Why do mathematicians study maths?
Why do you love maths? When Fields Medal winner Maryam Mirzakhani was asked what she found most rewarding about doing maths, she said:
Of course, the most rewarding part is the "Aha" moment, the excitement of discovery and enjoyment of understanding something new – the feeling of being on top of a hill and having a clear view. But most of the time, doing mathematics for me is like being on a long hike with no trail and no end in sight.
The average (non-problem solver) math student is like someone who goes to the gym three times a week to do lots of repetitions with low weights on various exercise machines. In contrast, the problem solver goes on a long, hard backpacking trip. Both people get stronger. The problem solver gets hot, cold, wet, tired and hungry. The problem solver gets lost, and has to find his or her way. The problem solver gets blisters. The problem solver climbs to the top of mountains, sees hitherto undreamed of vistas. The problem solver arrives at places of amazing beauty, and experiences ecstasy which is amplified by the effort expended to get there. When the problem solver returns home, he or she is energized by the adventure, and cannot stop telling others about their wonderful experiences. Meanwhile, the gym rat has become stronger, but has not had much fun, and has little to share with others.
The explorer is the person who is lost.
Solving a mathematical problem is like doing a jigsaw puzzle, except that you don't know in advance what the final picture will look like. It could be hard, it could be easy, or it could be impossible to solve. You never know until you actually do it (or realise it is impossible to do). This uncertainty is perhaps the most difficult aspect of being a mathematician.
With any math problem, you hope and pray that you will be able to find a nice and elegant solution, and perhaps discover something interesting along the way. And you certainly hope that you will be able to do it in a reasonable period of time, that you won't have to wait 350 years for the conclusion. But you can never be sure.
For a long time, no pattern seemed to emerge, but suddenly it all became clear to me. The jigsaw puzzle became complete, full of elegance and beauty, in a moment that I will always remember and cherish. It was an incredible high that made all the sleepless nights worthwhile. For the first time in my life, I had in my possession something that no-one else in the world had. If you experience this feeling once, you will want to go back and do it again. This was the first time it happened to me, and like the first kiss, it was very special. I knew then I could call myself a mathematician.
What does this have to do with increasing students' perseverance?
What interests me here, and matches with my experience of doing maths, is that what makes it difficult is also what makes it worth doing. Most mathematician talk about being lost, and about the "Aha!" moment when they find a solution to a problem - and this feeling is what makes them want to solve more problems, and ultimately become mathematicians.
Do you give your students the chance to get lost?
Do you give them the chance to feel the elation that mathematicians feel?
Do you give your students the desire to want to do more maths?
I am not saying that our students should be in a perpetual state of confusion, and there is a huge place for practicing the basics. But I would argue that it is essential that we give students the chance to feel the emotions that come from solving difficult problems if we want them to become motivated and persevere when doing maths.
The science-y bit
I'm not a brain scientist, but I have read some research that backs up this idea. The hormone and neurotransmitter Dopamine controls our ability to feel, and act towards, rewards - it is linked to that "Aha" moment when we solve a problem. Scientists have shown Dopamine is in turn linked to our levels or perseverance - expected increases in levels of Dopamine will increase our chances of persevering with a task. This coincides with the feeling we get when solving a problem - we may be lost, but we know (or at least hope) that the "Aha" moment will arrive eventually - and this is what keeps us going when solving difficult problems (see the example below) .
On the flip-side of this, there is also evidence to suggest that not achieving goals will dry up the dopamine release. It is important that we do not give our students many problems that are impossible for them to solve, otherwise they will not gain that "Aha" feeling and lose motivation. I am not saying that they should be able to solve all the problems we give them with ease. What I am saying is that we should give them problems they can access, but that pose them some difficulty - but they should definitely have a reasonable chance of success. There is a fine line between giving them problems that are too easy and too hard - knowledge of our students, and the problems we set them, is key here.
To give you an idea of this feeling of being lost, and the subsequent "Aha" feeling in action, look at this STEP I question (2006):
The first two parts are relatively simple for someone with a knowledge of quadratics. We can solve part (i) by considering the y-intercept of the graph, and part (ii) by considering the relationship between the coefficients of the quadratics and the solutions (x-c)(x-d)=0.
Part (iii) is much more interesting/difficult. The first half is OK if we notice that calculus tells us that function is increasing. But what on earth is going on with the last part? What is the relevance of the curious expression 4p^3 + 27q^2? Understanding the significance of this expression is the key to understanding this question, but where has it come from? Here is that feeling of being lost... Let's enjoy it and try something out.
Let's just consider each condition at a time. Differentiating to find stationary points, and considering the condition p<0, tells us there are two stationary points, one positive and negative (why?). Considering q<0 and drawing the graphs gives us three cases:
What does this teach us?
In this post, I have discussed a key reason why mathematicians (and I) study maths, and argued that we need to give our students the same feeling of being lost to give them the "Aha" moment, which in turn will motivate them to persevere with more complicated maths problems. I used an example of my experience in solving a STEP problem to show this in action.
Of course we need to give students time to practice the basics, but I would argue that we also need to give our students the chance to experience the (positive and negative) emotions involved with solving difficult maths problems - with the caveat that we do not want our students to be lost all the time, with little chance of success. We need to get the Dopamine flowing!
I think this has a very important consequence when teaching students to take the STEP papers, in particular. They are difficult; there are long periods in the wilderness, they require strong problem solving strategies, a wide range of mathematical techniques, and a huge dose of perseverance. The chances for some Dopamine are few and far between - so perhaps they best way to teach students how to do well on STEP is not to dive straight in with STEP past papers, but instead give students a spattering of less complex but equally interesting problems to solve, and build from there.