M208 Pure Mathematics

I still have the plague (man flu) and I am having to take some heavy duty painkillers for a chest infection.  This has caused a certain haziness which has made any mathematics study difficult, but not impossible.

I have continued to try and refine my study methods that I will deploy when M208 begins in January 2012.  The books should arrive at the end of December, around xmas time, but I have managed to find enough of the unit materials, to not only get a small head start, but also to allow me to prime any areas that I may find difficult next year.

If I were to say which are my favourite areas of M208 mathematics, they would probably be, in the following order, (favourite first):

Group Theory
Analysis
Linear Algebra

I find all the groups stuff very interesting and with such amazing applications from discovering new particles to solving the Rubik's cube.

Just a general note about M208.  I fear that if one were to stumble across the OU website and take the 'Are You Ready for M208' diagnostic knowledge check; then one could very easily get caught in an elephant trap, if one then assumed that the test is a fair representation of the mathematical maturity required, to study the course.

It is true that you can start the course with the knowledge from a good A level, but having looked at all the M208 study material in detail, last month; there are some very challenging topics and it goes into wonderful depth and breadth, around these topics.

I am not sure I would have coped, had I not started reading Brannan et al, last month and therefore turning a 9 month course into a 12 month course by getting a head start.

The rough list of main topics covered is as followed:

Real functions and graphs
Graph sketching
Hybrid functions
Curves from parameters
Sets
Functions
Proof
Binomial theorem
Geometric series identity
Number systems
complex numbers
Modular numbers
Equivalence relations
Symmetry in R^2
Representing symmetries
Group Axioms
Proofs in group theory
Symmetry in R^3
Groups and subgroups
cyclic groups
isomorphisms
groups and modular arithmetic
Permutations
Conjugacy
Subgroups
Cayleys theorem
Cosets and Lagrange's theorem
Vectors and conics
Matrices and vectors
Vector Space etc
Linear transformations
Eigenvectors
Conics and Quadrics
Inequalities
Bounds upper and lower etc
Sequences
Series
Continuity
Conjugacy
Homomorphisms
Kernels and images etc
Group actions
Orbits and stabilisers etc
Limits and continuity
Differentiation
Integration
Power series

Wow, all that in 9 months, 7 coursework assignments and one 3hr exam.

I can't wait!

Dicta-phones and highlighter pens!

I'm currently in bed sick, after 5 days of glandular fever, which is not pleasant.  As such, I have only managed a few hours of study this week and I have concentrated further on honing my study method, rather than worrying about what to study.

Last week I decided to ditch the idea of recording study material and then playing it back to myself.  However, I have further expanded that idea and this week I have used the dicta-phone again, but I have recorded short questions, based on the material, followed by a recorded answer.

I have found that listening to the question, retrieving the information, and then answering the question, has been a useful memory tool.  It has allowed me to effectively learn some of the 'filler' material in a unit, that isn't necessarily directly tested by doing practise exercises, yet is important to know.

I have also used the highlighter to clearly define what I need to remember, and then strictly, only re-read the highlighted areas, when I have revised the unit.  This seems to be working well and is cutting down revision times.

Anyway, I'm too sick to do much else this week.  Hopefully I'll be better soon.

Study Methods (Mathematics)

Alrighty,

This past week or so, I have been testing out some different approaches to studying mathematics.  I hope to find a groove that I can develop and then duplicate for future mathematics courses over the next 3 years.  I have tried a couple of minor tweaks, whilst I have been studying Brannan's Analysis.  Some examples are below:

Test 1 (4 days of study time)
Chapter 1,  p.1- 29
2 x skim read (to gain overall structure of chapter)
2 x detailed read, no examples attempted (to become used to new terminology and start to embed the flow of the subject material)
1 x recording on dictaphone, all rules, theorems and definitions.
1 x playback of recording
Selected 1 x example for each type of new problem introduced.
Attempting selected problems (6 in total), repeated each day, at the start of a study session.
1 x video lecture on topic (Notts University Analysis series on U tube)


Test 2 (4 days of study time)
GTA1 Symmetry M208 (Pure Mathematics)


2 x skim read
1 x detailed read, with highlighter pen, in hand
4 x detailed read of highlighted text only
Selection of 1x examples for each different type of problem
Attempting selected problems (12 in total), repeated each day, periodically throughout the day



I haven't done a scientific test of which aspects of these two test periods is the most effective.  I have simply gone on feel.  I want to develop a personal set of tactics that fit my current lifestyle and work / life balance; whilst maximising the amount of information retention.

The early results are thus:

Recording and playing back a dicta-phone, didn't really work for me.  I will probably ditch that.

The skim read was good and stopped me getting bogged down with examples, whilst giving me an overall picture of the chapter.

The detailed read was good, but it was difficult to stop my self getting tied up with examples.  That discipline will come though with practise.

Selecting one example for each area was a good idea.  It allowed me to repeat each one, eventually from memory.  This will help with later exams, allowing me to memorize problem structures, rather than relying on an annotated handbook, which will save time on the day.

The Notts University year 2 Analysis lectures, are a real find.  They are clear, concise and presented by a lecturer who is clearly a very good teacher, rather than just an academic who is begrudgingly forced to teach undergraduates.  No clouds of chalk dust here.

I have also started to experiment with some new software that I am trialling.  It is called MindGenius and is some very advanced and intuitive mind-mapping software.  It allows for rapid creation of mind-maps of study chapters, almost as quick as you can type.  As a dyslexic, I have needed to study using mind-maps, for years, as they suit my non linear way of thinking and aid my memory recall.

I will report more on MindGenius next week.
ps: I've put the link here for ease of finding it, I don't earn any money from it.

Other maths studied this week
Factor theorem
Remainder theorem
Graphing quadratics
Proof of identities


Total time spent: 36hrs

A Mathematical Interlude...

I have composed a quick test that can be used, to determine whether you been studying maths with the O.U , for too long:

1.  Read the following joke.  If you laugh, then you have done too much studying.  I recommend that you take 6 weeks off.

Joke:

Q:   What's a rectangular bear?
A:    A polar bear after a coordinate transform!


That is all.

My Aims, Update

Following on from a comment last week, I have decided to give a quick update on my aims and goals, of this experiment.  This is an update to my first ever post that I wrote back in February 2011 and I think that I will now provide an update and overview of each year, as I progress.


The comment that led me to writing this post, was from a learned fellow blogger and physicist.  He had commented that my ambition had ostensibly altered path, and was curious to know whether this was the case.

Well, my aim hasn't altered, but my understanding of my chosen field has developed, and this is causing the slight nudging of direction, to achieve that aim.

I think that what is happening, regarding aim and ambition, is that when I made my decision at the start of the journey, I had a 'layman's' understanding of the different areas of physics and maths.  I knew clearly in my head what I wanted to achieve i.e. learn lots of maths and apply it to some of the groundbreaking areas of theoretical physics.

Hence the theme of 'from GCSE maths to...'. And  this aim definitely hasn't changed.  What has happened, is I have been building up a better understanding and holistic view of maths and science in general, and the areas that I am interested in pursuing are, as rightly pointed out, more mathematical applications to physics, rather than physics, with maths thrown in.

I guess that is what I have enjoyed the most about this endeavour; I have found that my goals were initially broad and based on a low level of technical and professional knowledge, of the areas that I want to pursue.

It is only now since I have studied and discovered knowledge from various sources, that I am achieving a greater understanding of my proposed field of interest.  And as that base of understanding grows, and as I sample more of what maths and physics has to offer; I have no doubt that my field of interest will narrow further and the language that I use to describe my goals, will become more exact and discriminant, in its contextual meaning.

Another very important and undeniable reality that I have discovered, as part of the first steps of my journey; is that my best chance of success as a person who needs to continue to support a family and home throughout my studies, is to study mathematical physics.  This subject area lends itself more to either part-time, distance or self directed learning, as opposed to experimental or other areas of physics.

The reality is, that I can't study full time and loose my salary; and I suspect that to do a PhD in any physics that required me to be based 9-5 Mon-Fri at a lab, would not be realistic for me as a full time dad and employee.

Saying that, the plan is to try and reduce my hours at work once I start the masters either with the O.U or  K.C.L, for example; but I still need to feed a family of four!

I think that part of the experiment and what will hopefully be captured over the next few years in this blog; is the evolution and discovery of what works and what doesn't, for a person in my position.

I truly believe that whether I succeed or not, academically; the experiment will have been a success.  I am hoping that during that time, I will have created a record of the decisions made, the work studied, the ups and downs, and also, how it has shaped my aims, goals and ambitions.

One aspect that will be constant, is my dedication to study, my ambition and my time allocated to achieving my goals.  Also, one of the best aspects of blogging this experiment, is that the comments I receive, help enormously and make it a less isolated experience whilst distance learning.

Physics and Maths, Studied this Week.

This week, I have broken new ground by studying some of the course units of M208.  I have also been recapping and filling in some gaps in my knowledge, so that I don't falter because of a lack of basic manipulation skills.

With M208, Pure Mathematics, I have started to study the Unit GTA (Group Theory A).  I have not really studied Group Theory in any depth before, other than some fun in the summer with my Rubik's Cube, and applying some group theory to its solution.

The method of study that I am using for these 'pre'-read topics that I am looking at, is somewhere in between a light touch scan, and a full exploration of the material and examples.  My aim is to have lightly read each M208 Unit, before the module begins in January.  I am reading through once and then checking my understanding by doing one worked example, followed by one or two questions on my own.

The purpose of this style of studying, is to briefly understand the new concepts and then for them to quietly mature over the winter, followed by detailed practise in an attempt at mastery, when I study the unit proper, next year.

Just a quick word on the act of reading through a mathematics text.  I have decided to experiment over the winter with a few different methods of reading.  The first that I am applying to Group Theory A and Analysis A and B, will be the Feynman method.

Richard Feynman used a simple but effective method when approaching graduate texts for the first time.  He would start at the beginning and read until he got an example wrong or lost the thread of the text.  At that point, he would then go all the way back to the beginning of the text and read through again until he hit the next area of difficulty.  Each stopping point would be slightly further on than the previous one.

I am testing this method out, as I want to see how much time it takes to keep rereading a text and whether this saves revision time later on.  I presume that after five or six passes over the whole text in this way, most of the structure and concepts of the mathematics being studied, will be like an old pair of slippers by the end of the course.

In the meantime, I am also researching other methods of study and will perhaps attempt one or two more, before comparing the efficacy of these approaches.

Anyway, here is this week's completed study:

Time spent 14hrs.

Group Theory A
Intuitive ideas of symmetry
Formalising symmetry
Symmetries of a plane figure

Analysis - Brannan
Convergent sequences and the squeeze rule. Most exercises attempted. P.48 - 61

Other Maths Exercises (60 exercises completed, in total)
Remainder Theorem
Factor Theorem
Proving inequalities
Binomial Expansion

Next week, I will fully implement the Feynman method and look at some more Group Theory.

Physics and Maths, Studied this Week.

This week, I have concentrated on Brannan's fascinating text, 'A First Course in Mathematical Analysis'.  It makes up the bulk, if not all of the sections of Analysis in the Open University course  M208 Pure Mathematics, and is a significant part of the whole course.

So, getting a head start on this material, much of which is new to me, will help to ease any mid-course calamities.

Most O.U courses, as of 2012, will run from October until June each presentation.  This is 9 months of study which is then followed by a rest period of 3 -4 months, before the start of the next module.  However, I have decided to use the dead time, to get a head-start on the material.  This will hopefully help implant the knowledge, into the long term memory, as I will approach the material twice; once before the course starts and then later on during the module.

My only worry, is that I finish M208 with an exam week that runs from October 16th which overlaps with my next module's start date, being M337 Complex analysis, that starts early October 2012.  I plan to remedy this by pre-studying the first unit of Complex Analysis before Xmas this year, and then to recap just prior to the course starting in October 2012.

Anyway, here is this week's study.  Total time spent:  18hrs

Brannan - Analysis 

- Solving and proving inequalities p.10 - 19  including all exercises.
-  Least Upper Bounds and Greatest Lower Bounds p.22 - 30 Including some exercises.
-  Monotonic Sequences p.37 - 42 including all exercises.

Velleman - How to Prove It

- Operations on Sets p.34 - 54

Maths Skills Practice exercises. 80 exercises in total
                                           
 -  Irrational numbers
 -  Quadratic Equations and complex numbers
 -  Discriminant
 -  Root Coefficient relationships