# Impressive Maths Thinkers

It is easy to be a bit jaundiced about A level maths, when you see the drill and practice text books with the exam board’s logo on the front and student’s in permanent practice, practice mode because of the modular exams. Well, I had an excellent day in a non-selective, state sixth form college, working with two groups of charming A level maths and further maths students.

The good story is that I have a starter activity with a graphing calculator, where I set up a function, f(x) in symbolic mode hidden from the audience. I then set up the machine as a blank table, that reports the value of f(x) for values of x that the audience provide. The aim is for the audience to guess what the function is. After a couple of x and f(x) pairs are generated, I insist that they decide what f(x) will be (but NOT say anything) before giving the next value for x, we keep going like this until a clear view of the key details of the function emerges. I then allow them a quick glimpse of the graph (with no axes showing) they can see the ‘shape’ of the function. We then tie up the details before seeing the original function. Now, I have done this with a large number of teacher groups and I generally choose a quadratic with whole number roots. Teachers are very scared to commit themselves. They frequently seem concerned about doing other than finding a rule from the sequence in the table and are not willing to see the geometry. I worry that some teachers have become drawn into the desire for a clear, one-dimensional method, that they are now solving problems themselves this way.

Well, my sixth form students are doing MEI, which stands alone in sticking up for mathematical engagement. And it shows. I gave them a cubic. Quickly they got the y-intercept and went systematically to check for roots, then they checked extreme positive and negative values to be convinced they had found all the roots. When people have got it, I get them to say both x and f(x), so they can check, but others can keep going. The students were clear, precise and systematic. Very impressive. So, I gave the second group f(x)=2sin(x+1) with angles in radians. Same story, once they had the range of -2 up to 2, it was clearly periodic which they checked with increasingly extreme values and tied it down with f(-1)=0. Perfect. I will post some screen shot of how to set up this activity on an HP39/40GS which I was using. After showing the students how to do the activity, they created their own function to test their neighbour … and there were some pretty complicated functions in there, I can tell you. If these young people are the future teachers of maths, all will be well.