication tables. This is essential for dyslexic learners, for whom the ability to calculate six times seven is important and achievable, but the prospect of memorising the result is likely unrealistic. On the whole, these methods worked tremendously well, and seeing the lights coming on for students who had almost given up on learning mathematics remains among my proudest memories.1)I’d highly recommend Dorian Yeo’s Dyslexia and Maths to anyone keen on helping dyslexic students do well at the subject.

However, not all of our students were dyslexic. Some had processing difficulties, or specific issues with short-term, rather than long-term, memory. For such students, an ability to memorise facts by rote was among their primary academic strengths. How did we teach such students? Well, we focused on understanding the underlying logic of multiplication tables…

In other words, by trying to apply an approach that was so helpful for the dyslexic members of the class, we were failing other students. It was oddly easy to do so; images of draconian, cane-wielding pedants, chanting multiplication tables at cowering students, made it easy to miss the fact that this approach could actually help some of our students.

For it’s uncomfortable but undeniable that there are areas of mathematics in which rote learning can really, really help. Of these, one of the most under-appreciated is the degree to which a background of facts, learnt by rote, can actually help students to assimilate new ideas and concepts. I’ve touched before on the idea that one of the major difficulties in understanding a teacher’s explanation is all of the background processing that a student has to perform at the same time. Ask a class of Year 7s to add 5x to 6x, for example, and students who arrive easily at the ’11’ will have an enormous advantage in deciding what to do with the ‘x’.

What should be learned by rote? For some students, such as those with dyslexia, almost nothing. For students with an opposite set of difficulties, as much as possible. For most students, however, a set of key facts represents a happy medium: multiplication tables to 10 or 12, number bonds to at least ten, key decimal / fraction equivalences such as three quarters being equal to 0.75 and one-fifth converting to 0.2.

And if this post reads as a cautious defence of rote learning, at least in certain circumstances and for certain students, it remains no excuse for bad teaching. To teach rote facts requires as much creativity and sensitivity to the student as any other method. To pore over a list might be right for one or two students, but walking around the room will help many more, while others learn faster from games or even songs.

There are general principles to effective rote learning, but these are largely the same as elsewhere. Try not to muddle the mixture, or delude yourself about how well students really know the material. Make sure the student enjoys it – a sense of growing mastery should be compensation for their hard work.

Above all, teach rote learning in the way that your student learns. A kernel of unforgettable mathematical knowledge, once memorised, will help students in every aspect of the subject for many years to come. If it’s in your power to give such a kernel to a young learner just starting their mathematical journey, it would be a crime to keep such a gift to yourself.

Notes   [ + ]

1. I’d highly recommend Dorian Yeo’s Dyslexia and Maths to anyone keen on helping dyslexic students do well at the subject.