‘Practice makes perfect’ is a commonplace phrase, but like most commonplace phrases, it is an over simplification. Certainly an expert has to put in lots and lots of practice, but the psychology of successful exercising involves a particular flow of energy. The mathematics has, in some way, to inspire the learner to want to practice. If someone tells you to ‘get some more exercise’, there is rarely sufficient energy or desire to sustain exercise. This is particularly true of adolescents, while some may engage through a desire to please, many rebel against being told.
Here are two strategies to produce useful and effective exercise, based on the ‘principle’ that constructing your own examples is more engaging than dealing with pre-set pre-constructed examples from someone else. The idea, based on an observation of Caleb Gattegno, is that an expert uses a minimal amount of attention to perform what are for them routine tasks. This frees their attention to keep track of a larger goal, and to watch out for wrinkles or surprises. By contrast, a novice needs to attend to the details of procedures. So exercising that promotes the novice attending to details is of less value than exercising which draws the novice’s attention away from the details.
Invite learners to explore some generality, perhaps to find a relationship or to count the number of different ways of doing something. In the process of exploring, learners will have to specialise – construct their own examples – in order to find out what might be going on. They will have investment in those examples because their attention is directed to finding out something, and they will be practicing the work with those examples. For example:
In how many different ways can a given number be the least common multiple of two numbers?
In order to find out, you probably need to try some particular examples which you construct yourself, and you then need to do a fair few calculations of least common multiples. However, your attention is not on the lowest common multiples but on what your actions tell you about how many there might be in different situations.
In how many different ways can a unit fraction be expressed as the difference of two unit fractions?
Again, you probably need to choose some particular unit fractions, and then try to express them as differences of unit fractions, paying attention to how you find them rather than to the actual calculations.
In how many different ways can you write down a decimal number that lies between 2 and 3, which has no digit 5 but does have a digit 7, and is as close to 2.5 as possible?
Again you need to construct some examples, then become aware of the general principles you are using so as to decide what counts as ‘different’.
For each of these examples note the parameters that could be changed to make similar tasks for use over a period of time. Even better, get learners to do this!
Instead of asking learners to ‘do’ a set of exercises from a book or worksheet, ask then to make up a collection of examples which display ‘an easy one of this type’; a ‘difficult one of this type’; a tricky one of this type’; ‘a challenging one of this type’; ‘as few examples of this type as possible that show all the things that can happen‘. It is important that they also say in what way their examples are ‘of this type’, because recognising the type of task really matters in an examination.
If learners become familiar with this type of task, so that they begin to do it by themselves, they will have learnt a very powerful study device. They will be enriching their ‘example spaces’ which will in turn sensitise them to recognising types of tasks in the future.
If you try these strategies please send in a brief account of how they worked for your class. What questions did you come up with for Strategy 1? How did your students react to Strategy 2?
All the best,
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).