It isn’t that they can’t see the solution. It is that they can’t see the problem.
G. K. Chesterton
In an uncertain future, the ability to solve novel problems will become increasingly in demand. Now, some folk might, correctly, point out that we’re hard-wired to problem solve. We can’t not attempt to solve problems when we have an idea of what the answer might be. However, we tend to be much worse at solving problems when we have no idea what the solution might look like. Everyone, regardless of their ideological beliefs about education, will surely agree that problem-solving is a vital skill, so, obviously, we should put some effort into teaching students how to do it. The question is, how?
Last week I observed a maths lesson in which Year 10 students were given problems from the MEI’s Senior Team Mathematics Challenge, aimed at sixth formers, to solve. They were incredible! They spent almost the entire hour working diligently on what, to me, looked like utterly incomprehensible maths problems. And what’s more, they left the lesson proud and happy. Clearly this must be the future of maths teaching, right?
Some context: these students are seriously able mathematicians; they all took their GCSE in Year 9 and got an A*. This year they are spending some of their maths lessons on a bespoke problem-solving course. You might well expect that this would be a relatively straightforward endeavour – how could these students not excel at problem-solving in maths? Apparently though it’s been a hard slog. At the beginning of the year, the teacher warned her class, “This is going to make your heads hurt and you are going to hate me.” Sure enough, it did, and they did.
The fact that they were talented mathematicians who knew a lot about maths, didn’t seem to help. When confronted with problems like “In how many different ways can 2015 be written as the difference of the squares of two positive integers? “they took one look and said, I can’t do that.
Certainly when I look at that problem, my mind boggles. I have literally no idea where to start. There’s nothing for me to cling on to and I slip right off. All I learn from being given this sort of question is that I can’t do it. Failure is encoded, making sure that the next time I see such a problem I already know I can’t do it.
These students had an advantage because they knew they were good at maths, but they had little experience of solving this kind of problem. Being able to solve problems is any domain of knowledge is about recognising the deep structure of the problem. There are, apparently, a finite number of structures for most maths problems and, with time and experience, you can learn to recognise them. If you recognise the structure of the problem and you know how to do the maths, solving the problems is trivially simple.
These Year 10 mathematicians have been learning a series of steps, distilled from the wisdom of George Polya’s 1960s classic, How To Solve It:
Step 1: Understand the problem
- What is the unknown?
- What is the condition you are trying to meet?
- What information is given?
- Is any information redundant?
- Do we expect one answer, or many, or none?
- Draw a diagram
- Introduce suitable notation
- Separate, and write down, the various bits of the condition
Step 2: Creating a plan
- Have you seen it before?
- Have you seen a similar problem, maybe in a different form?
- Do you know a related problem?
- Do you know a theorem that could help?
- Can you solve it directly, or do you need a ‘stepping stone’?
- Look familiar? Think of a similar problem having the same or a similar unknown.
- Could you use the result?
- Could you use the method?
- Can you imagine a more accessible related problem?
- Can you imagine a more general or a more specialised problem?
- Breaking it up: Keep only part of the condition and drop another part.
- How far does that determine the unknown?
- How can it vary?
- Can you derive anything useful from this data?
- Stepping stones: If I could find this then I could solve the problem
- Can you find anything useful from the given data?
- What data would you need to find to determine the unknown?
- Could you change the unknown, or the data, or both so that the new unknown and the new data are nearer each other?
- Oops I forgot!
- Did you use all the data?
- Did you use the whole condition?
- Have you taken into account all the definitions?
Step 3: Carrying out the plan
- Have you checked each step for accuracy?
- Can you see clearly that each step is the correct thing to do?
- Can you prove that it is correct?
Step 4: Looking back
- Can you check the result?
- Can you check the argument? (Push the limits?)
- Can you derive the results another way?
- Can you derive the result more elegantly? (At a glance)
- Can you use the result, or the method, for some other problem?
It seems to me that while these steps are tailored to fit mathematical problems they could be easily adapted to fit the demands of problem-solving in other subject areas. Experts will have internalised these four steps and will often be able to ‘just do’ whatever needs to be done to solve a thorny issue in their field of expertise. Novices are novices precisely because they have yet to go through this process.
Back to the maths lesson. Most of the students spent the vast majority of the hour working in silence on their chosen problem. Their body language was relaxed and confident. They had learned how to apply what they knew about maths to a wide range of problem types and what to do when they encountered an unfamiliar problem.
The teacher had predicted that one student would struggle more than the others. Despite being one of the most able mathematicians in the group she still often feels helpless in the face of difficulty. Her body language was markedly different: she seemed almost to squirm as she grappled with the problems. I was told that in most lessons, she gives up quickly and then gets another student to explain the answer. Then, when she’s told the answer, she feels stupid for not having seen the solution herself. Accordingly, we planned a strategy to prevent her from relying on others for help and luckily it paid off. The teacher made one or two brief interventions to assure her that she was heading in the right direction and she was independently able to solve two of the problems. As the lesson progressed, her posture changed. By the end of the class she was smiling and joking. The hope is that the this experience of having encoded success will transfer to future lessons.
So, can we teach problem-solving? Yes, undoubtedly we can. These students demonstrated just how much they’d learned in the few weeks they’d been taking this class. But, the ability to solve problems is dependent on knowing a lot about the field you’re attempting to solve problems in. The steps above would be insufficient to get me (and most students) to solve most maths problems as I just don’t know enough mathematics. Yet.
NB this is a blog about problem-solving, not a blog about mathematics.
After the lesson, the teacher gave me this problem to solve:
I have 10 green socks and 4 blue socks. It is pitch black and I want to get a matching pair of socks. What is the fewest number of socks I must take in order to guarantee that I have a matching pair?*
I’m pleased to say I managed to work out the answer, and with my (admittedly small) success came a sense of confidence that I might be able to work out ever so slightly more difficult problems.
*Do please give your answer below and don’t forget to show your working out! 🙂