The fast-approaching Christmas break is often a period of revision for students – some may be preparing for mock exams and others taking time to recap topics covered in their first term. At this point, I find myself asking what it is all for, and how revision might be transformed into learning for the longer term, as well as serving the short term. The immediate goal of achieving some desirable result on an examination is a bit like the immediate goal of achieving some desirable result in athletics. Achievement comes not from simply repeating the athletic event over and over; instead, you work on the component parts. This means not only training behaviour, but awakening desire in the learner to integrate useful actions into their functioning. It is often not enough to hope that repeated practice will somehow turn into effective actions in the examination.

Of course there has to be some desire on the part of the student…teachers cannot do the learning for students. However, awakening that desire can be done more or less effectively. For example, recommending frequent repetition of standard tasks is much less motivating than getting students to engage with the material, perhaps by inviting them to ‘work on’ the types of questions the teacher expects to be in the examination.

By ‘working on’ I mean treating a collection of tasks as a mathematical object. What can vary in a question and still be answered using the same technique? Getting students to make up their own similar questions which they think are ‘easy’, which they think are ‘hard’ and which they think are ‘interesting’ (or at least novel in some way) is much more likely to equip them to recognise the ‘type’ of a question on an examination than simply ‘doing’ more and more questions from some prepared source. It is always useful for students to say in what way they see their version as ‘similar to’ the original, because sometimes it is not at all obvious! How general a question can they formulate based on a set of exercises? Can they ‘do’ a general one? Can they make up a question which shows the sorts of things that can happen in such a question type? Comparing ideas with peers is a great way to become aware of aspects that could be varied in ways you might not have realised.

Getting students to ‘work on’ question types helps them recognise the type when it appears on an examination, so that an appropriate action becomes available to be enacted (we sometimes say ‘comes to mind’, but that assumes that ‘mind’ includes behaviour as well as cognition).

Of course, there is also the matter of developing speed, an important part of facility. This does perhaps require repetition, in the same way that an athlete repeats the same action over and over in order to internalise it so that attention is not required. But the energy required to do this comes from the desire to gain facility, thus harnessing the student’s own motivation. But, as in the case of Karate Kid, the best thing to do is to divert attention away from the intended goal. The mark of an expert is that they make actions look easy and use a minimum of attention to direct that action. The same applies equally well to mathematics. To integrate an action into your functioning is to reduce the amount of attention required to carry out that action.

In mathematics, the way to release attention is not to ‘do’ a sequence of repetitive tasks, but rather to make use of the action you are trying to internalise in some context where you must keep a larger goal in mind. So if you want to internalise the process of adding fractions, you could, for example, find how many unit fractions (¹⁄₂, ¹⁄₃, ¹⁄₄, …) you need to add together until the total is greater than 2 (or 3, or 4 …). Here, attention is on the result, and it is the result that the student wants in order to decide whether they have added enough of them.

Other exercises could include:

• How many fractions of the form ³⁄₄, ⁵⁄₆, ⁷⁄₈ etc. do you need to multiply together to get a result less than ¹⁄₂?
• Given four digits, what is the result of adding all the fractions that can be made from those four digits?
• What is the smallest and the largest result of adding two fractions whose numerators and denominators are assigned from the four given digits?

Again, you could engage students with the creation of such tasks as well as the execution. The possibilities are endless and depend only on the students recognising some aspect of a routine that can be varied, and, importantly, deciding themselves to vary it.

As another example: given a ‘word problem’, in how many ways can you provide a different context for the same calculations? What is it about the structure of the word problem that makes it amenable to that sequence of calculations? Can you construct a context and a word problem in which you have to use each of the four operations once in order to reach a resolution?

John Mason

John Mason worked for the Open University Mathematics Department, where he designed two of the mathematics summer schools, contributed to numerous courses, and then helped form and run the Centre for Mathematics Education. He has written numerous books and booklets, as well as research articles and book chapters, the best known being ‘Thinking Mathematically’ (1982, 2010).