A colleague has invented a pop-up classroom in a dustbin that he can take to refugee camps and wheel from place to place so that ‘school’ can be taken to the children rather than expecting the children to walk to the school through the tented random cities. Lessons have to be stand-alone but worthwhile so that children who only get one or two lessons can benefit, as well as children who get lessons more regularly. This got me thinking: suppose there are children who have some elementary mathematics knowledge – what content from the secondary curriculum would be most worthwhile if you could only teach individual, unconnected, pop-up lessons, and what would those lessons be like? You can assume that your ‘class’ will have several different languages, and very little English, and a hugely varied mixture of mathematical knowledge.
One obvious route to go down would be the kind of ‘everyday’ mathematics that you assume they will meet wherever they eventually settle: money, measuring, reading timetables, journeys, speed, estimation. These ‘preparation for life’ skills are obviously worthwhile, but are also picked up and developed as survival skills by children growing up on the margins. Terezinha Nunes and her colleagues showed that street children in Brazil had relatively sophisticated knowledge of proportional reasoning when they needed it in order to earn meagre livings (Nunes, Schliemann & Carraher 1993) and many other anthropological studies have shown how the mathematics necessary for the demands of everyday life develop within the culture that makes them necessary. One lesson may not be long enough to introduce formal notations and methods for estimations and calculations that are already done informally or mentally. And if you only have one lesson, your ‘class’ might hope to learn something they are not learning in their daily lives.
You could teach them something that would prevent them from being cheated – that would be useful. Percentages require some care to avoid being cheated; situations in which there is an increase of, say, 10% followed by a decrease of 10% do not get you back to where you started and I can imagine setting up some haggling games that bring this to light. It would also be possible in one lesson to do enough that the students could continue to think about it even if you never saw them again. This situation is not intuitive and you can build up to it by simple physical models showing that, for example, increasing something by a half can be ‘undone’ by a decrease of a third of the new amount; increasing something by a third can be ‘undone’ by a decrease of a quarter of the new amount; etc.
When I thought about this more I realised that teaching refugee children how to avoid being cheated might add to a negative view of the world, and it might be better to do something that shows mathematics in a life-enhancing light. Life is enhanced by beautiful things so one direction would be to show something that is mathematically beautiful. But showing children what is ‘wow!’ for you does not necessarily make it ‘wow!’ for them. Life is also enhanced by our inner feelings of power and creativity. If I know about unit fractions, and I know that a decrease by a third ‘undoes’ an increase by a half, and a decrease of a quarter ‘undoes’ an increase by a third, then I know how to ‘undo’ any unit fractional increase – generalisation is power.
This turned my ‘one lesson’ challenge into the question: ‘what can be done in a pop-up mathematics classroom in one lesson that gives students some worthwhile experience of their own creativity and power?’
Suppose I had a ball of string, some sticks, a rubbish plastic bag and some soft ground. I could draw a circle on the ground, and with the string and a stick mark out any angle in a semicircle. I could cut a right angle template from the plastic. I could show that my angle is a right angle by fitting the template into the angle. I could then challenge them to do the same and see if they too can make a right angle. After a few of these it will begin to look as if anyone using any point can make a right angle so I could change my challenge to ‘is it possible to make one which is not a right angle’. Maybe we could then move on to making a string angle based on two points of the circle that are not on a diameter. We could cut a template for the angle out of the plastic … you can guess how I might move forward from that. Within one lesson several circle and angle theorems can be discovered and feelings of power can be enhanced through the students’ own creativity and reasoning. Another use for this ‘kit’ could be to mark equidistant points round the circle and link them with string: what do you make with three points? four points? etc. and then what do you make with twelve points if you join up every third point, every fourth point, every fifth point and so on. Within one lesson they could experience something that is worthwhile and mathematical even if they never return to circle geometry.
It is a truism in teaching that whenever you think about how to teach well in one particular situation, you end up with ideas you can use in any lesson. So my challenging question could be recycled as ‘what can be done in any mathematics classroom in one lesson that gives students some worthwhile experience of their own creativity and power?’
All the best,
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.
Nunes, T., Schliemann, A. D., & Carraher, D. W. (1993). Street mathematics and school mathematics. Cambridge University Press.