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
Sunday 28 August 2011
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!
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