Oh my! I have spent the last three weeks, using up every available moment of my time, studying my geometry and number theory courses. I hadn't realized just how much new stuff there is to learn; and not only learn, but manipulate and mould, into something that can be used in my TMA questions.
It has sunk in that number theory, is really, really hard. I mean, not just hard, but insanely hard. I'll try and explain a little. I am currently staring at my TMA sheet, in which it has taken me 20hrs, to answer 3.5 questions out of 10. I have looked at two of the questions and not only can I not work out an answer for them; I can't even recognize which bit of the course, they are supposed to be testing.
This isn't just some changing of symbols used, that is throwing me; this is a case of me tackling this stuff without a full and comprehensive background in mathematics, at A level and 1st year university level maths. These questions are calling on stock pieces of mathematical knowledge, that I am shaky on, combined with high level abstraction.
All I have been able to do, is work, work and work some more. I have kept an accurate note of my time spent studying this month; and I am about on track with my studies and TMA work; but it has taken me an average of 30hrs per week, to get there.
And the worrying thing is, I still feel like I don't really know the material all that well.
All I can do is maintain this level of study for the entire course; and I won't give up. I can't really say more than that, at this point.
This is the hardest that I have ever worked, in my life; and I am going to make it count for something.
Tuesday, 29 October 2013
Saturday, 5 October 2013
Division
Last year, I completed a T.M.A for my pure mathematics course with the Open University, in which one of the questions asked for a simple proof of a number theory example. I forget exactly what the example was, but it involved proving that a certain expression was divisible by 12, or some other integer.
At the time, I had a bit of a brain freeze and struggled with the question for some days. It consumed my days and nights until I finally gave in and rang my tutor, begging for salvation. I though, at that point, that I wasn't cut out for number theory.
It wasn't until he asked some very probing questions, that we finally got to the bottom of where my difficulties lay. It was the simple fact, that I didn't quite understand what was meant by 'divisible by'.
What? Nonsense, I hear you say! Every six year old knows how to divide two numbers. But the simplicity hid a rather elegant idea, in that, to divide two integers, and get another integer, is quite difficult to prove or understand, from axiomatic principles.
This was born out when I delved into my first book of number theory, last week, and came across... you guessed it... a proof for the Division Algorithm (...how to divide two numbers).
Its not a short proof, but it is elegant, none-the-less. If I had my LaTex editor working, I would write it below. I might do that next month.
The division algorithm is really quite lovely and it is one of the few gems in this new course, including my favourite so far, the proof of general mathematical induction; and my least favourite, the rather puzzling, second principle of mathematical induction. I say it's puzzling, as I haven't quite managed to figure out why, and when, you need to apply it to a problem, rather than just using the first principle.
I'm working on that one.
At the time, I had a bit of a brain freeze and struggled with the question for some days. It consumed my days and nights until I finally gave in and rang my tutor, begging for salvation. I though, at that point, that I wasn't cut out for number theory.
It wasn't until he asked some very probing questions, that we finally got to the bottom of where my difficulties lay. It was the simple fact, that I didn't quite understand what was meant by 'divisible by'.
What? Nonsense, I hear you say! Every six year old knows how to divide two numbers. But the simplicity hid a rather elegant idea, in that, to divide two integers, and get another integer, is quite difficult to prove or understand, from axiomatic principles.
This was born out when I delved into my first book of number theory, last week, and came across... you guessed it... a proof for the Division Algorithm (...how to divide two numbers).
Its not a short proof, but it is elegant, none-the-less. If I had my LaTex editor working, I would write it below. I might do that next month.
The division algorithm is really quite lovely and it is one of the few gems in this new course, including my favourite so far, the proof of general mathematical induction; and my least favourite, the rather puzzling, second principle of mathematical induction. I say it's puzzling, as I haven't quite managed to figure out why, and when, you need to apply it to a problem, rather than just using the first principle.
I'm working on that one.
Thursday, 26 September 2013
First Impressions
Well, I think I like my new courses. Number Theory isn't as heavy on content, as I first thought, and I have been able to manage to stay on track for my first 1.5 weeks of study. The main thrust of the material, is the mastery of Mathematical Induction as a way of providing simple proofs for integer problems.
I struggled with induction, when I studied M208 Pure Maths; but I think this was because I didn't allow enough time to complete practice problems, on the subject. However, I have taken head of the warning in the material for the number theory course, which states that Induction must be mastered, asap.
So, I have spent hours doing problems and redoing them, until the method started to stick. Having said that, I have pretty much struggled with the algebraic step in the last part of all of these proofs. Now, I considered myself to be at an advanced intermediate level, when it comes to algebraic manipulation; but these examples in the texts, make huge jumps in the algebraic reasoning, that I have struggled to follow or replicate (even by punching the expressions into Wolfram Alpha).
I definitely need to ask my tutor for some advice, about that. I do hope that I have not reached a natural ability ceiling, that hinders my progress on the course.
The groups course has been quite heavy on material volume, in the first book. You only get 1.5 weeks per book, where as, you get 2 weeks for a similar amount of material in number theory. Most of the material introduced, was basic concepts and language, that will be needed for the course; however, there was a cheeky little chapter on Affine geometry, which I started off hating, but then when I got all the examples right, without referring to the answers; I started to love it. How fickle!
Also, I have to say, that I find it soothing, rather than boring, to manipulate tiling's. I know that people who previously took the course, felt that it was all a bit dull. I find that they are all so geometrically neat and satisfyingly symmetrical; just the right amount of order needed, after a torturous and chaotic day at work!
I am adopting a new strategy for my studies this year. That is, I have decided to complete all of the practice examples in the books, having a good go at them, before referring to the answers. In my last few courses, I had started to get into the habit of skimming the examples, kidding myself that I was saving time, or leaving myself extra practise examples for exam time. I think it was a false economy and harmed my understanding of some of the more difficult concepts in analysis and linear algebra.
Lets see if it does me any good, this year.
I struggled with induction, when I studied M208 Pure Maths; but I think this was because I didn't allow enough time to complete practice problems, on the subject. However, I have taken head of the warning in the material for the number theory course, which states that Induction must be mastered, asap.
So, I have spent hours doing problems and redoing them, until the method started to stick. Having said that, I have pretty much struggled with the algebraic step in the last part of all of these proofs. Now, I considered myself to be at an advanced intermediate level, when it comes to algebraic manipulation; but these examples in the texts, make huge jumps in the algebraic reasoning, that I have struggled to follow or replicate (even by punching the expressions into Wolfram Alpha).
I definitely need to ask my tutor for some advice, about that. I do hope that I have not reached a natural ability ceiling, that hinders my progress on the course.
The groups course has been quite heavy on material volume, in the first book. You only get 1.5 weeks per book, where as, you get 2 weeks for a similar amount of material in number theory. Most of the material introduced, was basic concepts and language, that will be needed for the course; however, there was a cheeky little chapter on Affine geometry, which I started off hating, but then when I got all the examples right, without referring to the answers; I started to love it. How fickle!
Also, I have to say, that I find it soothing, rather than boring, to manipulate tiling's. I know that people who previously took the course, felt that it was all a bit dull. I find that they are all so geometrically neat and satisfyingly symmetrical; just the right amount of order needed, after a torturous and chaotic day at work!
I am adopting a new strategy for my studies this year. That is, I have decided to complete all of the practice examples in the books, having a good go at them, before referring to the answers. In my last few courses, I had started to get into the habit of skimming the examples, kidding myself that I was saving time, or leaving myself extra practise examples for exam time. I think it was a false economy and harmed my understanding of some of the more difficult concepts in analysis and linear algebra.
Lets see if it does me any good, this year.
Tuesday, 17 September 2013
Go...
They have arrived. I now have the first few month's worth of study materials for both Number theory and The O.U Groups course.
The Groups course is full of groovy plastic overlays and lots of different handbooks and equation sheets; yikes!
I have already started both courses, completing my first hour of study, on each of them yesterday. It's always difficult to say which will be the more challenging course; however, going solely on first impressions - I would say that the Groups course looks a little easier than the Number theory course. Time will tell, as to whether this first impression, lasts.
The Groups course is full of groovy plastic overlays and lots of different handbooks and equation sheets; yikes!
I have already started both courses, completing my first hour of study, on each of them yesterday. It's always difficult to say which will be the more challenging course; however, going solely on first impressions - I would say that the Groups course looks a little easier than the Number theory course. Time will tell, as to whether this first impression, lasts.
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