This week’s blog is about my efforts to think through the place of problem-solving in teaching, and in relation to the development of conceptual understanding and knowledge of techniques. When the introductory blurb to the national curriculum was written, the first draft put the aims in this order:
To us – the team who were giving expert advice on the mathematical qualities of the curriculum – this order did not make sense. Either mathematical problem-solving should come first, with fluency, conceptual understanding and reasoning as the tools necessary to solve the problems, or mathematical problem-solving should come last as the purpose for developing fluency and reasoning. You now know that the second view prevailed, but not because problem-solving provides the purpose. Instead, the view of the people who made the final decision was that problem-solving is about application. This is a very traditional view of mathematical learning, i.e. that learners develop techniques and some conceptual understanding and then, after some practice, apply what they have learnt.
The chickens and eggs in my title are therefore the problems of mathematics and the concepts and techniques of mathematics. Some of the most exciting lessons I have seen start with the presentation of a problem that appears to be beyond the reach of the students, but provides a shared goal towards which the class creeps bit by bit.
One example of this is an historical approach towards finding a way of calculating the area of a circle by successive calculations based on the areas of inner and outer squares, regular pentagons, hexagons and so on. The necessary techniques of area finding, and the reasoning of convergent limits, have a purpose and it really doesn’t matter if some students already know about π – the chase is still interesting. In fact, if some students do know π then the students themselves have a way to check the correctness of their area calculations.
Think also of the value of this task as generating a need to find a general formula for the area of a regular polygon – to make this easier you could use a spread sheet. The crux of this problem is to find out something about the area of a circle by working intensively with area using geometry and trigonometry. However, these techniques won’t just emerge naturally from the problem-solving process, there has to be some teaching input as and when appropriate. To extend the chicken and egg metaphor a long way beyond its shelf-life – there needs to be some planned midwifery going on rather than telling everyone beforehand what they might need to solve the problem.
Another example where a mathematical problem gives birth to the necessary mathematical techniques is the question:
‘How do we know if different algebraic expressions are equivalent?’
For example, students finding a general formula or the number of squares in the nth term of a spatial growing sequence are likely to create expressions that look different, according to the way they ‘see’ the growth.
Possible generalisations include: n2 + (n + 2); n(n+1) + 2; (n+1)2 – n + 1; (n + 1)( n+ 2) – 2n. Each of these arises from a particular way of seeing how the shape is built up, so we know that they must be equivalent as they are describing the same structure. To show this from the algebra it is necessary to learn some techniques concerning brackets and like terms. Or students can be given the task of deducing some technical rules of algebra, knowing that these expressions are equivalent.
I have written in an earlier blog about the value of studying quadratics, and how the different forms of quadratics can be used in a similar manner. For example, ‘y = (x + 2)(x – 3) and y = x2 – x – 6 give the same curve using a graph plotter – why?’
Appropriate problems may be lurking in your textbooks. In the Extension Mathematics series by Oxford University Press, each section starts with a ‘Problem 0’ that may be an appropriate problem in this sense. Creating your own, or as part of your subject team, is also useful because in order to do so you have to keep asking ‘Why is this concept or technique in the curriculum?’ in terms of ‘What mathematical problems does it help to solve?’
Maybe you can contribute some ideas to this blog? My main point, which I hope I have illustrated in these musings, is that problems do not need to be applications.
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.