In my last blog I set a task, which was to find out and explain what happens in this situation:
Given a set of consecutive natural numbers from 1 to 2n, choose any n of them. Arrange these n numbers in ascending order. Next to them, in one-to-one correspondence, arrange the remaining numbers in descending order. Find the absolute difference of each pair of numbers, and find the sum of all these absolute differences. Then explain why the total is n2. This is called Proizvolov’s identity.
The first part of this task is an investigation, from which a conjecture can be made and tested after a few trials for different values of n. The explanation is novel and involves a way of seeing the process of totalling. Until this is ‘seen’, many ways of exploring can be pursued, usually involving construction of a simple case (choosing numbers 1 to n as the ascending set) and then swapping numbers around and explaining how such swaps do not alter the total. Another kind of simplification is to look at the structures of cases where n = 1, 2, 3, …. Neither of these approaches yields a convincing general argument. I am not going to give the game away here – explanations and proofs can be found on the web when you get desperate!
The point of this elaboration is to show that problems of this kind (i.e. for which a fresh way of ‘seeing’ makes the explanation obvious) offer intrigue and challenge, but do not make a good training ground for mathematical problem-solving. The usual heuristics suggested by Polya (trying simple cases, making and testing conjectures, making a plan based on knowledge, thinking about similar situations, breaking it into steps…) do not work except to add to the frustration and provide fruitless practice of elementary arithmetic. You have to have a fresh way to see the situation in order to explain it directly.
I am writing this blog after the International Congress on Mathematics Education, during which much was said about problem-solving. Research about how to ‘teach’ problem-solving falls into several types:
- focus on students’ behaviour, resilience, reflection, willingness to try multiple pathways, working in groups, gender differences, and so on,
- focus on word problems in which the student has to imagine a situation and decide which operation to perform,
- focus on word problems in which students are trained to notice key words or phrases and act accordingly,
- focus on heuristics (make a plan, carry out the plan, reflect on outcomes, …) and whether students adopt these ways of behaving,
- focus on problems of a particular kind, or that require the application of a particular concept or procedure.
Most studies are short term and success is tested only with the kind of problems that fit the focus of the study – probably not Proizvolov’s identity then. To me, these directions of research seem not to be massively helpful for teachers, because of their specificity. Most teachers know that a problem-solving mindset develops over time as an attitude towards mathematics.
I am now going to suggest one kind of problem that helps towards that development, and why it helps. The problem is to find a value or values for x.
I saw this problem being offered to a class of students at the end of year 9. It was being given in a spirit of challenge. They were assured that they knew all the mathematics they needed to know and that several stages were involved and it might take some time. As a whole class, they were asked to contribute anything they knew that might be useful. All the ideas were listed: rules of indices, 4 = 22, methods for transforming equations, and so on. Initially they did not anticipate needing methods for solving quadratic equations, but several students called out later remarks like: ‘Oh it’s quadratic equations’. The solution took the class a whole lesson to reach, orchestrated by the teacher.
Solving it is neither a matter of trying out several values until a pattern shows itself, nor a matter of trial and adjustment. It is neither open-ended, nor dependent on a sudden genius idea. Instead, students have to collect a toolkit of likely bits of knowledge to apply, and apply them until the problem seems simpler and/or they gain some insight. This problem is neither routine, nor open, nor ‘low threshold; high ceiling’, but requires prior knowledge that has to be collected together and coordinated for a closed purpose. The Japanese textbooks I have seen have many such problems, and they have been developed and refined over decades. This means teachers are using the same problems that were set for them when they were in school; they know the problems and their classroom-potential deeply. In the absence of a historical collection, what do we do? Over to you for ideas! Because this is an OUP blog, I start by directing you towards Tony Gardiner’s textbooks where some such problems lurk, and there are many to be found in NRich, but do you ever use the UK Maths Trust and Kangaroo Maths questions as whole class problems?
A final word – I am not saying that ‘low threshold; high ceiling’ tasks (realistic problem solving, puzzle situations, and investigations) have no value. All these types of task have value and a place in classroom life. I am saying that this other kind, in which known mathematics can be coordinated to achieve a closed and challenging outcome, helps the development of mathematical problem-solving in a different, but important, way.
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.