Wednesday 31 August 2016
Paying attention to detail
Just sharing what I did today with one of my Mathematical Studies classes as a warm up exercise for the new term. We played a few rounds of this 'picture this game' with a view to reflecting on how students look at mathematics problems. The idea of the game is very simple. There are two large packs of cards and each card has on it an image which is a close up photograph of a well known object. The photos are so close up that it is not at all obvious what the 'object' is. The goal of the game is obviously to try and figure it out. It is both incredibly satisfying and frustrating and, I think, a lovely analogy for reactions we can all have to mathematics problems!
Frustratingly difficult - depending on which card you get first, it can be really tough because there is nothing that gives it away. Often it takes a few goes to get used to it..... try these two shown below........
Blindingly obvious - Then, once you know what it is, you often can't believe that you couldn't see it before!
Satisfying - and eventually you begin to get the hang of it and an eye for the key features of the images that give it away and experience a wonderfully satisfying certainty that you know what it is. It is a similar feeling to the one you get when you first see the 'magic eye' pictures.
Click below to see the answers......
Well, I think what happens to us when we look at these pictures is fascinating, particularly when you have an instinctive reaction to what you see. For example, maybe, like me, you first thought the toothpicks were pencils. Once you have an idea in your head it is very difficult to shift that mindset and imagine it being something completely different. Because we are not seeing 'the big picture' we are prone to making snap, unjustified judgements about what it is. Unless we check ourselves and try to look from a different angle, we are at the mercy of these judgements.
I think this happens to students when they tackle exam questions and they go on to make related mistakes. Often when you give the paper back and point out the important feature of the diagram/question that they missed, they experience clarity and say 'I can't believe I didn't see/do that the first time' - much like you can't believe you didn't see the toothpicks in the picture above (perhaps you did, but believe me there are others there you wont get first time!). Bridging the gap between the first 'instinctive response' to the information shown and the 'clarity' becomes a significant goal. Paying attention to key details that give the bigger picture away is crucial and often the reason students do well (or not as the case may be).
I found myself recounting to a class today, the lovely picture that Andrew Wiles paints of his exepriences with mathematics that goes something like this.....
When you first confront a new bit of mathematics, it is like going in to an unknown room of a house that is pitch black. To start with, you have to move carefully around the room and build a map of the furniture in it - as you do so, you effectively turn the light on in the room and can see it clearly and the door that leads out the other side so you can repeat the process in the next room until eventually, you have illuminated the whole house perfectly.
WIles was more eloquent and emotionally charged given that his house was 'Fermat's last theorem', but the analogy holds with the card game and classroom mathematics. Those moments of illumination and clarity are priceless right!
So I think the exercise is very valuable and I will get the odd card out for a while and warm students up with this idea. Lets illuminate the toothpicks as much as we can this year I say!