I read a brief summary of a paper a couple of weeks ago (in TESPro 24/2/12) about ‘mathematical resilience’. The paper is by Sue Johnston-Wilder and Clare Lee, and in it they describe how students should learn to talk like mathematicians, possessing the language necessary to express their understanding – or lack thereof – any given topic. The more they talk like mathematicians, the more they feel that they are capable of understanding maths. ‘Mathematical resilience is about…confidence in what is understood and knowledge about what to do if you do not understand. Pupil’s confidence increased, and they felt supported in answering a challenging task and knew what to do to overcome any barriers that they encountered.’
We have all felt the frustration when a student comes to you and says ‘I don’t get it’, when you know full well that they have simply looked once at the question and given up immediately. You know as a teacher that they could have got at least some of the way by simply trying something, or asking the people around them what approaches they were taking, but they didn’t. They just gave up.
I want my learners to be resilient, to be unafraid to try things. That is why I use puzzles.
I have ‘Puzzle Wednesdays’ (which is the day I see all but 1 of my classes -they have puzzle thursdays…!)
On Puzzle Wednesdays, each class, regardless of ‘ability’ will have a go at some puzzles. I set a puzzle of the week (which I now upload into the POTW section on this blog) which contains two puzzles. One is usually visual/spatial, and one is more numerical in nature. The level of arithmetical understanding I know a class possesses will normally determine which puzzle I give them to try, but those that get the answer sooner will be given the other to do as well. Students often choose to work together, which is one of the best bits for me. They are talking like mathematicians without me even needing to ask 🙂
I also have a class set of puzzle cubes (http://bit.ly/xpljWo) which I think teach the following skills: perseverance, logic, reasoning, spatial awareness, and the understanding that just because you think you are good at maths does not mean you can automatically solve this the fastest…. (e.g. humility!) I use these occasionally, although I always end up being the one to put them back together! The students have not got bored of them in all the time I’ve been using them, as there are 32 different ones to try, and you don’t tend to remember how you did them anyway.
I have a large collection of puzzles on the site too – http://www.ilovemathsgames.com/puzzles.html
Puzzle Wednesdays takes up a lot of my teaching time with each group. I sometimes only do it for half a lesson, but that is still 12.5% of their maths lessons for the week. I don’t do this because of the new emphasis in the syllabus on problem solving, and the new styles of question that are appearing in exam papers (although it helps!) I do this because I believe that being a mathematician is not about being able to do the hardest arithmetic, but about being an incorrigible problem-solver. It should bug you if there is a problem you can’t seem to solve. It should be eternally present in the back of your mind until you can crack it.
I use the analogy with students that you could have a large toolbox, full of tools like a saw, hammer, drill etc, all of which you know how to use. Having this toolbox does not mean you can automatically build a garden shed. You probably couldn’t build the shed without them, but more is required. How should the bits fit together? How will you make sure it doesn’t leak? All of this requires thought, and a whole different skill set.
You can learn all the tools of arithmetic, and possess all those skills, but unless you are unafraid to try out problems with different approaches (without someone telling you what steps to take), you are not a mathematician.
To be a mathematician, I would say that the most important skill you need is imagination.
What happens if I…? But what if I changed this bit…? How can I model this situation…?