If I gave you two points A (4, 4) and B (-4, 4) then I imagine you would be able to find the equation of the circle that had AB as its diameter.
In fact, you could probably do it without too much of a problem; but it’s worth just stopping to think how you would go about it and what knowledge or skills you would be drawing upon.
If I then asked you to find the equation of another circle, which still has A and B on its circumference, but not on its diameter, then I hope you would agree it’s a little more complicated. I could narrow it down by asking that the coordinates of the centre be integers, and that the radius be an integer too. Now can you find the equation?
(Stop reading here for 10 minutes and come back when you’ve found one… or two!)
So – what made it harder? Why was it more of a “problem” rather than a “question”? And, more importantly, what did you do to compensate?
Did you think about what information you had, and what extra information you needed?
Did you consider what skills and tools you would be using?
Did you draw a sketch?
Did you label some unknowns and create some equations?
Could you find a solution in more than one way?
Finally – after thinking about all of that, how would you actually teach those approaches to your students?
As maths teachers, most of us are very good at role-modelling our working and describing our thought processes. Usually, however, this is on a question or problem we already know the answer to (not always, I admit, especially if you’re looking at a Further Maths problem!)
If we’re brave, however, we can describe what we’re thinking when we tackle an unseen problem – this is really useful for students to see, especially if we “go down a dead-end” or make some mistakes. Students can see, then, that this is all part of mathematical problem solving, and we don’t always (or even often) find a nice, neat solution the first time around.
Another useful approach is to share a structure or system for solving problems. There are plenty to Google but perhaps the most popular one is George Pólya’s, which he describes in his book “How to Solve it”. In summary, he suggests some of the following strategies:
- Draw pictures or diagrams
- Use a variable
- Be systematic
- Solve a simpler basic version of the problem
- Guess and check (trial and error)
- Look for a pattern
More recently, Blackpool Research School ran a project in schools about teaching problem solving skills to Year 7 maths classes using the structure of:
- Planning my approach
- Monitoring my progress
- Evaluating my success
Each phase was supported with key questions and prompts for the pupils to use.
Why not try some of these structures or approaches in your classroom?
Finally, I’ll leave you with one more question about problem solving: is it more useful to answer several problems in one way, or to answer one problem in several ways?
More details about the Blackpool Research School can be found on their website.
Once the project is complete, you should be able to find it on the EEF website.