After a two-week holiday at the beginning of June in Germany, I have slowly got back into the swing of things in the last three weeks by starting to do the further exercises in M208. I have been attempting to do these questions with only the aid of the handbook and nothing else, just as we will be expected to do in the exam. For the most part, I seem to be managing ok but it is clear that the handbook is sometimes a bit turgid when you are trying to figure out how to answer a question and additional underlinings and notes seem essential. And yes, I must admit, that in some cases I have written out complete answers to particular questions in the back. Why? Well, either I have a complete mental block over some questions, and I need this sort of example to assist me, or the required answer has such detail that I need a template to refer to.
So I have complete answers written out for 1) finding the image of a function (mental block), 2) proving the continuity of a function using the epsilon-delta definition (template), 3) proving something by mathematical induction (template) 4) (orthogonally) diagonalising a matrix (template) and 5) finding the image of a homomorphism (mental block). I am sure there will be others. So is it cheating? Well, you are allowed to do this, so my view is that it isn't and you should grasp every opportunity that is offered.
Occasionally there are some interesting problems in these further exercises. Take for example GTA3 Ex 4.8. I hope the OU doesn't mind if I quote the question - (a) Write down an element of (the symmetric group) S5 of order 6 and hence find a cyclic subgroup of S5 of order 6 (b) Given that S5 contains exactly 20 permutations of order 6, find the number of cyclic subgroups of S5 of order 10. I liked this question because it really made me think and I was quite pleased with myself when I figured it out!
I notice that some of my other fellow students are spending time writing stuff out on cards, memorising theorems etc. I just can't force myself to do this, although I wish I was that well organised. Instead, I am relying on practise and that unarguably helpful organ, my gut (!) which keeps me informed about how it thinks a question should be answered!! I hope it doesn't fail me in the exam.
