What do learners think mathematics is actually about?
Their opinion is likely to be developed from the sorts of tasks they are given to do in lessons: getting well known answers to collections of tasks assigned by teachers. Do they ever see someone thinking mathematically – and realise that that is what is happening? Are they ever in the presence of someone who does not know ‘the answer’ and is using their mathematical thinking to try to find out? How often do they see someone explicitly struggling to make sense of some mathematics, and have their attention drawn to the actions that help make progress?
How is a learner supposed to know what to do when they are stuck if they never see anyone stuck getting themselves unstuck? It is perfectly reasonable for learners to imagine that mathematical thinking emerges from the pen when placed on paper, if all they see are efficient solutions to routine tasks. It is also perfectly reasonable for learners to decide that their pen is ‘not mathematical’, and to use this as a personal reason for giving up and declaring themselves to be ‘not mathematical’.
Sometimes it can help if students are in the presence of mathematical thinking: the teacher publicly thinking mathematically. This means verbalising uncertainties, making conjectures, then testing and modifying them. The art of the teacher is to keep learners anticipating or wondering what will happen next, and to be sufficiently self-aware to draw attention explicitly to how their attention shifted, how this or that action – specialising, generalising, checking, etc. – came to mind.
Teachers often tackle a problem by getting suggestions from the learners as to what to do next. What really matters with worked examples is to know, not simply what to do next but how the person knew what to do next. So the teacher’s role is not only to manage the suggestions, but to draw attention to what is effective, and to get learners to become aware of what was effective.
One way to prepare for being mathematical in public is to work with colleagues on the same or similarly parallel problems. One colleague draws attention to mathematical actions as they occur, while the others try to verbalise what they are attending to and in what way.
Where might you get useful problems for being mathematical in public?
One source might be learners themselves who are encouraged to notice opportunities to ask questions about things that happen in the material world. Another is to be on the lookout for puzzling situations. There is no need to always proceed to a solution: it is enough to make some reasonable conjectures. This is how mathematicians conclude – if only temporarily– their work on a problem: with some examples and some conjectures based on the insight gained while doing those examples.
If you have some ideas for useful problems for being mathematical in public, do send them in along with your thoughts about what mathematical actions you attended to whilst solving them.
All the best,
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).