Hi first let me apologise for not posting earlier I'm currently working on the so called 'Straightforward but Tedious' process of deriving the Friedmann equations (the ones that form the basis of modern cosmology) from Einstein's Theory of relativity. The process is certainly tedious whether or not it is straightforward is another matter. I'll keep you posted. Anyway in the mean time here are my thoughts on the two maths courses I did this year. These were

MST121 Using mathematics and MS221 Exploring Mathematics.

As essentially entry courses for the degree in mathematics they try to cover most of the stuff that would be expected for an entry into a typical university. So roughly equivalent to the old A level Pure and Furthe Pure maths courses with a little mechanics and statistics thrown in. Without boasting for me most of the course material was revision mainly to get back into thinking about stuff which I hadn't done for years. My colleagues Neil and Nilo have already described these courses in some detail and there is not much I can add to their descriptions. For those who want a blow by blow description I refer to their blogs.

However here is a quick overview. I think the courses are best taken together as a unit with MST121 in October and MS221

The courses are split into 4 blocks and the material in the MS221 blocks A, B and C build on the corresponding blocks in MST121. Whereas block D in MST121 covers basic statistics and block D in MS221 provides an introduction to pure maths of which I will be getting a lot more of when I do M208.

Block A is concerned with sequences, functions, coordinate geometry trigonometry and recurrence relations

the stuff on recurrence relations was new to me and one of the more interesting parts of the course especially the stuff about Fibbonacci series and the golden ratio. It's that sort of maths which on the one hand is quite straightforward but has amazing applications. However solving a second order recurrence relation can be quite tricky to get correct and I'm sure I lost a couple of marks in the exam. For those contemplating doing MST221 next year it really is worth mastering the procedure for rotation of a conic section as this question always seems to crop up in the second half of the exam. (Of course I didn't heed this advice of which more later)

Block B is concerned with vectors matrices and their application to geometric transformations and also another topic which was new to me the fixed points of an equation and how they govern the behaviour of a function when it is iterated. Again a fixed point question seems to come up all the time and it is worth getting to the stage where you can invert a 2x2 matrix, find it's inverse, eigenvalues and corresponding eigenvectors.

Block C on calculus was my favourite but that is because I have loved calculus ever since I first came across it when I was about 15 or 16. I think anyone who hasn't got a basic understanding of calculus at the level of this course doesn't deserve to be called properly educated. A bit strong perhaps but without a grasp of calculus you have no idea how physics or engineering works. Those in the Arts world often call physicists and engineers philistines as they aren't familiar with the main works of Shakespeare are just as philistine themselves if they don't know how calculus works . If I have a slight criticism of this unit I feel that they could have made an attempt to include material on Second order differential equations instead of Taylor series. It is second order differential equations which form the backbone of most physics modelling enshrined in the simple harmonic oscillator, The thing that made me really interested in mathematical physics was the realisation that the same basic differential equation described both mechanical and electrical oscillations. Taylor series whilst important could really be left to a level two course whereas an introduction to second order differential equations with an explanation of some of their applications would be much more useful. Especially for those people for whom MS221 will be their last maths course. Anyway those who want a taste should down load the units of MST209 available on Open Learn. Anyway despite getting my best marks for the TMA's in this part I totally bombed the long exam question as I had a mental block. So whilst calculus is interesting doing it under pressure is not so good and it is not recommended to do the long calculus question.

MST121 Block D Statistics Most people found this boring however I have to say I thought it gave a good overview of the process of hypothesis testing. This has had quite significant social implications in recent years especially when it comes to testing the efficacy of certain proposed medical statistics. In brief hypothesis testing is a method whereby tests can be devised to test the validity or otherwise of a certian proposed treatment. It enables a statistic such as the mean from two samples to be compared, In medical tests two groups of people of the same size are compared one is given a new drug say whilst the others are given a placebo. If the number of people who improve after being given the drug is significantly higher than those who are just given the placebo then the efficacy of the drug treatment is deemed worthy of further investigation. More often or not there is no statistically significant difference between the two samples and thus the treatment can be ruled out. In the current climate their appear to be all sorts of quack remedies such as homeopathy or other so called alternative medical treatments. By performing a hypothesis test the validity of these claims can be rigourously checked, Needless to say when these tests are performed the treatment is generally deemed statistically insignificant. The point is that this methodology is a powerful tool and politicians and people who make policy decisions should be aware of this. Had a proper analysis of the claims of Dr Wakefield who claimed that there was a link between the MMR vaccination and autism been undertaken much needless scaremongering and agony for many parents would have been avoided. However as a consequence of the claim many people were put off subjecting their children to the MMR jab and as a consequence we are now facing the rise in measeles a disease which had been effectively erradicated.

The point being that with tools like hypothesis testing soft sciences such as the social sciences can be raised to the status of real science as they have a method of falsifying claims. Unfortunately the unit in MST121 did not really mention this application of statistics. Had it done so then perhaps people would not have found it so boring.

Unit D MS221 Introduction to Pure Maths this covered complex numbers a brief introduction to number theory group theory and methods of proof including induction

This is an attempt to introduce some of the thinkning behind pure mathematics I was familiar with complex numbers, group theory and mtethods of proof. As a taster the group theory was Ok but (and I suspect) this will become more so when I do M208 and M336 there was no real rationale behind why group theory is important. Yet it has had a profound impact in physics via the use of representation theory as applied to particle physics and quantum mechanics. However the part that is necessary for this is representation theory and this I believe is not taught in any of the OU maths courses. However group theory as a mathematical structure is quite profound in iteself I refer you to Nilo's site for those interested in this aspect.

As for number theory I really don;t get it and the course did not really convince me give me a complicated integral and I'll get my pen out and do my best to solve it. Show me the analogy between vector spaces and function spaces (of which more later ) and I'll be really impressed, On the other hand I just can't get excited by relationships between prime numbers or congruences. The application to coding given in this unit was quite tricky easy to make mistakes and fiddly. I'm afraid I couldn't resist at a tutorial getting my tutor to work through an example on the board I was relieved to find that he made more or less the same errors that I would have done.

Ok so overall impressions as a warm up to the real stuff and practice in doing TMA's these courses are great however the real stuff starts at level two I can't wait. As a sideline I was really lucky in having a really good tutor and a friendly tutorial group I have made lots of friends including Nilo (who I got to know via a link on Neils blog) , Duncan and Neil this is the first OU course where I became convinced of the value of tutorials and the friendships that follow. I would like to thank my tutor Alan Borthwick and all the friends I made on MS221.

Block B

Very interesting review Chris. I particularly liked the comment about philistines in the arts.

ReplyDeleteI was very impressed to read that your tutor was Alan Borthwick, the 'star' of Algebra Workout! I hope he brought his magic board with him to help demonstrate algebraic manipulation! :)

Yeah Alan's an amazing guy really committed to helping his students get through the course and full of energy and enthusiasm.

ReplyDeleteDespite their evident age, I quite enjoyed his Algebra Workout videos, and was disappointed that they didn't continue into M208. That's not to say the M208 videos aren't informative in their own right. The CD sections however, I had an issue with...

ReplyDeleteNeil (hoping to have his name attached this time)

This seems to be the definitive post on MST121 and MS221. ;-) - I am studying 2nd order differential equations -right now! Prof. Mattuck (MIT 18.03) repeatedly says: this is why you had to calculus, linear algebra, group theory: to understand DE's. Fascinating stuff, I am sure I am going to like MST209. - I am really sorry that you feel so bad about number theory. ( Have you seen the movie Pi? Besides an exciting thriller about a mathematician on the run for orthodox Jewish number theorists working on the bible codes it's about how integers rule physics. - This is a pic from the movie. http://mathematics-diary.blogspot.com/2006/04/fibonacci-sequence.html - I am somewhat jealous because you seem to have found your research project already. I am still searching... ( Nilo )

ReplyDeleteI wanted to post a comment on Neil's blog about MathCad, but I don't think he has 'enabled' comments yet. ( Is he building his own blogger software? - Or have I plain missed it?! ) - I wish the both of you were on Facebook. I was sceptic too, but you do meet people you wouldn't have met otherwise. Already made friends with three OU students of which one is going to do M208 + M337 next year. Facebook has extensive possibilities to protect your page and it has a secured access.

ReplyDeleteHi Nilo glad you are enjoying differential equations as the guy says you finally find what all that calculus was aiming for. Can see you becoming an Applied mathematician/ Physicist yet :). If you want something a bit more have a look at the Cambridge maths site I gave you a link to. You will amazed that you can already do some of the 1st year differential equation problems especially those involving Bernoulli's equation.

ReplyDeleteMST209 is fine as far as it goes but it's main omission is solution by series I think that is covered in some the third level applied courses but I'm not sure about it. I suspect the MIT lectures will cover that anyway.

Especially as you are keen on representation theory. I guess number theory gets better the more you get into it. As for me having a research project it's more trying to get an understanding rather than anything original so it's not really research as such. Although presumably ideas will follow once I have the basic understanding. I guess I'm quite parasitic really.

Finally Neil is using his own code as far as I can tell. However you can contact him via his OU blog. I'll think about face book but at the minute blogging is a first initial step. I also need to look at that Latex stuff you sent me so I can type my equations in here.

Best wishes Chris

Hi Chris

ReplyDeleteGreat review! I agree with Nilo--definitive.

I also agree about the missing differential equations, frankly I think that we could have dropped Taylor Polynomials and done these instead. [I found TPs really boring]

@Nilo

Yes my blog is handrolled and although I can, and have, build/t my own commenting system I prefer to use the OU one. You'll find me in the blogs.

And I agree [again!] that it's a shame that Chris didn't like the number theory, for me the best part of the course.

And everyone OU should read http://learn.open.ac.uk/mod/oublog/viewpost.php?post=50087 from one of my fellow M257 coursemates. A wonderful example of an 'odd' proof.

Aw ra best...

neil