A couple of months ago, I was in a meeting and doodling away when I accidentally sketched this shape:
I wondered what its area would be, thinking it would be a trivial problem. Five minutes playing around with it didn't yield anything so I put it to one side. I revisited it several times, each time not making any progress. I knew that I could take a purely algebraic approach (using co-ordinate geometry) but where's the fun in that? I was looking for a beautiful proof. Then one day, I took the problem out again and suddenly saw the answer:
And there have been many times when the solution to a problem suddenly materialises after I have taken a break from the problem. Once I was stuck on this problem from a Number Theory course I was taking as part of my Masters of Mathematics for Teachers at the University of Waterloo:
Show that for any positive integer, n, there exists n consecutive values, none of which are prime.
For example, suppose n=4, I have to show that there are 4 consecutive numbers which aren't prime. I can do that: 24, 25, 26 and 27 are four consecutive numbers which aren't prime. But I had to prove that it works not just for 4 consecutive numbers but for any amount of consecutive numbers. I worked on this for quite some time but without making significant inroads so I took a break, and as it was late went to bed. My son woke me up at half past three in the morning asking for a glass of water. I got this for him, tucked him in and then headed back to my bed. In the six steps it took me to get back to my room, in the middle of the night, I suddenly saw the answer. In fact I saw it so clearly that I knew that I didn't need to write it down anywhere.
And I know that I am not alone in this. In the wonderful book about the great mathematician Paul Erdos, The Man who Loved Only Numbers another great mathematician, Ron Graham, explains how he had a "...flash of insight into a stubborn problem in the middle of a back somersault with a triple twist."
I would love to know the neurological reasons why this happens. Is it a case of the brain thinking too hard about the problem (as result missing some vital information) and then, after a break, a rejuvenated, more relaxed brain sees what should have been seen all along?
Whatever the reasons for this, as a teacher, I need to model what I do when I get stuck. I need to get students to understand that sometimes the best way to crack a problem is to leave it alone. I can tell my students that if they get stuck with a question (on a test for example) to leave it, do some other questions and then come back to it: they might then take a fresh, more productive approach to the problem.
But also, in the same way that English teachers will habitually share with their students what book they are currently reading, maybe we as Maths teachers can share with our students what Maths problem we are currently working on. This, for example is what I am working on right now (courtesy of the University of Waterloo CEMC's Problem of the Week.