In this post I’m going to give a few thoughts about how I think chess can help students develop mathematically. I’ll also provide an idea for scaffolding students through problems or proofs.
Below is a Beginner level Chess problem from www.chessproblems.com. With black to move first, can you figure out a way for black or white to win within the next 10 moves? (Have a think about this for a minute before moving on)
If you figured it out, well done. If not, here is a solution for black to win within 2 moves.
This is an interesting example as the checkmate requires black to sacrifice the queen – a move which many beginners may not see on their reluctance on giving up their most powerful piece. A good chess player always looks towards checkmate and a checkmate is all the more sweet when the other player thinks for a moment that they have gained the advantage by taking the queen, only to be crushed with defeat some moves later.
Even if you don’t play chess, you probably have some idea of the fact that each piece moves according to certain rules. Solving a problem such as the one above, a chess player has to put together a logical sequence of moves (which are based on the games fundamental rules) which result in a checkmate. This is of course exactly what a mathematician (famous example being Euclid) would do to devise a proof in Mathematics based on axioms and other theorems. Hence there is an obvious connection between playing chess and Mathematics; namely that it helps develop a logical approach and increases ones patience and persistence when solving problems. Whilst I can only draw on my own experiences, I do genuinely believe that consistently playing chess from 9 years old onwards has helped to build up, what Dan Meyer has termed, my patient-problem solving ability in Mathematics.
More often than not, when a chess problem is posed, you’re given a configuation and asked how to reach checkmate in a certain number of moves. Indeed, the problem posed on chessproblem.com asked how black would win in 2 moves. How does this connect to Mathematics teaching and learning?
I thought it interesting that the person is told how many moves it should take to reach checkmate. What if we told students how many steps it would take to answer a question? or how many lines it would take to prove a theorem? Whilst this would not always be appropriate, I could imagine a scenario in which this could help students to solve a problem. Take for example a question posed by a teacher during GCSE revision:
This problem is easily solved in 3 or 4 steps.
4 steps: find area of square, find area of circle, subtract circle area from square area, divide by 4.
3 steps: find area of quarter square, find area of quarter circle, subtract quarter circle area from quarter square area.
Whether giving the number of steps needed would help is something I’m not too sure of – certainly worth a try though.
Maybe this would be more useful when proving theorems. I remember at times pondering during my undergraduate studies how long a proof was going to take and often convincing myself that it wasn’t quick and easy to prove. This was until the next day when I would see a simple 4 line proof and get frustrated that I hadn’t persisted long enough. I believe that, if on some occasions, I’d been given the number of steps in the proof, it may have increased my persistence and patient problem solving further. It could be argued that this wouldn’t align with a multiple solutions perspective of mathematics. However, it does not prescribe how the proof should be constructed, only that it has been done before in 4 lines of argument. This connects with the common occurence of Mathematicians being able to solve a problem after it has been announced that someone has found a solution.
In terms of chess and Mathematics, I believe that chess does promote patient problem solving from a young age. In terms of how chess problems are posed, maybe there are circumstances in which providing the number of steps in a problem or proof could be the only scaffold required.