At higher level GCSE, it is possible to get a grade B having got 60% of the paper wrong. Since this is the benchmark for moving on to an A level course, there could be a concern that students could decide, say to avoid learning algebra and concentrate on geometry and statistics in order to get the B (or vice versa). In the main, the questions that they choose will have one or two steps at most to a solution, or if more are needed then guidance will be offered in the form of question structuring. In these circumstances, more extended A level questions, where the mathematics required may cover more than one area, would prove a significant culture shock. From 2012, GCSE questions will contain less structural guidance, although some exam boards are keen to assure their customers that this will not make the exams harder. Instead, teachers will explain to students what the structure should have been. The issue, really, is that we live in a BiteSize world, but Maths is not a BiteSize subject. Actually engaging mathematically starts with a problem you care about and is followed by tenacity, insight and skill. Ofsted inspectors demand that students show progress within the space of a 50 minute lesson. In mathematics this is nonsense. Ideas are complicated and take time. Going backwards in the first hour would be unsurprising and frequently beneficial, so you make the mistakes you can later reflect on. But this would generate an unsatisfactory grading and it is no surprise that teachers feel unable to take the risk of allowing students opportunities to engage.
At the first session this year of the ATM/MA London Branch, of which I am secretary, we spent an enjoyable Saturday morning engaging with problems from the ATM’s now historic publication ‘points of departure‘. It is good to be reminded of how insights into a mathematical problem come unexpectedly and the solution arrives by a meandering path. On our table of 4, we found three completely different routes to essentially the same insight. So, now that the Key Stage 3 SATs tests have gone, the space has opened up to allow students time and space to be more creative. One possibility that is underexplored in schools is to use mathematical puzzles to help students sustain activity. A good puzzle seeps into the brain and won’t let go. A classic is the T-puzzle. After all, how difficult can it be to put 4 simple pieces of wood together in a ‘T’ shape. I spent a very happy hour in the Stockholm Interactive Science Museum with my daughter solving this. The sense of satisfaction at the end is palpable. Then again, we used to use the Tower of Hanoi puzzle for students to get top grades in GCSE coursework (remember that?) because you can prove by induction a rule for the least numkber of moves for a given number of discs. However, before you do the algebra, you have to get under the skin of the structure of the puzzle and again, that feels so good when you work it out.
So, start with a simple puzzle and sophisticated mathematical thinking and indeed mathematics comes out. Giving learners plenty of chance to be grabbed in this way seems a better route to sustaining their activity on those tougher maths problems which need grit and determination. Minimum step GCSE questions will seem easy and A level more than possible, when you have the desire to get to the end.