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Let’s begin with a puzzle:
I encountered this particular mathematical problem (by Peter Lijedahl) earlier this year at a meeting designed to provoke discussion, and expose elements, of ‘mathematical thinking’. The phrase ‘mathematical thinking’ is one that is used increasingly as something that we as educators and designers want to encourage, deepen, and develop in students - but it is very hard to quantify and observe. As we become more experienced problem solvers (and problem selectors!) we may forget what it’s like to be faced with a completely new mathematical problem and therefore the essential elements of early mathematical thinking which may be noticeable in our students.
So: it’s worth taking the time to examine our own thinking when we encounter new problems, to perform some forensic metacognition. If you can, spend five minutes reacting to the problem and a further five reflecting on your reaction.
Interestingly, when I did this, what I found particularly problematic about the problem above was that I didn’t like it!
I also came to that decision frighteningly quickly. Having read it through and established the ‘rules’ of the game, I was already dismissing it in my mind as ‘not my sort of problem’. This was only reinforced when I managed to convince myself (incorrectly, I might add) that it was impossible to devise a strategy in order to locate the treasure at all, let alone in 30 days. So, problem dismissed.
All was well until sharing and collaboration amongst our groups was encouraged. At this point one colleague proudly presented a representation of a strategy that informed her that it was possible to locate the treasure in the allotted time. Playing the role of your typical, sceptical student, I was unconvinced by her explanation. I couldn’t see how her solution dealt with the problems that I had encountered – but my curiosity was piqued enough to want to return to the problem in my own time later.
As is often the way, I didn’t return to it until several weeks later when another colleague, excited by the problem, wanted to discuss it with me. At this point something interesting happened. First of all, his excitement was infectious: he made me feel that there was more to the problem than I had first anticipated. Secondly, he didn’t care that I hadn’t got very far with the problem myself, he simply asked some questions, and encouraged (at times, forced) me to write some things down. His first question was simply: What if there were only four doors? My response was hesitant so he drew four lines on the board, to represent the four doors, and asked ‘what if you opened this door (pointing to the first door and drawing an X)?’
I began to respond verbally and he stopped me. He drew another set of four lines underneath and asked me to use these to record the possible places the treasure could now be (assuming that it was not initially behind that first door).
I was quite smug at this stage as the treasure could still be behind any one of the doors and so my initial scepticism was clearly justified. With limitless patience, he asked me what would happen if I opened a different door first instead. I selected the second door, reasoning that by symmetry I would then have dealt with all possible starting points.
This was different. The treasure could now only be in three locations. Curiosity taking over, I continued using this representation to consider the effect of opening subsequent doors and then, substantially more interested, I disappeared to my office to work further on extending the problem.
Thinking back on this experience, it’s not the mathematical problem itself that is interesting, but the fascinating way in which I moved from dismissing it and essentially refusing to work on it, to a place where I found it interesting and successfully developed a strategy for a solution.
I was left wondering two things: what exactly was it that I had ‘seen’ in the problem that made me take an instant dislike to it, and what was it about the interventions of my colleagues that changed my mind? Students and teachers make snap judgements all the time about whether or not a mathematical problem is worth pursuing. As teachers and designers we attempt to provide interventions that will support and encourage pursuit of the problem without revealing too much, but this is inevitably a tricky business. What was striking to me was that it took so little to shift my attitude towards the problem, and in this case the most important step was finding a way of representing it that allowed all of the information to be stored on the page rather than in my head.
This leads me to another thought: I play an online version of Scrabble and recently noticed that there is a feature called ‘word strength’ that, when turned on will give you an indication of how good your proposed score is relative to your possible plays on the board.
Since activating it, I have spent considerably longer searching for words that will at least score into the green end of the scale, where previously I would have just submitted a lower scoring word. The word strength feature doesn’t tell me where to go; it just tells me if it’s possible for me to do much better. That’s enough for me to have significantly improved my resilience in looking for good plays and therefore my success in the game. The realms of possibility have been sketched out for me so I have some sort of mental satnav for the problem – for me, immensely helpful in scoping for my own expectations for solutions and consequently allocating mental resources. How might we do this with mathematical problems? What might it look like?
Can you think of a mathematical problem that you have immediately dismissed as ‘not my sort of thing’? What do you think it would it take for you to pick it up again?