Straight 100s

Well I managed to do it. Today I received my assessment back for TMA07, the last in this course, and I scored 100/100. That means I have got a straight 100's for all my M208 TMA's and I feel that I have managed to achieve something here. Even if I don't do very well in the exam, at least I can look back at this and say "I did do this!". I am very happy!

Up and Down

Last week I started working though M208's specimen paper and I found it quite hard going. I can't face sitting down to 3 hours of work at one go, so I have been doing a question at a time and timing myself. It was a bit of an eye opener. The first 12 questions (part 1) carry 70 percent of the marks and ideally I would like to spend 10 mins each on these questions leaving an hour for the longer part 2 questions. However my timings varied from 7 mins up to 23 mins and one question I had to abandon because I got in a stew over it. The need for speed also caused some terrible errors. I then went on to do questions 13 and 14 and came a cropper in both as I couldn't complete them and was lost as to what to do.

So a bit of a dip in my confidence followed! I had a rethink about my approach and decided that accuracy and "more haste less speed" was required. So this week I have begun the part 1 questions from the 2007 paper and things have improved a bit. Timings are much the same but the accuracy has got better and I think this is the best I can hope for. If I can get through all of the part 1 questions and have a good stab at one part 2 question then I will be happy. So confidence has gone up a bit again.

The problem with trying to speed up is that once you start to make mistakes, you start to doubt yourself and what you are doing. As panic starts to creep in, so your mind tends to go blank and it all becomes a muddle. My attitude then will be one step at a time and to stay focused and hope that I can keep this up for 3 hours, much like I did for MS221.

Not long now

It has been over a month since I last posted anything here and I thought it was time for an update. The course itself is drawing to a close as, by now, we should have finished all the coursework (bar the last two TMA's). As such I am beginning to feel that slight frisson of anxiety about the impending exam in October and it is starting to get me interested in past papers.

One of the things I did do in the last week or so was to read through the specimen exam paper carefully and see if there were questions that I didn't know how to approach. This was quite useful as it highlighted some holes in my knowledge about infinite quotient groups, which I have since rectified. Also, if there are methods which I suddenly think to myself "how the hell do you do that again" I have been looking them up and adding yet more examples and notes to my handbook. One of these was the counting theorem which I would like to get off pat before the exam and I think I will have to test myself with some examples.

Generally things are going ok. I am keeping up my 100% score on the TMA's which keeps me happy and I am still ploughing on with the further questions in the course text. I stopped doing these at random and instead went for the more consistent approach of tackling a block at a time. I have done about half the questions now and they keep highlighting areas where the material hasn't sunk in very far. Strangely enough, the material I feel most comfortable with is analysis as I think I have a good feel for what's involved. Some parts of group theory and linear algebra still have the ability to cause me problems. Ideally, when I get to the exam, I would like to feel that there was no part of the paper that I couldn't do, so that it gives me confidence right from the start. I think my game plan for the first 12 questions will be to do them in order as I did in MS221. It will mean I can avoid wasting time deciding which to do first. I can't wait until around mid September when I will start doing the past papers properly and really testing out my knowledge.

Further Exercises

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.

Getting into revision mode

I have now got into revision mode by working on the further exercises of the course. There are about 100 sections in all with these additional exercises and so there is plenty to do. As planned, I am doing these at random - literally, as I am using a calculator to generate a random number which indicates which section I should tackle next. This avoids me favouring certain sections and avoiding others. It also means that I get used to working on any part of the course rather than just one particular bit. I am also working with just the Handbook and if I think it doesn't give me what I need I am adding extra notes (and this happens quite a bit).

I have now completed all the TMAs and am in the process of writing up the last two in neat. I did well on TMA03 scoring another 100%. I keep thinking that I must make a mistake at some point. I have to keep pinching myself when I think that I only dropped one mark in the TMA/CMAs for MST121 and none in the TMAs for MS221 and now I haven't yet dropped a mark for M208. It'll all end in tears at some point!!

This leads me to think sometimes that I have missed my vocation. The problem is that it is all too late now. If I wanted to take it all further, it would mean another 3 years or so of a degree, then some sort of masters or PhD, by which time I would be pushing late 50's. Who would want to give me a job then and how much would I realistically expect to achieve? I should have done this all 10 or 20 years ago.

Ach, well, I can still study. I have just acquired another book on number theory in a sale at Blackwells. This is Elements of Number Theory by John Stillwell. It looks more approachable than some of the other number theory books and it may get me deeper into this subject. Edinburgh library has a good selection of books and I was surprised to find such a technical book as A first course in general relativity by Bernard F. Schutz. I couldn't resist taking it out on loan even though I know I will probably only read a few pages. If only I had more hours in the day!

TMA06

I have now completed TMA06 and just have to write it up neat. There weren't any nasty surprises in this TMA and it all pretty much fell into place. One thing I found, though, was that the answers to the exercises for the AB books tended to make more assumptions about continuity and the behaviour of limits. I wanted to follow this example but found that I wasn't always happy making these assumptions. In the end I decided to add an appendix to my answers where I proved some of the assumptions I was making. Not surprisingly some of these proofs were nearly as long as the answers themselves, which probably explains why assumptions have started to creep into the text.

Wanting to get all the course work out of the way I have started TMA07. The first question was a bit of a tease which surprised me a bit, but I saw the light in the end.

I have started reading a few books now that I have more time. One is "Einstein, A Life in Science" by John Gribbin and Michael White and the other is "The Millennium Problems - the seven greatest unsolved mathematical puzzles of our time" by Kevin Devlin. The Einstein book is a biography and I enjoy these type of books. It is clear that the young Einstein didn't like authority, tended to avoid lectures at University and only wanted to study things that interested him. It is fortunate for us that this didn't ultimately affect his career!

The Millennium Problems interest me. In 2000 the Clay Mathematics Institute decided to offer seven $1 million prizes for those who could solve seven of the most currently difficult outstanding problems of current mathematical research. This was a bit like the 23 problems outlined by David Hilbert in 1900. One of the Millennium Problems, the Poincare Conjecture, has already been solved by Grigori Perelman but he rejected the prize! The problem that interests me the most is the Riemann Hypothesis because this is related to prime numbers.

The End

I have reached the end of the text for M208, so that's quite a milestone reached, and I won't have to tackle any new material from now on. I started working on M208 back in mid February 2011 and so it has taken me a year and three months to read through what should really take eight months, so I have been going at roughly half speed. I haven't done the further exercises yet, but I have read and worked through nearly every proof and, believe me, that was quite a chore. On the last book I gave up with reading the very last optional proof as I just couldn't be bothered!

I enjoyed AB4 on Power Series. It was a nice amble through Taylor polynomials, Taylor's Theorem, Taylor Series and manipulating power series. Nothing too difficult and all fairly straightforward, so it maybe a topic to choose in the exam.

So what now? Well, there is TMA06 and 07 to do. That will be my first priority. Then I will be keeping things ticking over for a while by doing the further exercises. I plan to do these at random, so that I have to practise working from different parts of the course at the same time. I will also just try and work from the Handbook and see how I get on. I suspect that at this stage I will be adding a few extra notes to this where I think there isn't enough information to get me going on a question. Then sometime around late August I will start turning my attention to past exam papers and practise getting questions answered quickly and honing those skills ready for the exam in October.

I have already started think about what I might do next and this probably won't involve the OU for now. I intend to start another blog about my own explorations into prime numbers. I have also thought that I might begin trying to understand some chapters from 'An Introduction to the Theory of Numbers' by Hardy and Wright. Then another avenue for me is, perhaps, to get to grips with Special Relativity which I felt I never really did when I was a physics undergraduate. The thing is there are so many choices of things to do that I will probably not know where to start! I still have all my undergraduate text books and it is tempting to delve into Quantum Mechanics again, or Electromagnetism, or perhaps finally sit down to 'Gravitation' by Misner, Thorne and Wheeler!