Having been around in the O.U universe for almost 10yrs, I have come across two distinct sets of students. There are those that see TMA's as a necessary evil; just one more thing to make you pull your hair out, when the deadline looms and you have the other pressures of life, competing for your time. But there is also the second set (to which I belong), that started out as TMA haters, but after completing a few of them, realised just how much learning actually comes from doing some mathematics, that requires you to properly apply what you have tried to learn, in the preceding weeks.
I believe, that TMA's are a vital and indispensable way, of forcing you to delve deeply into the mathematics, for a level of understanding that you just don't get from practising a few example questions. Now, you may have read the forum posts or looked at the course student reviews, and seen comments such as 'TMA's are formulaic', 'the TMA's were not really challenging', or even, 'the TMA's were a pain in the backside'.
But, reading these comments, you could easily be misled.
From my own perspective, as part of my experiment to learn maths and physics through distance learning and self-study; I have discovered, that no matter how much I apply myself to a particular area of study; I do not seem to obtain such a thorough understanding of it, unless it is tested via a TMA or other piece of coursework.
Exams, are different. I mean, they tend to be the 'lite' version of a TMA; lots of lighter questions to test general understanding and recall of methods and mathematical concepts. But, by its very nature, an exam can't dwell too much on one particular topic and in any great depth. Otherwise, they would have to be more than 3hrs long.
It is TMA's and other marked pieces of coursework, that quite often force you to re-read a unit. Back and forth, dipping in and out of pages, over many hours. Checking and re-checking answers. 'Did I calculate that accurately?', 'Why doesn't the answer look right?'. Sometimes going over a paragraph in a Unit, line by line, word by word.
It is this type of study, that allows you to have those 'aha!' moments, when you finally come face to face with what the mathematics is trying to say. Having said that; the only way to have this type of learning experience when completing a TMA, is to leave yourself the time to do it justice. I have, on previous courses, sometimes left myself with too little time for completing a TMA; only to resent its completion and gain no learning or pleasure from it, whatever.
It is therefore very important, to leave plenty of time for completing a TMA. I often try to allocate the whole 16 - 20hrs of one week, dedicated to its completion. I also often find, that if I am short of study time and my reading of a unit has fallen behind; it is often a better use of time, to stop trying to catch up with the unit and actually start the TMA. You can then catch up with the reading, as part of your research into formulating your answers on the relevant TMA questions.
I 'heart' TMA's; although I know there are those that would disagree!
Friday, 27 January 2012
Tuesday, 24 January 2012
M208, TMA01 Part 1, Completed.
Having had a good head start on the study materials of M208, I have now managed to complete the first part of the TMA01. It comprises of a relatively (deceptively?) straight forward exercise in sketching graphs.
This TMA covers the first intro book, and with the Unit book exercises possibly being more complex than what was required for the TMA, it hasn't quite scared me off from M208, just yet.
This is both good and bad. Good, because it is a bit of a confidence booster, before heading into group theory. Bad, because it only scratches the surface of some of the more complex graph sketching (combination and hyperbolic functions: I'm looking at you!), that is contained within the first unit.
I am not complaining though. After a week of struggling with equivalence classes, it made a change to actually have a vague idea of what I am doing.
All that it about to change, as part 2 of TMA01, hits all the difficult bits of the intro units. I plan to start that TMA, tomorrow. I am not sending my first TMA off to my tutor, just yet. I plan to attend the day school which is themed (graph sketching); as I don't want to drop any clangers.
From bitter experience, I know that tutors have their own styles of marking and their own vision of what they like to see, in each TMA. Once I get a feel for what my tutor wants in each TMA; I'll check that my first effort ticks all those boxes, before sending it in (about the 5th Feb should do it).
This TMA covers the first intro book, and with the Unit book exercises possibly being more complex than what was required for the TMA, it hasn't quite scared me off from M208, just yet.
This is both good and bad. Good, because it is a bit of a confidence booster, before heading into group theory. Bad, because it only scratches the surface of some of the more complex graph sketching (combination and hyperbolic functions: I'm looking at you!), that is contained within the first unit.
I am not complaining though. After a week of struggling with equivalence classes, it made a change to actually have a vague idea of what I am doing.
All that it about to change, as part 2 of TMA01, hits all the difficult bits of the intro units. I plan to start that TMA, tomorrow. I am not sending my first TMA off to my tutor, just yet. I plan to attend the day school which is themed (graph sketching); as I don't want to drop any clangers.
From bitter experience, I know that tutors have their own styles of marking and their own vision of what they like to see, in each TMA. Once I get a feel for what my tutor wants in each TMA; I'll check that my first effort ticks all those boxes, before sending it in (about the 5th Feb should do it).
Saturday, 21 January 2012
Rigorous Mathematics: A Disturbance in the Force
I think that it would be very dangerous to underestimate the 'introduction' part of the Open University course M208 Pure Mathematics. The intro consist of 3 books which cover the required knowledge, that you are expected to have under your belt; before embarking on the course.
It covers logic, set theory, functions, complex and modular arithmetic and is a sudden and violent introduction to a level of rigour, that you tend to only get at a Funeral home. I have found, coming straight off of the back of MST121 Using Mathematics, that there are some gaps in my knowledge base. They are little things that the course never really taught; Including the rigour that is required, to answer even the simplest of example questions on M208.
What has been a shock to me, and has been by far the most difficult part of the last 3 weeks of study; has been the way that you are required to explain results, that seem so bloody obvious. You can quickly get confused and muddled, trying to explain something that is ostensibly so simple, in such formal terms.
As an example, I struggled for hours, trying to prove that set A was contained in set B and set B was contained in set A. When I first looked at the question, I instantly knew what the correct answer was, and why it was so. But could I explain it in the convoluted mathematical way, required in the model answer? Could I hell. I have literally had to rote learn the structure of some of those phrases that seem to crop up, including those charming "hence's" and "therefore's".
It reminds me of the time that I learned to speak French. It took forever to get used to the way that a sentence was structured, and it took much practise to learn and then repeat those structures. The only problem is; that I have worked myself silly for the last 3 weeks and yet I am not quite confident,. that I can produce any good, exam style answers, from previously unseen questions.
It has been thoroughly enjoyable so far, but really quite taxing. It feels like I need to do 30hrs a week, not 15-20hrs, before I can gain that confidence, that I am seeking. I do think, though, that once I am used to the rigour; things will settle down and the 'newness' will not be causing such a disturbance within the force.
On the bright side, I now have a tutor and a set of 5hr day schools at Aston University on Saturday mornings. I am hoping that some tutor input, will allow me to shave some fat off my study regime and allow me to revise what I need for the exam, without any superfluous bits added on. This should help me settle down and gain some confidence as the months progress.
It covers logic, set theory, functions, complex and modular arithmetic and is a sudden and violent introduction to a level of rigour, that you tend to only get at a Funeral home. I have found, coming straight off of the back of MST121 Using Mathematics, that there are some gaps in my knowledge base. They are little things that the course never really taught; Including the rigour that is required, to answer even the simplest of example questions on M208.
What has been a shock to me, and has been by far the most difficult part of the last 3 weeks of study; has been the way that you are required to explain results, that seem so bloody obvious. You can quickly get confused and muddled, trying to explain something that is ostensibly so simple, in such formal terms.
As an example, I struggled for hours, trying to prove that set A was contained in set B and set B was contained in set A. When I first looked at the question, I instantly knew what the correct answer was, and why it was so. But could I explain it in the convoluted mathematical way, required in the model answer? Could I hell. I have literally had to rote learn the structure of some of those phrases that seem to crop up, including those charming "hence's" and "therefore's".
It reminds me of the time that I learned to speak French. It took forever to get used to the way that a sentence was structured, and it took much practise to learn and then repeat those structures. The only problem is; that I have worked myself silly for the last 3 weeks and yet I am not quite confident,. that I can produce any good, exam style answers, from previously unseen questions.
It has been thoroughly enjoyable so far, but really quite taxing. It feels like I need to do 30hrs a week, not 15-20hrs, before I can gain that confidence, that I am seeking. I do think, though, that once I am used to the rigour; things will settle down and the 'newness' will not be causing such a disturbance within the force.
On the bright side, I now have a tutor and a set of 5hr day schools at Aston University on Saturday mornings. I am hoping that some tutor input, will allow me to shave some fat off my study regime and allow me to revise what I need for the exam, without any superfluous bits added on. This should help me settle down and gain some confidence as the months progress.
Wednesday, 18 January 2012
Complexity, Clocks and Study Time.
For those familiar with M208, they will probably guess from the title of this post, that I am in the midst of studying I3 (Number systems). Specifically Complex numbers and Modular arithmetic.
It has been an interesting week, as coming off of the back of a good 2 weeks studying of Functions and Graphs and Mathematical language units I1 and I2; I am already starting to try and guess, which parts I will need to know perfectly, to make the exam easier in October.
With these units, there seems to be a theme whereby, the study material starts off with basic concepts, before descending into some crazy depth involving some nasty looking proofs, that will probably be extensively tested in the TMA's, but perhaps not so deeply in the exam. They are the sort of proofs that you can imagine took some poor mathematician, a life's work to figure out. So, I won't be too hard on my self, for not learning the proofs back to front in approximately 15 - 20hrs of weekly study. However, I am taking extra care, to ensure that I can follow the exercises in each sub-unit, that tests the use of such proofs in a practical question.
I did get a little bogged down this week in the middle of the complex numbers sub unit. I got muddled up when trying to work out other arguments of complex numbers, when given just Z^n = a
It's not that I don't understand the maths; I fully understood it at first reading. It is more that, following through a relatively complicated set of processes, to arrive at an answer, contains its own pitfalls when one is studying it late at night, after a busy day at work. I have just about got the hang of doing questions on this topic now; but, it has eaten into nearly 1/2 of my allotted study time this week. That has not left me much time to conquer clock arithmetic and equivalences; or to do another quick revision of I1 and I2.
I have obtained copies of all of the previous M208 exam papers since 2006, with some example solutions as well. It is comforting to see, that they all seem to follow the same pattern, with a high probability of certain questions or topics, being given their own question space. These include, manipulating some complex numbers to find the Argument, drawing some graphs and labelling correctly, some equivalence work and also doing a basic proof based on a logical statement about some numbers.
It does give one a little bit of hope, that the exam almost looks doable and certainly written to be passed by those that put in some work. Fingers crossed.
It has been an interesting week, as coming off of the back of a good 2 weeks studying of Functions and Graphs and Mathematical language units I1 and I2; I am already starting to try and guess, which parts I will need to know perfectly, to make the exam easier in October.
With these units, there seems to be a theme whereby, the study material starts off with basic concepts, before descending into some crazy depth involving some nasty looking proofs, that will probably be extensively tested in the TMA's, but perhaps not so deeply in the exam. They are the sort of proofs that you can imagine took some poor mathematician, a life's work to figure out. So, I won't be too hard on my self, for not learning the proofs back to front in approximately 15 - 20hrs of weekly study. However, I am taking extra care, to ensure that I can follow the exercises in each sub-unit, that tests the use of such proofs in a practical question.
I did get a little bogged down this week in the middle of the complex numbers sub unit. I got muddled up when trying to work out other arguments of complex numbers, when given just Z^n = a
It's not that I don't understand the maths; I fully understood it at first reading. It is more that, following through a relatively complicated set of processes, to arrive at an answer, contains its own pitfalls when one is studying it late at night, after a busy day at work. I have just about got the hang of doing questions on this topic now; but, it has eaten into nearly 1/2 of my allotted study time this week. That has not left me much time to conquer clock arithmetic and equivalences; or to do another quick revision of I1 and I2.
I have obtained copies of all of the previous M208 exam papers since 2006, with some example solutions as well. It is comforting to see, that they all seem to follow the same pattern, with a high probability of certain questions or topics, being given their own question space. These include, manipulating some complex numbers to find the Argument, drawing some graphs and labelling correctly, some equivalence work and also doing a basic proof based on a logical statement about some numbers.
It does give one a little bit of hope, that the exam almost looks doable and certainly written to be passed by those that put in some work. Fingers crossed.
Thursday, 12 January 2012
I Now Understand, the Binomial Theorem!
I don't just mean: rote learnt the theorem, punched in the a,b and n values and then crunched those variables.
I actually mean that having studied M208 Unit I2 this week, which included some set theory work based around counting finite sets and elements of finite sets; followed by a brief recap and proof of the Binomial Theorem. That I have been able to actually see, through the symbols, and to truly understand, at the basic level; how and why the Binomial Theorem actually works!
I think this is what I love about studying University level mathematics. It just seems to keep revealing such lovely surprises where and when you least expect them.
Take the Binomial Theorem as a perfect example. Anyone who has studied A-Level maths at school or as part of pre-university maths modules, will have met the Binomial Theorem before. One of its uses, is to be able to expand out polynomials such as (a+b)^2
However, M208 has taken a sprinkling of set theory, a dash of algebra and a soupcon of factorial magic, to show in very clear and simple terms, how many different ways there are, to put a set of different elements, into different combinations. i.e.
The set {1,2}, can be put in the order of either {1,2} or {2,1}
This then leads to counting the number of ways this can be done and how many separate sets, each main set can be split into. Following on from this, we can then begin to use this principle, to treat a polynomial as a finite set with (n) elements, which can then be simply expanded in full, using the Theorem.
This is, yet again, one of those realisations and personal moments, when the connections have fired between maths skills learnt 30+ years ago (counting objects and putting them in different order), 2+ yrs ago (expanding polynomials using formulas), to finally understanding (I mean really understanding), how it all connects and how it all fits together.
These little revelations have been keeping me so fired up this week, that I even took a detour when I was studying some number theory yesterday (Implications and equivalences), I actually pondered an 'I wonder?' moment and proved a simple statement, that I made up off of the top of my head, scrawling manically on my white board and then dancing around the kitchen when I proved it (this was very uncool, according to my daughter).
I think that is my first ever bit of creative mathematics.
Beautiful.
I actually mean that having studied M208 Unit I2 this week, which included some set theory work based around counting finite sets and elements of finite sets; followed by a brief recap and proof of the Binomial Theorem. That I have been able to actually see, through the symbols, and to truly understand, at the basic level; how and why the Binomial Theorem actually works!
I think this is what I love about studying University level mathematics. It just seems to keep revealing such lovely surprises where and when you least expect them.
Take the Binomial Theorem as a perfect example. Anyone who has studied A-Level maths at school or as part of pre-university maths modules, will have met the Binomial Theorem before. One of its uses, is to be able to expand out polynomials such as (a+b)^2
However, M208 has taken a sprinkling of set theory, a dash of algebra and a soupcon of factorial magic, to show in very clear and simple terms, how many different ways there are, to put a set of different elements, into different combinations. i.e.
The set {1,2}, can be put in the order of either {1,2} or {2,1}
This then leads to counting the number of ways this can be done and how many separate sets, each main set can be split into. Following on from this, we can then begin to use this principle, to treat a polynomial as a finite set with (n) elements, which can then be simply expanded in full, using the Theorem.
This is, yet again, one of those realisations and personal moments, when the connections have fired between maths skills learnt 30+ years ago (counting objects and putting them in different order), 2+ yrs ago (expanding polynomials using formulas), to finally understanding (I mean really understanding), how it all connects and how it all fits together.
These little revelations have been keeping me so fired up this week, that I even took a detour when I was studying some number theory yesterday (Implications and equivalences), I actually pondered an 'I wonder?' moment and proved a simple statement, that I made up off of the top of my head, scrawling manically on my white board and then dancing around the kitchen when I proved it (this was very uncool, according to my daughter).
I think that is my first ever bit of creative mathematics.
Beautiful.
Friday, 6 January 2012
Intensity, memory and mathematics
It's funny, but I noticed this week, that I have discovered some new and surprisingly subtle things about functions and other bits of maths, that I thought I had previously known, very well.
I can't quite put my finger on it, but in the last few days, as I have been working through the first week of M208 unit 1, book 1, repeatedly drawing graphs of different functions and combinations of functions; or just sitting with a pencil and thinking about why the parametric equations are constructed, as they are; I have found my brain making subtle but important connections between bits of maths knowledge, that I had stored away from previous months or years.
The outcome? Well, I am starting to realise that learning and using maths, is not just about learning a stack of techniques and theorems; but it can actually be understood conceptually and at a depth of understanding, that allows one to look at a problem and actually know what it is asking and how to approach an answer, without rote learning.
Why am I having this minor epiphany 3/4 of the way through week one? Well, I think it is because, for the first time ever in my study journey; when I haven't been working, eating or sleeping; I have managed to spend every spare minute of each day this week, pondering or doing mathematics in an almost manic and intense manner.
Why the intensity? I think it is because of the exam at the end of the course and a mild state of panic that is motivating me to do mathematics, rather than read, go to the pub or watch television.
What has surprised me, is that this has actually been one of the most pleasurable experiences in my maths journey, to date.
It hasn't been a slog. It hasn't been a chore. And most importantly, by doing many multiple sessions of 10 - 30 mins of study and exercises, I have been able to take advantage of how the memory seems to retain information best; i.e. little, often and with great intensity.
What could possibly go wrong :-)
I can't quite put my finger on it, but in the last few days, as I have been working through the first week of M208 unit 1, book 1, repeatedly drawing graphs of different functions and combinations of functions; or just sitting with a pencil and thinking about why the parametric equations are constructed, as they are; I have found my brain making subtle but important connections between bits of maths knowledge, that I had stored away from previous months or years.
The outcome? Well, I am starting to realise that learning and using maths, is not just about learning a stack of techniques and theorems; but it can actually be understood conceptually and at a depth of understanding, that allows one to look at a problem and actually know what it is asking and how to approach an answer, without rote learning.
Why am I having this minor epiphany 3/4 of the way through week one? Well, I think it is because, for the first time ever in my study journey; when I haven't been working, eating or sleeping; I have managed to spend every spare minute of each day this week, pondering or doing mathematics in an almost manic and intense manner.
Why the intensity? I think it is because of the exam at the end of the course and a mild state of panic that is motivating me to do mathematics, rather than read, go to the pub or watch television.
What has surprised me, is that this has actually been one of the most pleasurable experiences in my maths journey, to date.
It hasn't been a slog. It hasn't been a chore. And most importantly, by doing many multiple sessions of 10 - 30 mins of study and exercises, I have been able to take advantage of how the memory seems to retain information best; i.e. little, often and with great intensity.
What could possibly go wrong :-)
Tuesday, 3 January 2012
M208, First Thoughts...
Well, I have launched my study of Pure Mathematics M208 with the Open University; and I can say, straight away, that I am impressed with the quality of the unit books, that form the main text of the course.
I was able to launch straight into the first book of 24 that make up the entire course, which is entitled 'Real Functions and Graphs'.
The course notes state that it is a quick introduction to the topics that you should have met before, but then adds a little more detail to them, in preparation for the later blocks that cover analysis.
The content is fairly straight forward, although the more advanced graph sketching techniques of hybrid trigonometric functions, is something that I usually struggle with, so it is welcome practise. I have looked through 4 previous M208 exam papers and the specimen paper that was sent out with the course books. From what I can gather, there is usually just one 5 point question on sketching a graph, in each exam.
They also appear to usually have just 5 key method points to cover, so I would suspect that some of the more lengthy and difficult trigonometric sketches, involving up to 10 / 15 different calculations / decisions, may not come up in the actual exam.
However, I am sure that one will probably pop up in the first piece of coursework (TMA) that is due in March 2012. We shall see.
It has been 'so far so good':
2 days of study (6.4hrs)
Mindmap of unit 1 created and annotated with my study plan for this topic
1st pass of the whole unit done
Several exercises attempted (had to use the book solutions, to go through the hybrid/trig sketches)
I am feeling okay so far and now plan to use the rest of the allotted 18hrs of week one study, to repeat approximately 3/4 of the exercises, until they are second nature. I'll leave the other 1/4 for exam revision.
Also, If I can work out how to do it, I may post copies of my mindmaps, (without annotations), onto the blog.
I was able to launch straight into the first book of 24 that make up the entire course, which is entitled 'Real Functions and Graphs'.
The course notes state that it is a quick introduction to the topics that you should have met before, but then adds a little more detail to them, in preparation for the later blocks that cover analysis.
The content is fairly straight forward, although the more advanced graph sketching techniques of hybrid trigonometric functions, is something that I usually struggle with, so it is welcome practise. I have looked through 4 previous M208 exam papers and the specimen paper that was sent out with the course books. From what I can gather, there is usually just one 5 point question on sketching a graph, in each exam.
They also appear to usually have just 5 key method points to cover, so I would suspect that some of the more lengthy and difficult trigonometric sketches, involving up to 10 / 15 different calculations / decisions, may not come up in the actual exam.
However, I am sure that one will probably pop up in the first piece of coursework (TMA) that is due in March 2012. We shall see.
It has been 'so far so good':
2 days of study (6.4hrs)
Mindmap of unit 1 created and annotated with my study plan for this topic
1st pass of the whole unit done
Several exercises attempted (had to use the book solutions, to go through the hybrid/trig sketches)
I am feeling okay so far and now plan to use the rest of the allotted 18hrs of week one study, to repeat approximately 3/4 of the exercises, until they are second nature. I'll leave the other 1/4 for exam revision.
Also, If I can work out how to do it, I may post copies of my mindmaps, (without annotations), onto the blog.
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