Happy new year to all.
Having now done all the pre-study that I needed to, I have now prepared a detailed mind-map of the entire M208 course, drilled down to the section level, using my Mindgenius software. I plan to use this map, to hang off all the study material and study periods, as they happen. The software allows me to convert the map into a Gantt chart, to keep track of critical path items of study such as TMA's and important topics, so that I don't get side-tracked, if short of time.
I have also exported it to a word file which is now plastered all over my office walls. As each section is studied, I plan to measure how many study passes each section has had, noting any difficulties and also counting the repetitions of example questions attempted.
I have also created a 'proofs book', a rather nice handmade 'Moleskine' A5 journal, in which I plan to write down all of the proofs that I now encounter, from M208 and beyond. I hope to make this book a permanent feature of my coat pocket; browsing through it, in all of those moments of waiting that life creates, such as in queue's for buses and the like.
I am very excited and have made a pact with myself to commit 100% of my spare-time, to learning the contents of M208 this year. I have used this approach before, when studying for professional exams, years ago; so it is a well trodden path, that I hope will not have too many pot-holes in it! (wife allowing)
So, again, happy new year to all. The first big step on my quest for mathematical maturity, begins today.
Saturday, 31 December 2011
Tuesday, 20 December 2011
M208...Is Here
It arrived today. I feel like Christmas has come early. The smell of new books; the anticipation of reading them; the shear amount of them!
I had a bit of a shock as they have delivered the entire course in one package. They usually deliver half of it first and then the second half later in the year. So I did have a moment of panic when I saw all of the books, before realising that what was in front of me, was the entire course.
The plan? Quite simply to begin by lightly browsing through the books and getting a feel for the treasures inside; perhaps read the odd passage that takes my fancy. Before finally settling down to some serious business between Christmas and new year (depending on hangovers).
Oh my goodness, what have I done!
I had a bit of a shock as they have delivered the entire course in one package. They usually deliver half of it first and then the second half later in the year. So I did have a moment of panic when I saw all of the books, before realising that what was in front of me, was the entire course.
The plan? Quite simply to begin by lightly browsing through the books and getting a feel for the treasures inside; perhaps read the odd passage that takes my fancy. Before finally settling down to some serious business between Christmas and new year (depending on hangovers).
Oh my goodness, what have I done!
Monday, 12 December 2011
The Calm Before the Storm
I am having a little rest from study and posting for part of this month, until my M208 study books arrive, signalling what will be a heavy and pre-longed period of chaos! Not only will it be full steam ahead for M208 between the end of December 2011 until the exam in mid October 2012; but I will then start Complex analysis and also Number Theory and Logic, just before the final exam for M208.
Once those courses are complete, that will have been a full-on 19 months of study without any break. Ouch!
So, I am taking this brief lull in my preparation, to prepare for a traditional family holiday at home. I may even partake in a few bottles of Bishop's Finger and the odd mince pie. Yum.
Once those courses are complete, that will have been a full-on 19 months of study without any break. Ouch!
So, I am taking this brief lull in my preparation, to prepare for a traditional family holiday at home. I may even partake in a few bottles of Bishop's Finger and the odd mince pie. Yum.
Wednesday, 30 November 2011
MST121, Passed!
I received my final result today, for MST121 Using Mathematics and I am thrilled to be able to say that I have passed. I received 96% for the final piece of course work TMA04 and 82% for the CMA, giving an overall combined course result of 93%. I plan a critical review of the course content in MST121, in the next few weeks!
So, now I can start to relax properly into my exam prep for M208 Pure Mathematics. However, I received some bitter-sweet news yesterday when I found out that the Open University have now confirmed some course changes for the next two years, which affect me greatly.
The first of these, is that they have now postponed presentation of the new course M303 Further Pure Maths until October 2014.
This means that I can no longer realistically take that course, without extending my studies by a redundant year. Now, the OU have countered this by stating that they will provide an additional presentation of the course M381 Number Theory and Mathematical Logic, which will actually allow me to take that course.
This is pleasing, as I do have a hankering for Godel's Proof which forms part of that course material. I will now also need to take the separate course M336, Groups and Geometry, again a good course that I am looking forward to studying. The only downside with these courses is that it will cost slightly more money to do two separate courses and will mean two 3hr exams instead of one.
But the real pain in the backside of all this is M338 Topology. By not doing M303, I will miss some important Topological material, and the O.U's Topology course is in its last presentation in Feb 2012, which I am not ready to take. So, It leaves a gap in my studies with the O.U.
I don't know how much of a profile gap that will leave, and whether I can successfully plug it by buying the Topology course and self-studying it. We will have to see.
So, now I can start to relax properly into my exam prep for M208 Pure Mathematics. However, I received some bitter-sweet news yesterday when I found out that the Open University have now confirmed some course changes for the next two years, which affect me greatly.
The first of these, is that they have now postponed presentation of the new course M303 Further Pure Maths until October 2014.
This means that I can no longer realistically take that course, without extending my studies by a redundant year. Now, the OU have countered this by stating that they will provide an additional presentation of the course M381 Number Theory and Mathematical Logic, which will actually allow me to take that course.
This is pleasing, as I do have a hankering for Godel's Proof which forms part of that course material. I will now also need to take the separate course M336, Groups and Geometry, again a good course that I am looking forward to studying. The only downside with these courses is that it will cost slightly more money to do two separate courses and will mean two 3hr exams instead of one.
But the real pain in the backside of all this is M338 Topology. By not doing M303, I will miss some important Topological material, and the O.U's Topology course is in its last presentation in Feb 2012, which I am not ready to take. So, It leaves a gap in my studies with the O.U.
I don't know how much of a profile gap that will leave, and whether I can successfully plug it by buying the Topology course and self-studying it. We will have to see.
Wednesday, 23 November 2011
More on Dyslexia and Mathematics
I thought I would drop in another quick post about dyslexia and how it affects the assimilation of knowledge, and in this case, mathematics. I have been researching the most up to date theories on the causes of dyslexia and how it affects learners.
One thing that I have known for a long time, being dyslexic myself, is that there are significant advantages to being dyslexic when carrying out certain tasks or roles.
For example, I find that I am able to work on manipulating 3D abstract objects, in my head; rotating them this way and that. However, with linear objects such as 2D graphs or some areas of algebra, I find no such advantage.
The ability to work in 3D can have nice advantages when doing certain elements of group theory and are wonderful when it comes to geometry in 3+ dimensions. It has also helped the visualisation of the complex plane, when handling 'i', and its 'real' friends.
As part of my research, I have discovered that the latest neuroscience surrounding the root causes of dyslexia, gravitate towards the theory that a primary cause is found in the theories surrounding left and right brained thinking models.
I'll explain. As children, we used our right-brain more than the left, to begin with. The right brain area deals more in whole pictures and broad themes, able to get the gist of a subject or concept, but not with much detail. The left brain, is where the detail is processed and the more linear and analytical side of ones approach to a task or subject, is generated.
As children become adults, the use of the brain hemispheres starts to even up and although most people will end up slightly biased towards either big picture / creative thinking or towards detailed analytical thought; most people will generally have a bit of both.
However, with most dyslexics, the brain doesn't seem to make the transition from right brained to left, all that well as they move towards adolesence. This means that for a dyslexic, they can cope with the big picture and sometimes see connections where others do not. They can handle tasks better in 3D, but will perhaps struggle with linear and fine detail tasks and working memory.
So where does maths fit into all of this? Well, if a dyslexic was to play to their strengths when learning maths, they would firstly aim to learn the broad themes and structures of a unit or topic, then followed by work on the detailed mathematics underneath. By working in this order, it allows a dyslexic to 'hang' all the detail from the broad branches that have been learned first, and provide structure to aid the retention of detail.
So is it impossible for a dyslexic to learn the finer details of maths sufficiently? Well, no. However, it takes a huge amount of effort to rote learn any mathematical facts, proofs or theorems, unless they can be pictured as part of the subject as a whole.
How does one do this? Well, I am using mind mapping software to lay out the subject, unit by unit, into a large and explorable mind-map. This is then followed by diligent study of the detail and exercises. I don't know if such a system of learning would suit a non-dyslexic and I know that there are some of my non-dyslexic peers who prefer to work in algebraic terms rather than geometric pictures or models.
I guess we are all different but I see my dyslexia as an advantage. I wouldn't quite go as far as one of my dyslexic friends, who described all non-dyslexics as 'Muggles'; but I think I understand where he is coming from, and perhaps for some types of maths, dyslexia is a gift rather than a hindrance.
One thing that I have known for a long time, being dyslexic myself, is that there are significant advantages to being dyslexic when carrying out certain tasks or roles.
For example, I find that I am able to work on manipulating 3D abstract objects, in my head; rotating them this way and that. However, with linear objects such as 2D graphs or some areas of algebra, I find no such advantage.
The ability to work in 3D can have nice advantages when doing certain elements of group theory and are wonderful when it comes to geometry in 3+ dimensions. It has also helped the visualisation of the complex plane, when handling 'i', and its 'real' friends.
As part of my research, I have discovered that the latest neuroscience surrounding the root causes of dyslexia, gravitate towards the theory that a primary cause is found in the theories surrounding left and right brained thinking models.
I'll explain. As children, we used our right-brain more than the left, to begin with. The right brain area deals more in whole pictures and broad themes, able to get the gist of a subject or concept, but not with much detail. The left brain, is where the detail is processed and the more linear and analytical side of ones approach to a task or subject, is generated.
As children become adults, the use of the brain hemispheres starts to even up and although most people will end up slightly biased towards either big picture / creative thinking or towards detailed analytical thought; most people will generally have a bit of both.
However, with most dyslexics, the brain doesn't seem to make the transition from right brained to left, all that well as they move towards adolesence. This means that for a dyslexic, they can cope with the big picture and sometimes see connections where others do not. They can handle tasks better in 3D, but will perhaps struggle with linear and fine detail tasks and working memory.
So where does maths fit into all of this? Well, if a dyslexic was to play to their strengths when learning maths, they would firstly aim to learn the broad themes and structures of a unit or topic, then followed by work on the detailed mathematics underneath. By working in this order, it allows a dyslexic to 'hang' all the detail from the broad branches that have been learned first, and provide structure to aid the retention of detail.
So is it impossible for a dyslexic to learn the finer details of maths sufficiently? Well, no. However, it takes a huge amount of effort to rote learn any mathematical facts, proofs or theorems, unless they can be pictured as part of the subject as a whole.
How does one do this? Well, I am using mind mapping software to lay out the subject, unit by unit, into a large and explorable mind-map. This is then followed by diligent study of the detail and exercises. I don't know if such a system of learning would suit a non-dyslexic and I know that there are some of my non-dyslexic peers who prefer to work in algebraic terms rather than geometric pictures or models.
I guess we are all different but I see my dyslexia as an advantage. I wouldn't quite go as far as one of my dyslexic friends, who described all non-dyslexics as 'Muggles'; but I think I understand where he is coming from, and perhaps for some types of maths, dyslexia is a gift rather than a hindrance.
Monday, 14 November 2011
Sometimes the simplest mathematics, is the most satisfying.
As I prepare for M208, Pure Mathematics, in January; I often take some wonderfully interesting and short diversions into areas of simple maths skills. These areas of interest include manipulating fractions or logarithms; messing about with inequalities or complex numbers; or simply factorising polynomials.
It can be very relaxing, something akin to doing Sudoko or crossword puzzles, glass of wine in hand. However, it all has a serious purpose. Whilst I suspect that there is less use for basic maths skills, as one approaches the dizzying heights of group theory etc. I do still find, that being deft at the handling of numbers and formulas, is like having a big fluffy comfort blanket.
Perfect for those dark winter nights!
It can be very relaxing, something akin to doing Sudoko or crossword puzzles, glass of wine in hand. However, it all has a serious purpose. Whilst I suspect that there is less use for basic maths skills, as one approaches the dizzying heights of group theory etc. I do still find, that being deft at the handling of numbers and formulas, is like having a big fluffy comfort blanket.
Perfect for those dark winter nights!
Tuesday, 8 November 2011
Mind-Genius and Mathematics
What I am really enjoying about my distance learning journey, is the fact that I am discovering new things about myself all the time. I have been able to try out and eliminate, any unhelpful learning strategies, that don't help me to achieve my goals.
When I studied maths with the O.U, last academic year; I discovered that whilst I found the details and elements of the actual maths, relatively easy to handle, intellectually; I did find that I would struggle to remember set solution methods to problems and also I would also keep loosing track of how one bit of maths, fitted in with another.
This led to some wasted study time, having to revisit maths I had already examined, because I hadn't gained a holistic view of the subject.
Now, I have dyslexia. And, the experts say, and I tend to agree with them, that dyslexics need to think in terms of connections, and that they learn more effectively, if they are able to 'hang' details from the branches of the whole subject. That is, in order to keep track of the details and to learn them completely, we need to know the bigger picture, and how these details slot into that image.
One of the best ways for this to be done, is through the use of mindmaps, as developed by Tony Buzan many years ago.
However, the problem with mindmaps is that they can be a devil to produce. This tends to mean that the creation of the map, can be a pain in the backside, that gets in the way of your study subject.
However, I have been trialling a piece of software called MindGenius 4, which is an important discovery for me, particularly as it fits my 'dyslexic' way of viewing the world.
I have enclosed a link for info purposes only, as I don't receive money for sending you there. There are alternative mindmap software's that I have tried, but they aren't as easy to use, as this one. I would encourage anyone to shop around and find what fits for them.
I won't bore you with the details, as anyone that is intrigued, can download a 30day trial for free and try it out. But, it basically allows you to create mindmaps, as quick as you can type and press return. It also collects and orders into mindmaps, any references and research material. Thus, it is also useful for creating a thesis or essay.
What this has allowed me to do, is to split my study time and study focus, between three aspects. These are; gaining a holistic view of a study unit, learning the detail and also doing example questions. I have found that learning the whole before delving into the detail, has meant that by the time I come to do the detail, I have a solid grasp and good memory retention, of the subject matter in hand.
Just as an example to illustrate my method, is my study of M208, Intro block A:
1. Mindmap created of the main subsection topics, including any definitions of theorems or other equations. (Time taken a very rapid, 1hr).
2. Entire unit book scan read and all facts and sentences that need to be remembered or studied, are highlighted. (Again, a rapid 2hrs. The art is not to get bogged down in the text; just keep moving and highlighting!)
3. Review of the mindmap is now done for 10 minutes before each subsequent study period.
4. Mindmap review followed by careful revision of the highlighted text (four periods of 2hrs each).
5. Detailed review of the mindmap (1hr).
6. Getting to grips with the example questions from the Unit. The aim is to repeat all the examples until I can achieve 100% accuracy without referring to the answers.
All in all it should take me 14hrs per week, allowing another 2-4hrs for sticking points or units that are more taxing.
Simples!
Monday, 31 October 2011
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!
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!
Monday, 24 October 2011
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.
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.
Tuesday, 18 October 2011
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
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
Wednesday, 12 October 2011
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.
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.
Tuesday, 11 October 2011
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.
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.
Monday, 10 October 2011
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.
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.
Saturday, 1 October 2011
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
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
Thursday, 29 September 2011
My experiment, so far...
Another week and apart from a miserable cold, I have managed to visit the O.U library, get a steer on what I should be studying, and adjusted my study load appropriately.
I found that I was getting too far into geometry and that I can leave some of my planned study in that subject, until next year and the year after.
I also reflected on some comments that were left on my blog this week, by a rather brilliant, learned and certainly not failed physicist, Chris. He said, tongue in cheek, that I was starting to transmogrify into a pure mathematician. I guess he based this astute comment on my recent fascination with some of the pure mathematics topics that I have recently encountered, and my lack of Open University physics courses that I plan to be examined in, over the next 3 years.
To answer this observation, I would say that my plan hasn't changed; rather, through detailed study and research, my approach has changed. Following my in-depth studies of the life and works of Rodger Penrose, and my experience of university level mathematics and physics; I feel, that to understand mathematical physics as easily as possible and with a wide range of knowledge and tools; whilst tackling the outlier abstract topics that exist; I believe that I will be best placed to succeed, by following the path below:
1. Absorb the building blocks of mathematics. E.g. not just learning the mechanics of an Einstein-Rosen Bridge, or Loop Quantum Gravity, but actually knowing how it all 'ticks' at the purist of mathematical levels.
2. Understand all undergraduate level, mathematics.
3. Learn both pure and applied mathematics at taught MSc level.
4. Then apply all of this knowledge to a theoretical physics PhD.
I believe that any plan to learn a bit of physics and a bit of applied maths at undergraduate level, will be okay; but wouldn't allow for the depth of understanding required, for a PhD in the field. Some may argue with that; but I know that I need to personally start at the basement, before moving up. One could argue that bothering with any pure maths topic is a waste of time when attempting an experiment such as mine. However, I would disagree with that assertion.
Don't get me wrong; I am not going to be neglecting physics over the next few years; rather, I will study it alongside the maths, until I know how to handle those numbers!
You only have to look at some of the most cutting edge theoretical or mathematical physics that is currently being worked on; and you will see that, to even understand it, you need a solid base of group theory, combinatorics, linear algebra, topology and a sprinkling of calculus.
If there is one thing that I have truly learnt from the first year of my experiment in perseverance by trying to convert my GCSE in Maths, into a PhD in Theoretical Physics; is that I love mathematics, I think I'm okay at it, and I can't wait to study all of the maths that I can squeeze in, over the next 10 years.
As my first-year sixth-form physics teacher once said, before his untimely death under the wheels of a lorry, 'mathematics is the language of physics; and without it, you are up S**t creek, without a paddle'.
Good old Mr. Curley.
I found that I was getting too far into geometry and that I can leave some of my planned study in that subject, until next year and the year after.
I also reflected on some comments that were left on my blog this week, by a rather brilliant, learned and certainly not failed physicist, Chris. He said, tongue in cheek, that I was starting to transmogrify into a pure mathematician. I guess he based this astute comment on my recent fascination with some of the pure mathematics topics that I have recently encountered, and my lack of Open University physics courses that I plan to be examined in, over the next 3 years.
To answer this observation, I would say that my plan hasn't changed; rather, through detailed study and research, my approach has changed. Following my in-depth studies of the life and works of Rodger Penrose, and my experience of university level mathematics and physics; I feel, that to understand mathematical physics as easily as possible and with a wide range of knowledge and tools; whilst tackling the outlier abstract topics that exist; I believe that I will be best placed to succeed, by following the path below:
1. Absorb the building blocks of mathematics. E.g. not just learning the mechanics of an Einstein-Rosen Bridge, or Loop Quantum Gravity, but actually knowing how it all 'ticks' at the purist of mathematical levels.
2. Understand all undergraduate level, mathematics.
3. Learn both pure and applied mathematics at taught MSc level.
4. Then apply all of this knowledge to a theoretical physics PhD.
I believe that any plan to learn a bit of physics and a bit of applied maths at undergraduate level, will be okay; but wouldn't allow for the depth of understanding required, for a PhD in the field. Some may argue with that; but I know that I need to personally start at the basement, before moving up. One could argue that bothering with any pure maths topic is a waste of time when attempting an experiment such as mine. However, I would disagree with that assertion.
Don't get me wrong; I am not going to be neglecting physics over the next few years; rather, I will study it alongside the maths, until I know how to handle those numbers!
You only have to look at some of the most cutting edge theoretical or mathematical physics that is currently being worked on; and you will see that, to even understand it, you need a solid base of group theory, combinatorics, linear algebra, topology and a sprinkling of calculus.
If there is one thing that I have truly learnt from the first year of my experiment in perseverance by trying to convert my GCSE in Maths, into a PhD in Theoretical Physics; is that I love mathematics, I think I'm okay at it, and I can't wait to study all of the maths that I can squeeze in, over the next 10 years.
As my first-year sixth-form physics teacher once said, before his untimely death under the wheels of a lorry, 'mathematics is the language of physics; and without it, you are up S**t creek, without a paddle'.
Good old Mr. Curley.
Tuesday, 27 September 2011
M208 and M337
I took another trip to the Open University Library in Milton Keynes, today. I wanted to closely examine the course material of M208 Pure Maths and also M337 Complex analysis. The purpose of doing so, was to try and get an idea of where my difficulties may lay with different units or topics, whether I need to brush up on anything that I'm not planning to study, make sure I am not studying too widely or deeply, and finally to estimate how long it is likely to take me to complete each unit booklet. This will then let me set my self-study plans for next year, without over burdening myself.
It was a morning very well spent. Not only do I love the O.U library as a place to sit and study, with its clean lines and light and airy spaces; but it is also interesting to leaf through other course materials, that are on display.
I was particularly interested in looking at courses such as M338 Topology, and a few other level 3 modules. The reason being, that some of the course materials from these dying level 3 courses, is being used in M303 Further Pure Maths, which I will be taking in October 2013. Again, it was a scoping exercise. I found the level 3 course material to be fairly accessible. And, although I don't have the required knowledge yet, to understand some of the topics and units; I know that they look very well written and certainly doable, if you have completed the pre-requisite courses.
Following my visit, I have decided to continue reading through the Brannan Intro to Analysis, but I am now reducing my study of Spivak's Calculus, to just reference checking topics such as the epsilon-delta, limit definitions etc. I am also going to reduce my Brannan Geometry reading, to now only include Conics, Quadric surfaces, transformations and some of the group theory that is contained within.
I am also going to replace some of that work, with some number theory practice, repeating exercises from Brannan and from online sources, as well as continuing to read the M221 units.
I also plan to re-visit a book that I picked up in August called, 'How to Prove It', by Daniel Velleman. This is because, the only part of the M208 that I feel I need the most practice at, is the minor use of theorem proving that features in the appendices and also in the units of Number theory and some of the analysis units.
I plan to confidently be able to prove some very minor results, by the time I complete M208, so this book will lay the foundations well and will actually go well beyond what M208 requires. Even so, I am always looking 3yrs ahead, to those post-graduate days which is really not that far away.
Whilst I was at the library, I also glanced at all of the MSc Mathematics modules. The only one that has taught materials in a style similar to OU undergraduate modules, is the M820 - Calculus of Variations and Advanced Calculus. It was tomb like and very heavy in places. Interestingly, the alternative 'pure' MSc module, M823 Analytical Number Theory (I), didn't look half as complicated as M820. It uses a set book as the main text and on reading the specimen exam paper and solutions provided, along with the TMA booklet; it looked a much more intriguing course than M820. I guess the shock would come when you study Analytical Number Theory (II), which looks like a beast!
Finally, whilst at the library, I happened to overhear a lively discussion between some faculty staff. They seemed to be discussing the unavoidable financial cuts and the possible effects on O.U content. The gist was, that they were discussing whether a paradigm shift from well written unit booklets and tutor supported learning, to one where students read set books and then presumably just took exams, would cut costs whilst keeping student numbers.
Interesting.
It was a morning very well spent. Not only do I love the O.U library as a place to sit and study, with its clean lines and light and airy spaces; but it is also interesting to leaf through other course materials, that are on display.
I was particularly interested in looking at courses such as M338 Topology, and a few other level 3 modules. The reason being, that some of the course materials from these dying level 3 courses, is being used in M303 Further Pure Maths, which I will be taking in October 2013. Again, it was a scoping exercise. I found the level 3 course material to be fairly accessible. And, although I don't have the required knowledge yet, to understand some of the topics and units; I know that they look very well written and certainly doable, if you have completed the pre-requisite courses.
Following my visit, I have decided to continue reading through the Brannan Intro to Analysis, but I am now reducing my study of Spivak's Calculus, to just reference checking topics such as the epsilon-delta, limit definitions etc. I am also going to reduce my Brannan Geometry reading, to now only include Conics, Quadric surfaces, transformations and some of the group theory that is contained within.
I am also going to replace some of that work, with some number theory practice, repeating exercises from Brannan and from online sources, as well as continuing to read the M221 units.
I also plan to re-visit a book that I picked up in August called, 'How to Prove It', by Daniel Velleman. This is because, the only part of the M208 that I feel I need the most practice at, is the minor use of theorem proving that features in the appendices and also in the units of Number theory and some of the analysis units.
I plan to confidently be able to prove some very minor results, by the time I complete M208, so this book will lay the foundations well and will actually go well beyond what M208 requires. Even so, I am always looking 3yrs ahead, to those post-graduate days which is really not that far away.
Whilst I was at the library, I also glanced at all of the MSc Mathematics modules. The only one that has taught materials in a style similar to OU undergraduate modules, is the M820 - Calculus of Variations and Advanced Calculus. It was tomb like and very heavy in places. Interestingly, the alternative 'pure' MSc module, M823 Analytical Number Theory (I), didn't look half as complicated as M820. It uses a set book as the main text and on reading the specimen exam paper and solutions provided, along with the TMA booklet; it looked a much more intriguing course than M820. I guess the shock would come when you study Analytical Number Theory (II), which looks like a beast!
Finally, whilst at the library, I happened to overhear a lively discussion between some faculty staff. They seemed to be discussing the unavoidable financial cuts and the possible effects on O.U content. The gist was, that they were discussing whether a paradigm shift from well written unit booklets and tutor supported learning, to one where students read set books and then presumably just took exams, would cut costs whilst keeping student numbers.
Interesting.
Monday, 26 September 2011
Calculus can be Simple
At times, I found the OU's treatment of differentiation, a little contrived. It was okay; and is probably the proper approach to learning the elements of calculus, preparing the way for Analysis at level 2 study.
However, I think much more emphasis should have also been given to teaching the mechanical, practical rules of differentiation and integration, in a much more utilitarian way.
I would like to have seen a separate booklet containing hundreds of progressive quick differentiation and integration problems, that could help towards mastery of the mechanical act of 'doing' the calculus. I believe that there are budding scientists or engineers who don't care for the minutia of Analysis, but instead need to use it as a tool in their trade, who may be put off by a lack of 'practical' exercises.
Anyway, I thought I would share an extract of the book 'Calculus Made Easy, by Thompson, Gardner. It just shows how simple calculus can be made. What is striking about this extract, is that the OU tried to explain the exact same element of learning, in about 800 words, whereas Gardner does it in about 100 words. Also, on first reading of the OU text, I didn't grasp the concepts and it took several hours of self study, to practice and attempt mastery of the technique. I read the Gardner extract in approximately 4 minutes and was able to apply it immediately. Here it is:
'Sometimes one is stumped by finding that the expression to be differentiated is too complicated to tackle directly.
Thus, the equation: \(y = {({x^2} + {a^2})^{\frac{3}{2}}}\) is awkward for a beginner.
Now the dodge to turn the difficulty is this:
Write some symbol such as 'u' for the expression \(({x^2} + {a^2})\)
then the equation becomes \(y = {u^{\frac{3}{2}}}\)
Which you can then easily manage; for \(\frac{{dy}}{{du}} = \frac{3}{2}{u^{\frac{1}{2}}}\)
Then tackle the expression \(u = ({x^2} + {a^2})\)
and differentiate it with respect to x, thus, \(\frac{{dy}}{{du}} = 2x\)
Then all that remains is plain sailing, for \(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \times \frac{{du}}{{dx}}\)
\(\frac{{dy}}{{dx}} = \frac{3}{2}{u^{\frac{1}{2}}} \times 2x\)
\( = \frac{3}{2}{({x^2} + {a^2})^{\frac{1}{2}}} \times 2x\)
\( = 3x{({x^2} + {a^2})^{\frac{1}{2}}}\)
And so the trick is done.'
Calculus Made Easy (1998)
Wonderfully simple. Oh, how glad I am that I found this book.
Tuesday, 20 September 2011
'Real-World' Maths text books
The dust has now settled following the completion of MST121 Using Mathematics, and I have now begun to put together my study for the interim period between MST121 and M208 Pure Maths in late January. The main theme that is going to run through this period of revision, reflection and breaking new ground; is the use of 'real world' text books.
The Open University study materials are written like a set of very good lecture notes which are quite different from the type of text's that one encounters in mathematics text books. The text books often gloss over algebraic manipulations, as they plough through the solutions. This can mean that if your algebra manipulation is not very good, then you will struggle to understand the text and will not get very much out of it.
I say that OU notes are easy to use, however, I do sometimes find them rather laboured which can actually be at the expense of clarity. I have encountered several topics that I have not really understood in the OU text, only to then glance at my copy of Brannan or Spivak, and they have explained in one paragraph, what the OU has tried to explain in a whole chapter.
Sometimes, brevity can mean clarity.
The reason that I am practising reading other textbooks, is two fold. Firstly, it makes sense to use two or three approaches to learning a subject, as one of them is bound to make sense. Secondly, postgraduate maths, requires the ability to sift through other people's work, because they are either used as the set book, or when you are required to write your dissertation, you need to be able to scan quickly, large amounts of the stuff and understand it.
I have found the Brannan books (Intro to analysis and the Geometry text), to be short, to the point and relatively easy to follow. I understand that large parts of the Brannan text, are used in courses such as M208 and I am sure I recognise parts of his Geometry text, in the course notes for M336 Groups and Geometry, which I browsed at the O.U library last month.
Even so, it is helpful to study the style and brevity of these texts, for future proofing my skills. The only worry that I have, is the horror stories that I have heard, where massive mathematics texts are riddled with errata. I can just imagine me banging my head off of a large brick wall after trying to understand a problem for hours, only to find out that the solution was written incorrectly in the appendix!
Anyway, here is my study for this week. Total hours 16.
Brannan: A First Course in Mathematical Analysis - Pages 1 - 9 Real numbers.
Brannan: Geometry - pages 1 - 10 Conic sections.
Gardner: Calculus Made Easy - Pages 184 - 209 Partial differentials / integration
Practice maths skills:
Advanced factorisation
Expansion of polynomial products
Review of Linear functions
The Open University study materials are written like a set of very good lecture notes which are quite different from the type of text's that one encounters in mathematics text books. The text books often gloss over algebraic manipulations, as they plough through the solutions. This can mean that if your algebra manipulation is not very good, then you will struggle to understand the text and will not get very much out of it.
I say that OU notes are easy to use, however, I do sometimes find them rather laboured which can actually be at the expense of clarity. I have encountered several topics that I have not really understood in the OU text, only to then glance at my copy of Brannan or Spivak, and they have explained in one paragraph, what the OU has tried to explain in a whole chapter.
Sometimes, brevity can mean clarity.
The reason that I am practising reading other textbooks, is two fold. Firstly, it makes sense to use two or three approaches to learning a subject, as one of them is bound to make sense. Secondly, postgraduate maths, requires the ability to sift through other people's work, because they are either used as the set book, or when you are required to write your dissertation, you need to be able to scan quickly, large amounts of the stuff and understand it.
I have found the Brannan books (Intro to analysis and the Geometry text), to be short, to the point and relatively easy to follow. I understand that large parts of the Brannan text, are used in courses such as M208 and I am sure I recognise parts of his Geometry text, in the course notes for M336 Groups and Geometry, which I browsed at the O.U library last month.
Even so, it is helpful to study the style and brevity of these texts, for future proofing my skills. The only worry that I have, is the horror stories that I have heard, where massive mathematics texts are riddled with errata. I can just imagine me banging my head off of a large brick wall after trying to understand a problem for hours, only to find out that the solution was written incorrectly in the appendix!
Anyway, here is my study for this week. Total hours 16.
Brannan: A First Course in Mathematical Analysis - Pages 1 - 9 Real numbers.
Brannan: Geometry - pages 1 - 10 Conic sections.
Gardner: Calculus Made Easy - Pages 184 - 209 Partial differentials / integration
Practice maths skills:
Advanced factorisation
Expansion of polynomial products
Review of Linear functions
Wednesday, 14 September 2011
Physics and Maths, Studied this Week.
The last 2 weeks have been taken up primarily with completing TMA04 and CMA42, for MST121. Now that I have completed this course, I am very satisfied with how it has all gone and I am looking forward to starting M208 (Pure Maths), in January.
In preparation for that course, and to make good use of the time between now and January (Only 18 study weeks left), I have put together a study plan, which I have already started yesterday.
The plan consists of the following work each day:
3 pages of slow, detailed study and completion of the exercises in 'A First Course in Mathematical Analysis' Brannan.
1hr of repeated maths exercises of all maths that I have done to date, right from handling polynomials, through to calculus.
2 pages of detailed study and completion of exercise in 'Geometry' Brannan.
2 pages of detailed study and completion of exercises in 'Calculus 3rd Ed.' Spivak.
I know that 7 pages per day of textbook study, might not sound a lot, but I am really going for critical analysis and understanding the concepts, rather than a whistle-stop tour. Also, I find that reading maths textbooks, is an art that needs to be learned, before building up to any significant volume. I hope that this approach between module presentations with the OU, will stand me in good stead for future postgraduate work.
Also, each week, I will do some themed topic study on various sections of maths and physics, depending on what takes my fancy. For example, last night, I dipped my toe in the partial differential waters and did a small exercise on that topic.
I am just missing the groups books for MS221, which I had hoped to use to gently introduce my brain to the subject in a scholarly way. I'm not sure whether to buy those books or whether there is an alternative good intro textbook, that I could pick up instead.
I plan to also re-read Gödel's Proof, re-read the Hitch-hikers Guide trilogy of four books (had a weird dream last night about the three breasted whores from Eroticon 6, But I won't go into that here ;-). Finally, I'll be carrying on with the Feynman Lecture series.
This is going to be a wonderfully exciting period of enlightenment.
In preparation for that course, and to make good use of the time between now and January (Only 18 study weeks left), I have put together a study plan, which I have already started yesterday.
The plan consists of the following work each day:
3 pages of slow, detailed study and completion of the exercises in 'A First Course in Mathematical Analysis' Brannan.
1hr of repeated maths exercises of all maths that I have done to date, right from handling polynomials, through to calculus.
2 pages of detailed study and completion of exercise in 'Geometry' Brannan.
2 pages of detailed study and completion of exercises in 'Calculus 3rd Ed.' Spivak.
I know that 7 pages per day of textbook study, might not sound a lot, but I am really going for critical analysis and understanding the concepts, rather than a whistle-stop tour. Also, I find that reading maths textbooks, is an art that needs to be learned, before building up to any significant volume. I hope that this approach between module presentations with the OU, will stand me in good stead for future postgraduate work.
Also, each week, I will do some themed topic study on various sections of maths and physics, depending on what takes my fancy. For example, last night, I dipped my toe in the partial differential waters and did a small exercise on that topic.
I am just missing the groups books for MS221, which I had hoped to use to gently introduce my brain to the subject in a scholarly way. I'm not sure whether to buy those books or whether there is an alternative good intro textbook, that I could pick up instead.
I plan to also re-read Gödel's Proof, re-read the Hitch-hikers Guide trilogy of four books (had a weird dream last night about the three breasted whores from Eroticon 6, But I won't go into that here ;-). Finally, I'll be carrying on with the Feynman Lecture series.
This is going to be a wonderfully exciting period of enlightenment.
Monday, 12 September 2011
MST121, TMA04 Finished
Finally. After 9 months of work, it is done. TMA04 and CMA42 both completed and sent in to the OU. I have now officially completed MST121 and can now have a short break of a few days, before I kick start my prep work for M208 Pure Maths, in January 2012.
In total, I have completed 50 points this year, at level 1. This now gives me a grand total of 250 pts; of which, 30 are at level 3, 60 at level 2 and 160 at level 1.
2012 will see me taking on:
M208 Pure Mathematics (60pts at level 2)
M337 Complex Analysis (30pts at level 3)
Plus one 30pt Level 3 course (TBC)
2013 will consist of:
M303 'Further Pure Maths' (60pts at level 3)
This is just my O.U studies and doesn't include lots of other exciting maths and physics, that I have planned.
Regardless of anything else, my minimum expectation, is to complete my degree and kick start some post graduate maths in 2014.
As part of my prep for my next course, M208, I have obtained a copy of the course MS221 (Exploring Mathematics), which I intend to self-study for the next 16 weeks. I also plan to set up a daily regime of additional mathematics work based around the Analysis and Geometry books by Brannan, alongside some basic proof work from, 'How to Prove It', by Velleman.
I'm also going to try and crack a sub-5min Rubik's Cube!
In total, I have completed 50 points this year, at level 1. This now gives me a grand total of 250 pts; of which, 30 are at level 3, 60 at level 2 and 160 at level 1.
2012 will see me taking on:
M208 Pure Mathematics (60pts at level 2)
M337 Complex Analysis (30pts at level 3)
Plus one 30pt Level 3 course (TBC)
2013 will consist of:
M303 'Further Pure Maths' (60pts at level 3)
This is just my O.U studies and doesn't include lots of other exciting maths and physics, that I have planned.
Regardless of anything else, my minimum expectation, is to complete my degree and kick start some post graduate maths in 2014.
As part of my prep for my next course, M208, I have obtained a copy of the course MS221 (Exploring Mathematics), which I intend to self-study for the next 16 weeks. I also plan to set up a daily regime of additional mathematics work based around the Analysis and Geometry books by Brannan, alongside some basic proof work from, 'How to Prove It', by Velleman.
I'm also going to try and crack a sub-5min Rubik's Cube!
Sunday, 11 September 2011
Am I mad?
I know I talked about economising my study time, in a previous post. But I just can't seem to do it.
I mean, I have spent the last week of over 24hrs total study time, working on just 10 points worth of TMA questions for MST121, that don't gain me a grade that is any higher than the rather vanilla 'pass' that they hand out for all of your efforts on this course.
It's not that I am struggling with the content, although the calculus is a little more complex than I expected. No, it is that I just can't seem to 'let it go'. Writing this TMA, has been like nurturing a small child. I am terrified to let it go, before I have made sure that I have done everything within my power, to make it turn out well.
It just doesn't make any sense whatever.
I have worked out, that I could score just 59% on this assignment, and still gain an overall course score of 85%.
So why, oh why, do I have this drive, no, compulsion, to dissect every last mark and use all of my time, books and brain power, to milk it of its precious marks?
I think I might know the answer, above and beyond a simple neurotic tendency towards perfectionism. It is more to do with me wanting to prove to myself, that I am intellectually capable, of mastering mathematics. Not just some of it, but all of it. Every last polynomial.
I have this strange notion, lurking in my limbic system, that if I can't ace the TMA's on a level 1 mathematics course (give or take the odd arithmetic mistake or lapse of concentration); then I am unlikely to be capable of completing a PhD that is mainly mathematics based.
I have tried to finalise my TMA, be sensible, and spend my time preparing for M208 in January. But I just can't seem to let TMA04 go, without licking my finger to wipe its cheek, before tucking its shirt in and sending it to the OU.
I mean, I have spent the last week of over 24hrs total study time, working on just 10 points worth of TMA questions for MST121, that don't gain me a grade that is any higher than the rather vanilla 'pass' that they hand out for all of your efforts on this course.
It's not that I am struggling with the content, although the calculus is a little more complex than I expected. No, it is that I just can't seem to 'let it go'. Writing this TMA, has been like nurturing a small child. I am terrified to let it go, before I have made sure that I have done everything within my power, to make it turn out well.
It just doesn't make any sense whatever.
I have worked out, that I could score just 59% on this assignment, and still gain an overall course score of 85%.
So why, oh why, do I have this drive, no, compulsion, to dissect every last mark and use all of my time, books and brain power, to milk it of its precious marks?
I think I might know the answer, above and beyond a simple neurotic tendency towards perfectionism. It is more to do with me wanting to prove to myself, that I am intellectually capable, of mastering mathematics. Not just some of it, but all of it. Every last polynomial.
I have this strange notion, lurking in my limbic system, that if I can't ace the TMA's on a level 1 mathematics course (give or take the odd arithmetic mistake or lapse of concentration); then I am unlikely to be capable of completing a PhD that is mainly mathematics based.
I have tried to finalise my TMA, be sensible, and spend my time preparing for M208 in January. But I just can't seem to let TMA04 go, without licking my finger to wipe its cheek, before tucking its shirt in and sending it to the OU.
Saturday, 10 September 2011
S197 How the Universe Works
I received a pass result this morning, for this Open University module. It was a 10 point level 1 course in elementary cosmology and particle physics. It was a very exciting course and went into more depth than I would have imagined for a level 1 course. I didn't receive any specific feedback from the results of S197, other than a pass, so I am not sure how well I did; but it was probably one of the most enjoyable course that I have completed, to date.
I have counted the course towards my certificate of higher education, which is now complete. I will claim it once the University offer it formally to me, which means that I then only have M208 Pure maths to complete, to achieve my diploma in higher education (I have previous humanities courses at levels 1, 2 and 3). I then only need 30 more points at level 3 (probably complex analysis), for a BSc and then another 60pts (M303 Further Pure maths), to complete one of my BSc Hons.
I did a quick calculation last night, on the back of a fag packet; and it looks like I should be into my postgraduate maths courses by October 2014, whilst continuing to study some physics and mathematical physics, at undergraduate level. I still haven't formulated how I will structure my study from 2014 onwards; but one possible option would be to continue to take an MSc in mathematics either with the OU or another college, such as Kings in London; whilst also taking undergraduate courses of study either formally, via self study, or both, along side the formal mathematics training.
Whether I need to complete an MSc in full (I suspect I will + it will give me a chance to cover all the physics I need to, alongside); or whether I can do a postgraduate diploma in maths and then hop onto an MPhil; I will need to think about that and seek some professional advice at the time.
One thing I am not short of is options, and there are so many permutations, that it is going to be lots of fun, which ever way I do it.
I have counted the course towards my certificate of higher education, which is now complete. I will claim it once the University offer it formally to me, which means that I then only have M208 Pure maths to complete, to achieve my diploma in higher education (I have previous humanities courses at levels 1, 2 and 3). I then only need 30 more points at level 3 (probably complex analysis), for a BSc and then another 60pts (M303 Further Pure maths), to complete one of my BSc Hons.
I did a quick calculation last night, on the back of a fag packet; and it looks like I should be into my postgraduate maths courses by October 2014, whilst continuing to study some physics and mathematical physics, at undergraduate level. I still haven't formulated how I will structure my study from 2014 onwards; but one possible option would be to continue to take an MSc in mathematics either with the OU or another college, such as Kings in London; whilst also taking undergraduate courses of study either formally, via self study, or both, along side the formal mathematics training.
Whether I need to complete an MSc in full (I suspect I will + it will give me a chance to cover all the physics I need to, alongside); or whether I can do a postgraduate diploma in maths and then hop onto an MPhil; I will need to think about that and seek some professional advice at the time.
One thing I am not short of is options, and there are so many permutations, that it is going to be lots of fun, which ever way I do it.
Friday, 9 September 2011
TMA04
TMA04, is a pain in the backside. 50% of it needs to be carried out using OU Stats and Mathcad. And, because of a software issue, my Mathcad just wouldn't behave itself. I therefore spent 3 or 4 days, trying to battle my way through it and sent a pleading email to my Tutor, for some advice. She couldn't give it, but did suggest a re-install. I un-installed, re-installed and... it works!!
All that wasted time faffing about. Anyway, I think I am now 75% of the way through the paper, with the Mathcad bit now complete.
The other problem with TMA04, is that they have put in, what I think, is an unfair question. Not quite a trick question, but not far off it.
I can't go into the detail, for fear of breaching University regs, but what I will say, is that the some of the answers require more algebra, trigonometry and problem solving skills, than I would have expected in a level 1 course of this nature. I do wonder whether it has been done purposely, to identify those that really have properly mastered ALL of the course concepts.
A horrible TMA and I'll be glad to see the back of it.
All that wasted time faffing about. Anyway, I think I am now 75% of the way through the paper, with the Mathcad bit now complete.
The other problem with TMA04, is that they have put in, what I think, is an unfair question. Not quite a trick question, but not far off it.
I can't go into the detail, for fear of breaching University regs, but what I will say, is that the some of the answers require more algebra, trigonometry and problem solving skills, than I would have expected in a level 1 course of this nature. I do wonder whether it has been done purposely, to identify those that really have properly mastered ALL of the course concepts.
A horrible TMA and I'll be glad to see the back of it.
Sunday, 4 September 2011
The Pareto Principle
After my CMA result this week; I had a chat with my wife. We discussed the result and I explained that it was low, compared to my other results, because I didn't like the statistics units (block D), within the course MST121.
As we discussed this issue, she asked me how long had I spent on studying Block D and completing the CMA. I told her that it had taken me 10hrs to read through all 4 books of block D and then a further 3hrs, to complete the TMA.
This was a total of 13hrs for studying rather than the 60hrs that the Open University timetable recommended (The block D and CMA, were given 7.5 weeks recommended study time, which is equivalent to 60hrs of study).
My wife then went on to ask me about the grading system that the OU runs. I explained that to score an equivelent to a 1st class degree, you must score 85% or above, average on your assignments and any exam taken.
My wife then asked me, what level of degree I needed, to be accepted onto a masters level degree programme. I explained that most of them for maths or physics, require a 2.1 honours degree but some, including the Open University, will accept a 2.2, but most employers or academic circles, would normally expect a 2.1.
My wife then asked me what this 'dropped' and rushed mark for the stats units, would ultimately get me, if it was translated into an end of course result. I explained that 88% would be equivalent to a solid grade 1 pass.
She then asked me how much study time I had spent on a set of units, that had achieved my TMA average of 94%, for the rest of the course. I reckoned that the study time required to read and pass, for example, the TMA02 with this score, had been in the region of 112hrs (not including my other non MST121 studies).
My wife then told me that she thought I was a little daft and how could I hope to pass a maths degree, if I couldn't see the blindingly obvious.
What she was suggesting, of course, is that I should evaluate what benefit I have derived from working for an additional 14hrs per week, to score an additional 6% in a TMA, when it would make no difference to the final outcome of the module or degree classification.
Hmm, she had a point. It then got me thinking about why I had spent so many hours slaving over this course. Well, I think it is for a few reasons. Firstly, I love studying. But as my father used to say, 'busyness is a symptom of laziness', meaning that sometimes we fill our hours with interesting yet unproductive tasks or events, to avoid the difficult or important. I wouldn't say that any of my studying has been wasteful or any form of procrastination, but I do accept that I have skewed on occasion into a few indulgences, that have perhaps distracted me from the task in hand.
This all got me thinking about the 'Pareto principle', or the '80% of the benefit, from 20% of the effort', theories.
Now, I love studying, it is what I do and will continue to do, so any 'fat trimming', will not be about reducing the time that I spend studying. Indeed I hope to one day, do it full time. No, this is more about maximising the next few years' output of work, whilst still achieving the same result (a good degree and acceptance onto a masters / PhD programme).
I have trawled the internet and looked at many University websites. Nearly (in fact, all bar one or two) ask for, what they describe as a good degree. They qualify the use of the word 'good', to mean a 2.1 Honours award.
In Open University world, that equates to a minimum score of 70% in the TMA's and the exams. Well, I know that by spending only 11% of the suggested time, I have managed to pass a CMA, with 88% of the marks. So, what is the benefit of me slogging unneccessarily over TMA's, to score in the 90's? I know that one of the other main reasons that I tend to do this, is that I am a perfectionist and that any dropped marks, feel like a failure (I'm sure a psychiatrist would have a field day with that little gem!)
But, I need to decide, what benefits I could derive, from reducing my study time per module, to even 50% of its current level:
1. I could take more modules per year and get it all done a lot quicker. If I took an extra 30 pt module this October, overlapping it with M208 Pure maths in January 2012. I would have an overlap of 6 months where I studied 90pts, and then an exam in June 2012, followed by an exam in October 2012. Doable!
2. I could shave off time taken to complete the degree and also tot up additional points, to take a Diploma, BA and a BSc Hons, by the end of 2013.
My only worry, is that potential PhD supervisors, may pick students with top drawer marks in the 90's, rather than in the 70's or 80's. Would they question a sudden dip in average TMA scores?
This is all certainly food for thought. And, I know that with my methods of using 100's of practise exam style questions and flash cards, that I can shut myself away and study well for an exam, without too much drama.
So the only question left really is, do I want to get a 'good' degree quickly and move onto more interesting post-grad stuff quicker? Or, do I want to score in the 90's for my own well-being and satisfaction, and take the slower route to success?
As we discussed this issue, she asked me how long had I spent on studying Block D and completing the CMA. I told her that it had taken me 10hrs to read through all 4 books of block D and then a further 3hrs, to complete the TMA.
This was a total of 13hrs for studying rather than the 60hrs that the Open University timetable recommended (The block D and CMA, were given 7.5 weeks recommended study time, which is equivalent to 60hrs of study).
My wife then went on to ask me about the grading system that the OU runs. I explained that to score an equivelent to a 1st class degree, you must score 85% or above, average on your assignments and any exam taken.
My wife then asked me, what level of degree I needed, to be accepted onto a masters level degree programme. I explained that most of them for maths or physics, require a 2.1 honours degree but some, including the Open University, will accept a 2.2, but most employers or academic circles, would normally expect a 2.1.
My wife then asked me what this 'dropped' and rushed mark for the stats units, would ultimately get me, if it was translated into an end of course result. I explained that 88% would be equivalent to a solid grade 1 pass.
She then asked me how much study time I had spent on a set of units, that had achieved my TMA average of 94%, for the rest of the course. I reckoned that the study time required to read and pass, for example, the TMA02 with this score, had been in the region of 112hrs (not including my other non MST121 studies).
My wife then told me that she thought I was a little daft and how could I hope to pass a maths degree, if I couldn't see the blindingly obvious.
What she was suggesting, of course, is that I should evaluate what benefit I have derived from working for an additional 14hrs per week, to score an additional 6% in a TMA, when it would make no difference to the final outcome of the module or degree classification.
Hmm, she had a point. It then got me thinking about why I had spent so many hours slaving over this course. Well, I think it is for a few reasons. Firstly, I love studying. But as my father used to say, 'busyness is a symptom of laziness', meaning that sometimes we fill our hours with interesting yet unproductive tasks or events, to avoid the difficult or important. I wouldn't say that any of my studying has been wasteful or any form of procrastination, but I do accept that I have skewed on occasion into a few indulgences, that have perhaps distracted me from the task in hand.
This all got me thinking about the 'Pareto principle', or the '80% of the benefit, from 20% of the effort', theories.
Now, I love studying, it is what I do and will continue to do, so any 'fat trimming', will not be about reducing the time that I spend studying. Indeed I hope to one day, do it full time. No, this is more about maximising the next few years' output of work, whilst still achieving the same result (a good degree and acceptance onto a masters / PhD programme).
I have trawled the internet and looked at many University websites. Nearly (in fact, all bar one or two) ask for, what they describe as a good degree. They qualify the use of the word 'good', to mean a 2.1 Honours award.
In Open University world, that equates to a minimum score of 70% in the TMA's and the exams. Well, I know that by spending only 11% of the suggested time, I have managed to pass a CMA, with 88% of the marks. So, what is the benefit of me slogging unneccessarily over TMA's, to score in the 90's? I know that one of the other main reasons that I tend to do this, is that I am a perfectionist and that any dropped marks, feel like a failure (I'm sure a psychiatrist would have a field day with that little gem!)
But, I need to decide, what benefits I could derive, from reducing my study time per module, to even 50% of its current level:
1. I could take more modules per year and get it all done a lot quicker. If I took an extra 30 pt module this October, overlapping it with M208 Pure maths in January 2012. I would have an overlap of 6 months where I studied 90pts, and then an exam in June 2012, followed by an exam in October 2012. Doable!
2. I could shave off time taken to complete the degree and also tot up additional points, to take a Diploma, BA and a BSc Hons, by the end of 2013.
My only worry, is that potential PhD supervisors, may pick students with top drawer marks in the 90's, rather than in the 70's or 80's. Would they question a sudden dip in average TMA scores?
This is all certainly food for thought. And, I know that with my methods of using 100's of practise exam style questions and flash cards, that I can shut myself away and study well for an exam, without too much drama.
So the only question left really is, do I want to get a 'good' degree quickly and move onto more interesting post-grad stuff quicker? Or, do I want to score in the 90's for my own well-being and satisfaction, and take the slower route to success?
Friday, 2 September 2011
CMA41 Result
I have received my MST121 Open University CMA41 result back, today. I scored 88%, which considering that I rushed through it because it was all about statistics (I hate statistics); then I am more than happy with that result. It is still in the pass 1 realms (85%+), which is where I need it to be.
When you actually add up all the stats work of MST121 including CMA41, part of TMA04 and part of CMA42; then it actually accounts for a large proportion of this course. A lot more than I imagined.
It still doesn't detract from my enjoyment of MST121, so far. Once I have completed TMA04 and CMA51 (very soon), then I will provide a critical evaluation of the entire course and some thoughts on how it sets one up, for further mathematics at University.
When you actually add up all the stats work of MST121 including CMA41, part of TMA04 and part of CMA42; then it actually accounts for a large proportion of this course. A lot more than I imagined.
It still doesn't detract from my enjoyment of MST121, so far. Once I have completed TMA04 and CMA51 (very soon), then I will provide a critical evaluation of the entire course and some thoughts on how it sets one up, for further mathematics at University.
Sunday, 28 August 2011
Physics and Maths, Studied this Week.
Well, a relatively quiet week, as I have only just got back from holiday; but I have started back in earnest, to finalise MST121 and await the results in December.
I also understand that my S197 How the Universe Works, course results are due around mid September. If I pass, I will have then picked up a certificate of Higher Education, as we enter the Autumn.
As I have started to tuck into TMA04, the last piece of coursework for MST121 (Using Mathematics), with the Open University. In doing so, I have realised that even though TMA04 repeats all of the sections covered on earlier TMA's, this repetition has greatly enhanced and cemented, my understanding of the course material. This surprised me somewhat, as I just thought that the course writers were being a little pedantic in going over it all again, in an attempt to fill out the module assessment strategy, in light of there being no final exam.
But I'm converted. It has been a pain to go back to all the earlier topics from previous TMA's, but it has made a real difference in my confidence and facility in using the concepts and techniques - even the f*****g Geese counting.
Anyway, here is this week's completed study:
MST121
TMA04 - completed final draft of Question 1
Other Maths skills
Square roots and arithmetic with them.
Quadratics and factorisation
Other Maths reading
How to Prove It. Velleman, Cambridge (A brilliant book to self-study maths logic and how to understand, copy and write your own proofs. Lots of practise, broken down into steps. I would 100% recommend this book to anyone who is likely to meet logic and proofs, as part of their studies. It is without doubt, the best book that I have read this year).
Total study time: 16hrs
I also understand that my S197 How the Universe Works, course results are due around mid September. If I pass, I will have then picked up a certificate of Higher Education, as we enter the Autumn.
As I have started to tuck into TMA04, the last piece of coursework for MST121 (Using Mathematics), with the Open University. In doing so, I have realised that even though TMA04 repeats all of the sections covered on earlier TMA's, this repetition has greatly enhanced and cemented, my understanding of the course material. This surprised me somewhat, as I just thought that the course writers were being a little pedantic in going over it all again, in an attempt to fill out the module assessment strategy, in light of there being no final exam.
But I'm converted. It has been a pain to go back to all the earlier topics from previous TMA's, but it has made a real difference in my confidence and facility in using the concepts and techniques - even the f*****g Geese counting.
Anyway, here is this week's completed study:
MST121
TMA04 - completed final draft of Question 1
Other Maths skills
Square roots and arithmetic with them.
Quadratics and factorisation
Other Maths reading
How to Prove It. Velleman, Cambridge (A brilliant book to self-study maths logic and how to understand, copy and write your own proofs. Lots of practise, broken down into steps. I would 100% recommend this book to anyone who is likely to meet logic and proofs, as part of their studies. It is without doubt, the best book that I have read this year).
Total study time: 16hrs
Saturday, 27 August 2011
Relative, Intellectual difficulty of University Mathematics
I have often pondered the question of whether mathematics, as a subject, increases in relative intellectual difficulty, as one moves from GCSE right up to research level mathematics / mathematical physics.
At first glance, I think most people would probably suggest that this is really a none question and ask what nonsense I am spouting "of course, maths increases in difficulty, as you move through the subject and it becomes more complex and, oh, so abstract."
However, if one were to imagine, say, an index of relative perceived difficulty from 1 to 10; and then went on to examine how a student would rate themselves, at each stage of their maths journey; I wonder how they would rate themselves over time?
Would their perception of the intellectual difficulty of their studies, relative to their intellectual ability at that moment, stay fairly stable?
Let me give an example of my own perception of the intellectual difficulty of mathematics, as I have moved from GCSE, to my current level of understanding and studies. I will then go on to discuss possible reasons for this and pose the question as to whether or not you would find a similar pattern in the general population.
My Perceived Intellectual difficulty index (1 easy - 10 difficult)
1988 - 1991
Pre- GCSE Maths (felt like I couldn't understand any of it, so switched off) --- Perceived difficulty: 10
1992
GCSE Maths (Grade C, but felt I could have got an A, but wasn't in the right stream to take the higher paper) --- Perceived difficulty: 5
2005
[Life event: Finally diagnosed with Irlen syndrome / Dyslexia, and treatment for Irlen's, improved study ability and motivation] Read this blog post I wrote, for full details:
2007
Pre-University self study (Equivalent to Further Maths A Level) --- Perceived difficulty: 8
2010
MST121 Using Mathematics (feels like I am coasting along) --- Perceived difficulty: 4
2011
Self Study via Standard text books (Number Theory / Group Theory) --- Perceived difficulty: 8
2011
Current Status (revisiting A Level skills and practising MST121, prep for M208 and M303) --- Perceived difficulty: 4
What does this all mean? Well, I can feel a definite pattern in my own path. Having done this exercise it seems very obvious to me now, that although higher maths is more complex and arguably more intellectually demanding; I find it no more taxing to meet for the first time, than some of the Pre-GCSE maths, that I met as a young teenager.
The only question that I can't personally answer at the moment, is whether this will continue, as I tackle advanced Undergraduate and postgraduate topics and texts, in the next 4yrs? Will those 8's turn into 10's, as I meet post grad maths for the first time? Will I be over-clocking my brain with 11's or 12's?
I think that as long as I can keep learning a chunk at a time, then I feel that I will cope intellectually with the later modules. But does this type of pattern occur in the general population? Without some research it would be impossible to say for sure; but I do wonder if it is the case that as long as you have average or above intelligence, that any person could apply themselves and achieve a certain level of university mathematics proficiency.
Unfortunately, I suspect that there is a ceiling to how high someone could reach, unless they were particularly gifted or very mathematically minded.
Do I have that ability? Not sure; not enough data yet. But I'll give it a bloody good go.
At first glance, I think most people would probably suggest that this is really a none question and ask what nonsense I am spouting "of course, maths increases in difficulty, as you move through the subject and it becomes more complex and, oh, so abstract."
However, if one were to imagine, say, an index of relative perceived difficulty from 1 to 10; and then went on to examine how a student would rate themselves, at each stage of their maths journey; I wonder how they would rate themselves over time?
Would their perception of the intellectual difficulty of their studies, relative to their intellectual ability at that moment, stay fairly stable?
Let me give an example of my own perception of the intellectual difficulty of mathematics, as I have moved from GCSE, to my current level of understanding and studies. I will then go on to discuss possible reasons for this and pose the question as to whether or not you would find a similar pattern in the general population.
My Perceived Intellectual difficulty index (1 easy - 10 difficult)
1988 - 1991
Pre- GCSE Maths (felt like I couldn't understand any of it, so switched off) --- Perceived difficulty: 10
1992
GCSE Maths (Grade C, but felt I could have got an A, but wasn't in the right stream to take the higher paper) --- Perceived difficulty: 5
2005
[Life event: Finally diagnosed with Irlen syndrome / Dyslexia, and treatment for Irlen's, improved study ability and motivation] Read this blog post I wrote, for full details:
Coping with Dyslexia
2007
Pre-University self study (Equivalent to Further Maths A Level) --- Perceived difficulty: 8
2010
MST121 Using Mathematics (feels like I am coasting along) --- Perceived difficulty: 4
2011
Self Study via Standard text books (Number Theory / Group Theory) --- Perceived difficulty: 8
2011
Current Status (revisiting A Level skills and practising MST121, prep for M208 and M303) --- Perceived difficulty: 4
What does this all mean? Well, I can feel a definite pattern in my own path. Having done this exercise it seems very obvious to me now, that although higher maths is more complex and arguably more intellectually demanding; I find it no more taxing to meet for the first time, than some of the Pre-GCSE maths, that I met as a young teenager.
The only question that I can't personally answer at the moment, is whether this will continue, as I tackle advanced Undergraduate and postgraduate topics and texts, in the next 4yrs? Will those 8's turn into 10's, as I meet post grad maths for the first time? Will I be over-clocking my brain with 11's or 12's?
I think that as long as I can keep learning a chunk at a time, then I feel that I will cope intellectually with the later modules. But does this type of pattern occur in the general population? Without some research it would be impossible to say for sure; but I do wonder if it is the case that as long as you have average or above intelligence, that any person could apply themselves and achieve a certain level of university mathematics proficiency.
Unfortunately, I suspect that there is a ceiling to how high someone could reach, unless they were particularly gifted or very mathematically minded.
Do I have that ability? Not sure; not enough data yet. But I'll give it a bloody good go.
Thursday, 18 August 2011
Physics and Maths, Studied this Week.
I have really concentrated this week on practise, practise, practise. Hundreds of exam type problems, that I have collected from text books, the internet and other sources. They have all been problems involving basic maths skills. The plan? To make them second nature. I feel this will make next years course of M208, Pure maths, much easier to navigate. If I can commit to memory and add fluency to handling of inequalities, polynomials, factoring, handling trigonometric identities; then I will be able to concentrate on the more challenging aspects of next years analysis, geometry and other subjects. I also want to aim for a grade 1 exam pass, so fluency will help with timing of exam questions and completion times.
I have now also mapped out my next 18 months of Open University courses. They are as follows:
January 2012 - M208 Pure Mathematics
October 2012 - M337 Complex Analysis (essential for further maths studies at Masters level)
October 2013 - M303 Further Pure Mathematics
Now, there are several comments that I need to make with regards to these course choices. Firstly, I am studying only 30 points level 3, from October 2012 until June 2013. The reason is, that I only need the 90 points from Complex analysis and further pure maths, to complete my Open Hons Degree. It is then straight onto M823 Analytical Number Theory (or its replacement course). Secondly, where is the physics? You might ask. Well, I have decided to study my physics subjects, away from the OU, as self guided study.
I need time to do this, and the above profile allows for this indulgence, whilst gaining my degree as quickly as possible. This will include studying MST209 Mathematical models and also M338 Topology, by purchasing the course texts, without registering for the modules. I will study these between October 2012 and October 2013.
This should leave me with several hundred hours of free time, to dedicate to my own course of scholarly pursuits, which include beginning to develop some embryonic postgraduate research skills and also mapping out a plan to get myself up to the level of passing the Cambridge University Mathematics Tripos Exam paper. I feel that if I can't pass that, then I have no business studying mathematical physics, at post graduate level.
I am very excited about next year, it feels like the real start to my journey. This year has almost been a dress rehearsal for 2012.
Study this week: 25hrs total.
MST121
Week off, no additional work (I still have TMA04 to start and the last 3 questions of the last CMA to finalise, but I needed a fortnight off, before embarking on this work, as it's yet more F***ing Geese populations, recurrence relations and dice rolling. Yawn)
Other Maths Practise
Inequalities
Quadratic Equations
Functions and Graphs
Simplifying monomials and polynomials
Multiplication of polynomials
Multiplication of polynomials, using formulas
Factorisation
Square roots
Four operations on fractions.
Other reading
Fermat's last Theorem (finished it off. It was very entertaining and such a happy ending)
How to Solve it - G. Polya
Next week I am going camping for a few days on the coast, so lets hope the weather is good. What I like about camping, is that once the kids are asleep, I can relax in a camp chair, in the cold and pouring rain, with a hot cup of coffee and a good maths text. It clears the mind and is truly peaceful.
I have now also mapped out my next 18 months of Open University courses. They are as follows:
January 2012 - M208 Pure Mathematics
October 2012 - M337 Complex Analysis (essential for further maths studies at Masters level)
October 2013 - M303 Further Pure Mathematics
Now, there are several comments that I need to make with regards to these course choices. Firstly, I am studying only 30 points level 3, from October 2012 until June 2013. The reason is, that I only need the 90 points from Complex analysis and further pure maths, to complete my Open Hons Degree. It is then straight onto M823 Analytical Number Theory (or its replacement course). Secondly, where is the physics? You might ask. Well, I have decided to study my physics subjects, away from the OU, as self guided study.
I need time to do this, and the above profile allows for this indulgence, whilst gaining my degree as quickly as possible. This will include studying MST209 Mathematical models and also M338 Topology, by purchasing the course texts, without registering for the modules. I will study these between October 2012 and October 2013.
This should leave me with several hundred hours of free time, to dedicate to my own course of scholarly pursuits, which include beginning to develop some embryonic postgraduate research skills and also mapping out a plan to get myself up to the level of passing the Cambridge University Mathematics Tripos Exam paper. I feel that if I can't pass that, then I have no business studying mathematical physics, at post graduate level.
I am very excited about next year, it feels like the real start to my journey. This year has almost been a dress rehearsal for 2012.
Study this week: 25hrs total.
MST121
Week off, no additional work (I still have TMA04 to start and the last 3 questions of the last CMA to finalise, but I needed a fortnight off, before embarking on this work, as it's yet more F***ing Geese populations, recurrence relations and dice rolling. Yawn)
Other Maths Practise
Inequalities
Quadratic Equations
Functions and Graphs
Simplifying monomials and polynomials
Multiplication of polynomials
Multiplication of polynomials, using formulas
Factorisation
Square roots
Four operations on fractions.
Other reading
Fermat's last Theorem (finished it off. It was very entertaining and such a happy ending)
How to Solve it - G. Polya
Next week I am going camping for a few days on the coast, so lets hope the weather is good. What I like about camping, is that once the kids are asleep, I can relax in a camp chair, in the cold and pouring rain, with a hot cup of coffee and a good maths text. It clears the mind and is truly peaceful.
Friday, 12 August 2011
Open University Maths, Compared with Other Universities
I am going to put a question out there, that I don't know the answer to at the moment.
Q. Are the mathematics courses offered by the Open University, both under and post graduate, of sufficient depth and breadth compared to other, traditional, University maths courses, in the U.K?
I ask this question, as I would be interested in debating this aspect about what is offered by the O.U. and whether it provides sufficient grounding for any budding mathematician, pure or applied.
I have found that their teaching material is very easy to follow and is challenging enough, to keep me interested and intellectually satisfied. It is both incremental in its content and yet I find that I am learning new concepts, each week.
However, I have noticed, that there are a lot of level 2 and 3 courses, that seem to start from the very basics, before moving into new material. For example, the Pure maths course M208, seems to go over abbreviated sections of the MS221 / MST121 material on graphs and numbers, for example.
Is this continual recap of basics at the beginning of each module, wasting time that could be spent fitting in more complex and deeper mathematics? Or, is it a necessary evil, for an institution that tries to cater for people who may dip in and out of their courses, and hence, need a quick recap of basic material, before they find their stride?
I wonder how much extra, and arguably important material, could be fitted into courses, without the slow start? One possible reason for this recap, could be that courses come and go, and often a pre-requisite course, may become unavailable after a few years. Therefore, the only way to fully prepare students for taking courses without these pre-requisites being available, is to provide the recap at the start.
As I read more mathematics for fun and in a scholarly way, I am discovering more and more amazing theorems such as Godel, Fermat and others. I know that the O.U are ditching their module that covered Godel's most famous theorem and I wonder how many other important parts of maths, are being left out?
I would be interested in hearing other people's views. I personally don't have an answer to this just yet.
Q. Are the mathematics courses offered by the Open University, both under and post graduate, of sufficient depth and breadth compared to other, traditional, University maths courses, in the U.K?
I ask this question, as I would be interested in debating this aspect about what is offered by the O.U. and whether it provides sufficient grounding for any budding mathematician, pure or applied.
I have found that their teaching material is very easy to follow and is challenging enough, to keep me interested and intellectually satisfied. It is both incremental in its content and yet I find that I am learning new concepts, each week.
However, I have noticed, that there are a lot of level 2 and 3 courses, that seem to start from the very basics, before moving into new material. For example, the Pure maths course M208, seems to go over abbreviated sections of the MS221 / MST121 material on graphs and numbers, for example.
Is this continual recap of basics at the beginning of each module, wasting time that could be spent fitting in more complex and deeper mathematics? Or, is it a necessary evil, for an institution that tries to cater for people who may dip in and out of their courses, and hence, need a quick recap of basic material, before they find their stride?
I wonder how much extra, and arguably important material, could be fitted into courses, without the slow start? One possible reason for this recap, could be that courses come and go, and often a pre-requisite course, may become unavailable after a few years. Therefore, the only way to fully prepare students for taking courses without these pre-requisites being available, is to provide the recap at the start.
As I read more mathematics for fun and in a scholarly way, I am discovering more and more amazing theorems such as Godel, Fermat and others. I know that the O.U are ditching their module that covered Godel's most famous theorem and I wonder how many other important parts of maths, are being left out?
I would be interested in hearing other people's views. I personally don't have an answer to this just yet.
Tuesday, 9 August 2011
Physics and Maths, Studied this Week.
Another week and I have spent the last period, solely concentrating on honing some basic maths skills, through lots of practise questions. I have had a complete week off from MST121, before I spend another few days completing TMA04 and effectively finishing the course.
A few days ago, I had the good fortune to visit the Open University campus in Milton Keynes. I spent the morning in the library which is an impressive building. They have the course material for every course that they run, on display. I therefore spent the morning leafing through the course books of every single maths course that they currently run.
It left me with the feeling that all of these maths courses look well presented and very do-able. Even though I have only just completed MST121, I was able to see how all of the level 3 courses, followed on nicely from either M208 or MST209 and all of the content was detailed, yet I definitely feel confident, that I could cope with most of it.
Whilst there, I also managed to get in the background whilst they were filming a course DVD on some sociology module. Not sure which one it was for, but I'm glad I had on my Sunday best, that day!
I then went to Oxford and spent the day going around the Colleges, Museums and drinking coffee. I left there with an amazing boost of my keenness to study. It is such an inspiring place and you can't help but feel affected by the backdrop.
Total Study time this week: 16hrs.
Maths Skills practise
Drawing graphs
Functions and graphs
Simultaneous equations in two or three variables
Simplifying polynomials
Inequalities
Other Maths reading
Geometry - Brannan
Fermat's Last Theorem - Simon Singh
ps: The Singh book is thoroughly entertaining and well worth a read. It follows the story of the disastrous solution provided by Wiles and then his subsequent 'year of hell', trying to put things right. It's a maths book that you can read with a bottle of wine, without worrying that you'll lose the ability to follow it, after the first glass; as it concentrates on the human story behind the incident.
A few days ago, I had the good fortune to visit the Open University campus in Milton Keynes. I spent the morning in the library which is an impressive building. They have the course material for every course that they run, on display. I therefore spent the morning leafing through the course books of every single maths course that they currently run.
It left me with the feeling that all of these maths courses look well presented and very do-able. Even though I have only just completed MST121, I was able to see how all of the level 3 courses, followed on nicely from either M208 or MST209 and all of the content was detailed, yet I definitely feel confident, that I could cope with most of it.
Whilst there, I also managed to get in the background whilst they were filming a course DVD on some sociology module. Not sure which one it was for, but I'm glad I had on my Sunday best, that day!
I then went to Oxford and spent the day going around the Colleges, Museums and drinking coffee. I left there with an amazing boost of my keenness to study. It is such an inspiring place and you can't help but feel affected by the backdrop.
Total Study time this week: 16hrs.
Maths Skills practise
Drawing graphs
Functions and graphs
Simultaneous equations in two or three variables
Simplifying polynomials
Inequalities
Other Maths reading
Geometry - Brannan
Fermat's Last Theorem - Simon Singh
ps: The Singh book is thoroughly entertaining and well worth a read. It follows the story of the disastrous solution provided by Wiles and then his subsequent 'year of hell', trying to put things right. It's a maths book that you can read with a bottle of wine, without worrying that you'll lose the ability to follow it, after the first glass; as it concentrates on the human story behind the incident.
Monday, 1 August 2011
Physics and Maths, Studied this Week.
We are now approaching the end of MST121 2011. I have completed the final coursework CMA41 and I have 3/4 completed the CMA51, the first part of the EMA (end of course assignment). Following that, there is just a few hours to spend on the TMA04, and then it is done.
Total time spent studying this week: 20hrs.
MST121
Completion of CMA41
Questions 1 - 20 of CMA51
Other maths problem practise
Simplifying polynomials
Simultaneous equations in three and four variables
Audio Books
Surely you're joking Mr Feynman
Maths Books
A first Course in Analysis - Brannan Chapter 1 review
Calculus 3rd Edition - Spivak Chapter 1
Godel's Proof - Nagel.
The teaching Company Lectures: Mastering Differential Equations
How computers solve differential Equations
Rubik's Cube
Best completion time: 20 mins
I have also bought a couple of Moleskine note books, that I intend to carry with me and start 'doing' mathematics, at odd moments during the day, as the mood takes me. I intend to record some key themes, in the front and then conduct some problem solving in the rear.
I am off to Oxford next week, for a day on the river and for a walk around the Colleges. I may even take my notebook with me and sit in a cafe pondering some maths. Nothing like getting in the mood!
Total time spent studying this week: 20hrs.
MST121
Completion of CMA41
Questions 1 - 20 of CMA51
Other maths problem practise
Simplifying polynomials
Simultaneous equations in three and four variables
Audio Books
Surely you're joking Mr Feynman
Maths Books
A first Course in Analysis - Brannan Chapter 1 review
Calculus 3rd Edition - Spivak Chapter 1
Godel's Proof - Nagel.
The teaching Company Lectures: Mastering Differential Equations
How computers solve differential Equations
Rubik's Cube
Best completion time: 20 mins
I have also bought a couple of Moleskine note books, that I intend to carry with me and start 'doing' mathematics, at odd moments during the day, as the mood takes me. I intend to record some key themes, in the front and then conduct some problem solving in the rear.
I am off to Oxford next week, for a day on the river and for a walk around the Colleges. I may even take my notebook with me and sit in a cafe pondering some maths. Nothing like getting in the mood!
Sunday, 31 July 2011
Infinity and Logic
This month, I have been taking a real interest in infinity and its effects on maths. It started out with me pondering the idea of the area approaching a limit, and the fact that you can have a finite area, split into infinite parts, represented by an infinite amount of real numbers.
From this, I picked up a copy of Godel's Proof, a book that tries to describe Godel's incompleteness theorem, in about 200 pages. And, I then discovered that Godel, among others, proposed the fact that if you identified sets of numbers, that were mind bogglingly large, you could just build new levels of infinity, on top of underlying numbers and then you could use these sets, to prove some of the maths problems that exist underneath.
It was then, that I stumbled on an article in the New Scientist periodical, this week. The piece was boldly entitled 'Ultimate Logic - so powerful it could wipe out mathematics as we know it.'
In a nutshell, the piece describes how a mathematician (Hugh Woodin), believes he has solved the continuum hypothesis (Is there an infinite set that sits between the countable infinity, such as counting the integers from 1 towards infinity and the 'continuous' infinity, such as when you split a sphere into infinite sections, when the whole is finite.)
The difference with his claim, is that he states that he has solved this maths problem, by using a new type of logic language and structure called 'ultimate L'. This method would allow extra steps of sets to be bolted on to the top of infinite sets, filling in gaps below sufficiently, to allow any lower mathematical problem to be solved. He makes the bold statement, that this new theory, allows him to provide 'a definitive account of the spectrum of subsets of real numbers and thus, proves Cantor's continuum hypothesis, as true; ruling out anything between the countable infinity and the continuum'!
Woodin, even claims, that this may overturn major parts of Godels incompleteness theorem and be a tool that actually, artificially, allows us to root out unsolvability in number theory. He doesn't go as far as saying that Godel's theories would be dead - but that you could 'chase it as far as you pleased up the staircase into the infinite attic of mathematics'.
This idea seems similar in theme, to the idea of calculus and limits. In that, yes, you might never be able to say what is actually happening at the limit its self, but you can get so close to it, that your results have the same effect as if you were actually at the limit.
This all reminded me of the amusing anecdote relating to Zeno's paradox. It is said, that when it was once explained to a student, that if you were trying to reach a girl on the other side of a room, that you would never actually get there, if you travel half of the previous distance travelled in discrete steps. The student's retort was thus;
'Well, I might not ever reach the girl, but I could get plenty close enough for all 'practical' purposes! It seems, that perhaps Woodin is saying; 'why try and reach infinity? when you can just get so close to it, that your results are almost the same, as if you were actually there?' He seems to believe, that he has created a new language for mathematicians, to solve these issue.
Could it be, that if we do end up with a radical departure from existing mathematical logic and ideas, to prove almost everything; that we will end up with two tiers of mathematics, such as happened in physics with 'classical' and 'quantum' theories?
From this, I picked up a copy of Godel's Proof, a book that tries to describe Godel's incompleteness theorem, in about 200 pages. And, I then discovered that Godel, among others, proposed the fact that if you identified sets of numbers, that were mind bogglingly large, you could just build new levels of infinity, on top of underlying numbers and then you could use these sets, to prove some of the maths problems that exist underneath.
It was then, that I stumbled on an article in the New Scientist periodical, this week. The piece was boldly entitled 'Ultimate Logic - so powerful it could wipe out mathematics as we know it.'
In a nutshell, the piece describes how a mathematician (Hugh Woodin), believes he has solved the continuum hypothesis (Is there an infinite set that sits between the countable infinity, such as counting the integers from 1 towards infinity and the 'continuous' infinity, such as when you split a sphere into infinite sections, when the whole is finite.)
The difference with his claim, is that he states that he has solved this maths problem, by using a new type of logic language and structure called 'ultimate L'. This method would allow extra steps of sets to be bolted on to the top of infinite sets, filling in gaps below sufficiently, to allow any lower mathematical problem to be solved. He makes the bold statement, that this new theory, allows him to provide 'a definitive account of the spectrum of subsets of real numbers and thus, proves Cantor's continuum hypothesis, as true; ruling out anything between the countable infinity and the continuum'!
Woodin, even claims, that this may overturn major parts of Godels incompleteness theorem and be a tool that actually, artificially, allows us to root out unsolvability in number theory. He doesn't go as far as saying that Godel's theories would be dead - but that you could 'chase it as far as you pleased up the staircase into the infinite attic of mathematics'.
This idea seems similar in theme, to the idea of calculus and limits. In that, yes, you might never be able to say what is actually happening at the limit its self, but you can get so close to it, that your results have the same effect as if you were actually at the limit.
This all reminded me of the amusing anecdote relating to Zeno's paradox. It is said, that when it was once explained to a student, that if you were trying to reach a girl on the other side of a room, that you would never actually get there, if you travel half of the previous distance travelled in discrete steps. The student's retort was thus;
'Well, I might not ever reach the girl, but I could get plenty close enough for all 'practical' purposes! It seems, that perhaps Woodin is saying; 'why try and reach infinity? when you can just get so close to it, that your results are almost the same, as if you were actually there?' He seems to believe, that he has created a new language for mathematicians, to solve these issue.
Could it be, that if we do end up with a radical departure from existing mathematical logic and ideas, to prove almost everything; that we will end up with two tiers of mathematics, such as happened in physics with 'classical' and 'quantum' theories?
Wednesday, 27 July 2011
Life's ambitions.
Iv'e been a bit sick this week, hence the lack of posts. Anyway, it hasn't stopped me from ploughing ahead, with my MST121 final coursework.
I have now completed all of the readings of new material from the course; and I have now completed and submitted, the CMA covering the Statistics chapters. It is now all about finalising TMA04 and the other CMA that covers the whole course material.
Also, I had a bit of a wobble this week, as I found myself struggling to bring forth, from my memory, the rules and identities for integration. Specifically separating the variables, and differential equations involving double angle formulas and trig identities.
It led me to ponder the question, as to whether I should be economical with what I am covering or learning, i.e. learn what I need to, just to pass the exams and TMA's, without worrying about trying to commit the entire course to memory. Or, alternatively, whether I should be fastidiously digesting all examples, all proofs and all learning points, to the point of mastery.
I guess it all comes down to what the end goal is. My end goal, is not just to pick up qualifications, but then it got me thinking; what is my actual goal? It is not as easy to identify, as one would imagine.
I decided to examine the steps carefully, and try to extract my reason d'etre, in the process.
It went something like this:
A. I Want to score maximum marks in all of my exams and TMA's.
Q. But why?
A. Because I want to be sure that I fully understand all of the concepts and material.
Q. But why? It's not required, to pass the course and gain a good (2.1) degree.
A. Because I want an excellent degree( 1st)
Q. Yes, but Why?
A. Because I want to know that I am intellectually capable enough, to study the subject at postgraduate level.
Q. But Why?
A. Because I want to eventually study a PhD.
Q. But Why? It's 10yrs of training, will cost a fortune to fund and the remuneration is not fantastic.
A. Because I want to produce something original, that no one else has ever produced. I also want to spend my twilight years, pottering about doing maths and physics research, surrounded by learned people; not retire, playing Golf and Sudoku.
Q. Ah, now we are getting somewhere! - So it is about creating unique legacy and also being part of a community of like minded people?
A. Yes - That's exactly it.
Q. So, most people feed that need to create something unique and to leave a legacy, by having a few kids - but you want to spend tens of thousands of pounds and thousands of hours, doing maths and physics that no ordinary person could give two hoots about?
It was at this point, that I told my inner voice to shut up, and I went down the pub, for a pint and a curry.
I have now completed all of the readings of new material from the course; and I have now completed and submitted, the CMA covering the Statistics chapters. It is now all about finalising TMA04 and the other CMA that covers the whole course material.
Also, I had a bit of a wobble this week, as I found myself struggling to bring forth, from my memory, the rules and identities for integration. Specifically separating the variables, and differential equations involving double angle formulas and trig identities.
It led me to ponder the question, as to whether I should be economical with what I am covering or learning, i.e. learn what I need to, just to pass the exams and TMA's, without worrying about trying to commit the entire course to memory. Or, alternatively, whether I should be fastidiously digesting all examples, all proofs and all learning points, to the point of mastery.
I guess it all comes down to what the end goal is. My end goal, is not just to pick up qualifications, but then it got me thinking; what is my actual goal? It is not as easy to identify, as one would imagine.
I decided to examine the steps carefully, and try to extract my reason d'etre, in the process.
It went something like this:
A. I Want to score maximum marks in all of my exams and TMA's.
Q. But why?
A. Because I want to be sure that I fully understand all of the concepts and material.
Q. But why? It's not required, to pass the course and gain a good (2.1) degree.
A. Because I want an excellent degree( 1st)
Q. Yes, but Why?
A. Because I want to know that I am intellectually capable enough, to study the subject at postgraduate level.
Q. But Why?
A. Because I want to eventually study a PhD.
Q. But Why? It's 10yrs of training, will cost a fortune to fund and the remuneration is not fantastic.
A. Because I want to produce something original, that no one else has ever produced. I also want to spend my twilight years, pottering about doing maths and physics research, surrounded by learned people; not retire, playing Golf and Sudoku.
Q. Ah, now we are getting somewhere! - So it is about creating unique legacy and also being part of a community of like minded people?
A. Yes - That's exactly it.
Q. So, most people feed that need to create something unique and to leave a legacy, by having a few kids - but you want to spend tens of thousands of pounds and thousands of hours, doing maths and physics that no ordinary person could give two hoots about?
It was at this point, that I told my inner voice to shut up, and I went down the pub, for a pint and a curry.
Friday, 22 July 2011
Physics and Maths, Studied this Week.
Okay, I have 3 pieces of work left for MST121, including two computer marked assignments. The first is CMA41, a 20+ multiple choice paper on Statistics, yawn. I have half completed this, but just need to read through MST121 Chapter D3 and D4, before I can finalise my responses this week.
The other pieces are papers on the whole MST121 course, covering all of the calculus, matrices, vectors, series etc.
I have also engaged in some fun diversions this week, including buying a Rubik's cube, which I plan to solve this month and then I will try and get my times down over the summer, to < 5min. I have also purchased a group theory book, that is based around using the theories, to solve the cube. It seems like a bit of a hoot, although a bit contrived in places. The book is called Adventures in Group Theory.
I have also now finalised my next year's Open University, course choice. I have decided to study, (drum role).....
M208 Pure Maths
This is instead of doing MST209, first. A fellow O.U student and blogger, Chris, has helped me with this next decision and was kind enough to send me some samples of the M208 course, to assist.
I have also done some in depth analysis of my skills, needs and plans, for the next 3yrs, and realised the following things:
1. I seem to find it much easier to understand maths applications, if I, know / master, the abstract maths theories behind those applications.
2. I know that I am wired up a little differently to most normal people (what ever normal is? / Is this part of my Asperger's or Dyslexia? / who knows?). As such, I have discovered this week, that I can rotate / translate and manipulate objects, in my head, whilst reading through the stuff on Group Theory. I don't know if many people can do this, but it seems to make that subject quite straight forward, for me. Thus, it is probably a nice quick win intro to honours level maths.
3. Doing applied calculus, without first doing the analysis behind the methods, is personally difficult for me. I just can't grasp the rules all that well, without de-constructing the nuts and bolts behind integration / differentiation etc.
4. Finally, I am almost certainly going to complete some of the O.U Maths MSc modules, as part of my studies; which, at the very least, means that I need to study M208 Pure Maths, M337 Complex Analysis and M303 Further Pure Maths. I can then fit in some applied stuff, after these modules.
I did think about possibly not doing M303 and replacing it with the final presentations of Number theory and Logic, with Groups and Geometry. Thus, by doing so, being able to work on the proofs of Godel's Incompleteness theorem. However, I am just not sure that this Brucie bonus, justifies the extra 3hr exam that would occur, if I replaced M303. Also, is such self indulgence in Godel, distracting me from making progress towards PhD? Never say never, but 303 looks favourite.
Anyway, this week's study has been:
Total 20hrs.
MST121
Chapter D2 Modelling Variation
CMA41 questions 1 - 16
Recap on Trigonometry identities
Other Reading
The Calculus Lifesaver, Banner Functions, Graphs, Lines, Review of Trig.
Feynman, Lectures Vol 1. Chapter 8. Motion.
Adventures in Group Theory Chapters 1-4.
M208 Intro part 1. Group Theory part 1.
S197 How the Universe Works
Finished and sent off EMA. Course now complete.
Last night, I watched, 'A Beautiful Mind', the story of Schizophrenic Mathematician John Nash. It was really sad!
Next week, Starting my EMA for MST121.
The other pieces are papers on the whole MST121 course, covering all of the calculus, matrices, vectors, series etc.
I have also engaged in some fun diversions this week, including buying a Rubik's cube, which I plan to solve this month and then I will try and get my times down over the summer, to < 5min. I have also purchased a group theory book, that is based around using the theories, to solve the cube. It seems like a bit of a hoot, although a bit contrived in places. The book is called Adventures in Group Theory.
I have also now finalised my next year's Open University, course choice. I have decided to study, (drum role).....
M208 Pure Maths
This is instead of doing MST209, first. A fellow O.U student and blogger, Chris, has helped me with this next decision and was kind enough to send me some samples of the M208 course, to assist.
I have also done some in depth analysis of my skills, needs and plans, for the next 3yrs, and realised the following things:
1. I seem to find it much easier to understand maths applications, if I, know / master, the abstract maths theories behind those applications.
2. I know that I am wired up a little differently to most normal people (what ever normal is? / Is this part of my Asperger's or Dyslexia? / who knows?). As such, I have discovered this week, that I can rotate / translate and manipulate objects, in my head, whilst reading through the stuff on Group Theory. I don't know if many people can do this, but it seems to make that subject quite straight forward, for me. Thus, it is probably a nice quick win intro to honours level maths.
3. Doing applied calculus, without first doing the analysis behind the methods, is personally difficult for me. I just can't grasp the rules all that well, without de-constructing the nuts and bolts behind integration / differentiation etc.
4. Finally, I am almost certainly going to complete some of the O.U Maths MSc modules, as part of my studies; which, at the very least, means that I need to study M208 Pure Maths, M337 Complex Analysis and M303 Further Pure Maths. I can then fit in some applied stuff, after these modules.
I did think about possibly not doing M303 and replacing it with the final presentations of Number theory and Logic, with Groups and Geometry. Thus, by doing so, being able to work on the proofs of Godel's Incompleteness theorem. However, I am just not sure that this Brucie bonus, justifies the extra 3hr exam that would occur, if I replaced M303. Also, is such self indulgence in Godel, distracting me from making progress towards PhD? Never say never, but 303 looks favourite.
Anyway, this week's study has been:
Total 20hrs.
MST121
Chapter D2 Modelling Variation
CMA41 questions 1 - 16
Recap on Trigonometry identities
Other Reading
The Calculus Lifesaver, Banner Functions, Graphs, Lines, Review of Trig.
Feynman, Lectures Vol 1. Chapter 8. Motion.
Adventures in Group Theory Chapters 1-4.
M208 Intro part 1. Group Theory part 1.
S197 How the Universe Works
Finished and sent off EMA. Course now complete.
Last night, I watched, 'A Beautiful Mind', the story of Schizophrenic Mathematician John Nash. It was really sad!
Next week, Starting my EMA for MST121.
Monday, 18 July 2011
Exciting Mathematics
Earlier on tonight, I had a quick chat on the phone, with my tutor. I wanted some advice on future course choices, as my tutor has extensive first hand experience of studying O.U maths and science courses, over the last few years, whilst also studying for a PhD.
As we were chatting, something she mentioned, got me thinking deeply about distance learning and the absorption of subjects that are as intellectually demanding as honours level mathematics. My tutor used an adjective to describe maths home study, that I have never heard used in that context before. The word that she used, was 'Exciting'.
I didn't pay much attention to such an unusual use of this word, during our conversation; but once I sat down for dinner, I pondered its use.
Exciting?..Yes...I agree!
Sitting on my own at night, with my books, my pencil and my exercise books; the thing that keeps me coming back for more, is that every time I learn a new piece of knowledge (especially an intellectually demanding one), I am excited by this.
I love learning new stuff. I guess that it seems a little counter-intuitive, though, to call it exciting. We all know that studying mathematics, is probably last on the list of classic past-times, that could be classed as exciting. Bungee jumping, yes; roller-blading, yes; Severn of Nine's Star-Trek uniform, yes; but studying?... Well, yes.
And besides, from that excitement also comes the risk of addiction, which any long-term distance learner will appreciate.
Should there be a public health warning on the side of each O.U textbook cover, such as: 'Warning, regular use may cause dependency.'? Maybe.
However, using a critical eye, I know that this 'excitement', can be short lived, particularly if the work is too easy or far too difficult. If the work is too easy, then this can be remedied by supplementary exercises and such like; but too difficult? Well, I guess that all of my outside reading, away from core O.U courses, has all been done as a - sort of - cushion. A buffer to ease any jolts of future difficulty, that may try and make the excitement, a distant memory.
Exciting, we like.
Brain melting, like a Pan-Galactic-Gargleblaster, we don't.
As we were chatting, something she mentioned, got me thinking deeply about distance learning and the absorption of subjects that are as intellectually demanding as honours level mathematics. My tutor used an adjective to describe maths home study, that I have never heard used in that context before. The word that she used, was 'Exciting'.
I didn't pay much attention to such an unusual use of this word, during our conversation; but once I sat down for dinner, I pondered its use.
Exciting?..Yes...I agree!
Sitting on my own at night, with my books, my pencil and my exercise books; the thing that keeps me coming back for more, is that every time I learn a new piece of knowledge (especially an intellectually demanding one), I am excited by this.
I love learning new stuff. I guess that it seems a little counter-intuitive, though, to call it exciting. We all know that studying mathematics, is probably last on the list of classic past-times, that could be classed as exciting. Bungee jumping, yes; roller-blading, yes; Severn of Nine's Star-Trek uniform, yes; but studying?... Well, yes.
And besides, from that excitement also comes the risk of addiction, which any long-term distance learner will appreciate.
Should there be a public health warning on the side of each O.U textbook cover, such as: 'Warning, regular use may cause dependency.'? Maybe.
However, using a critical eye, I know that this 'excitement', can be short lived, particularly if the work is too easy or far too difficult. If the work is too easy, then this can be remedied by supplementary exercises and such like; but too difficult? Well, I guess that all of my outside reading, away from core O.U courses, has all been done as a - sort of - cushion. A buffer to ease any jolts of future difficulty, that may try and make the excitement, a distant memory.
Exciting, we like.
Brain melting, like a Pan-Galactic-Gargleblaster, we don't.
S197 Finished! Cert H.E studies complete
At last, I have now completed and submitted my EMA for S197 How the Universe Works. It was an extremely enjoyable course and one, from which, I have learnt a lot.
The course really does give a whistle-stop tour of the Universe from its first few moments of Planck time, through to postulating the future of cosmology and the M-theory that is currently being worked on.
Considering that this was a level 1 course, It didn't dumb down the concepts or the detail needed on topics such as primordial nucleosynthesis or cosmic expansion. Of course, these topics were covered qualitatively, rather than with any mathematical rigour; but none the less, it was a very enjoyable diversion from my calculus studies.
A nice added bonus is that, if passed, it will complete another university certificate for me, on the very long journey towards a PhD. Although it will be superseded by a Dip H.E and a B.A, at the end of next year, it is still a nice one to put in the back pocket and to fill the baron, white space on a C.V.
The total of Cert H.E modules included for this qualification, are (nb: not full module titles):
A103 Humanities
L192 French
S194 Astronomy
S196 Planets
S197 Cosmology / Astronomy
That is 120 points at year 1 level
I have purposely aimed for a nice balance of the liberal arts rather than just a one sided science qualification at level 1. I am hoping that it looks better to any future employer and shows that I have a more rounded education.
I know, upon speaking earlier this year, to the course director of the BSc Astronomy at Lancashire; that she is one of many academics, who actively encourage students to make good use of any free choice modules in a degree profile, by exploring other non-science subjects. This is in the belief that it produces a more well-rounded student and person. I would certainly agree with that hypothesis.
(It has also helped me shout out more correct answers when watching University Challenge. My wife is mildly impressed)
Onwards and upwards. It is now full-on MST121 until September, with a little sprinkling of self generated maths study, in between.
The course really does give a whistle-stop tour of the Universe from its first few moments of Planck time, through to postulating the future of cosmology and the M-theory that is currently being worked on.
Considering that this was a level 1 course, It didn't dumb down the concepts or the detail needed on topics such as primordial nucleosynthesis or cosmic expansion. Of course, these topics were covered qualitatively, rather than with any mathematical rigour; but none the less, it was a very enjoyable diversion from my calculus studies.
A nice added bonus is that, if passed, it will complete another university certificate for me, on the very long journey towards a PhD. Although it will be superseded by a Dip H.E and a B.A, at the end of next year, it is still a nice one to put in the back pocket and to fill the baron, white space on a C.V.
The total of Cert H.E modules included for this qualification, are (nb: not full module titles):
A103 Humanities
L192 French
S194 Astronomy
S196 Planets
S197 Cosmology / Astronomy
That is 120 points at year 1 level
I have purposely aimed for a nice balance of the liberal arts rather than just a one sided science qualification at level 1. I am hoping that it looks better to any future employer and shows that I have a more rounded education.
I know, upon speaking earlier this year, to the course director of the BSc Astronomy at Lancashire; that she is one of many academics, who actively encourage students to make good use of any free choice modules in a degree profile, by exploring other non-science subjects. This is in the belief that it produces a more well-rounded student and person. I would certainly agree with that hypothesis.
(It has also helped me shout out more correct answers when watching University Challenge. My wife is mildly impressed)
Onwards and upwards. It is now full-on MST121 until September, with a little sprinkling of self generated maths study, in between.
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