M208
Pure Maths

I suppose that this is now getting serious—out of my comfort zone. Good.

Or, an impure boy’s continuing journey towards purity.

2011–10–12

end & elephant[s]

Well, course over—it’s that last post time again. This one may be a wee bit different from my others. This time I failed.

I woke up this morning and lay in my pit pondering upon the recent past and my immediate future. For about five minutes. Then I opened up my lappy and dived into some of the feeds that I’ve avoided reading for the last few weeks because I’ve been revising—AKA reading Plato. [That’s my current book.] Immediately I knew that I had a problem.

That’s the first elephant-in-the-room that I’ve been ignoring—I’ve forgotten why I’m doing this, this OU stuff. I say that it’s about me learning, it’s morphed into something else; I’ve become focused on the piece of paper. Which is why yesterday hit me so hard—not going to be such a nice piece of paper now.

I realized this when I realized that my lappy is a mess. I needed some techie stuff and it wasn’t there. Said lappy doesn’t even have XAMPP or NetBeans loaded; I’ve had lappy for months now and no programming has been done upon it. Even HTML-kit hasn’t been used in ages. [In fact when I tried using it it appeared to be broken.] I’ve done nothing for months.

I used to have fun programming, I still do, it’s just that I don’t seem to do it any more. That has to change. I joined up so that I could be a better programmer … I seem to have stopped. Which leads me to the second elephant: my drinking.

The functional in the functional alcoholic mix is becoming dominated by the alcoholic. I can deal with that, I’ve done it before. But it’s not something that I don’t want to tackle under stress. [Although when I won’t be stressed is an open question.] I now have a few months to fix me. And that has to be done.

Looking [and reading] back on this course I now see that it was the wrong course for me at this time, and hence perhaps the right one.

The last two paragraphs may seem disjoint and stupid. Perhaps they are but they say something to me: there’s no shame in failure unless you fail to learn the lessons from your failure. I intend to learn those lessons.

One lesson is, I think, that it is time for me to get back to hacking code. I’ll do the two maths courses that I’m signed up for next year, but I’ll be thinking of code.

Normally I sign these things off with a pretend commit, I haven’t had anything under version control for ages. So…

Gone coding…

2011–10–11

exam was…

A car crash. And I don’t even know how bad a crash it was at the moment. Here’s what happened…

  1. I was toddling along quite nicely, I felt, when, “you now have one hour left.” I was about half-way through the paper. [I don’t own a watch and I’m too blind to see the clock.]
  2. Panic set in, thought went out.
  3. As soon as I started to struggle with a question, I started on another one. [Or went back to one that I hadn’t finished.]
  4. “You now have fifteen minutes left.”
  5. I don’t know what I did in the last fifteen minutes, but if it gains me even a couple of marks I’ll be surprised.

I should scrape a grade 3 but even that is by no means certain. So what went wrong?

Me is the short answer. My last few posts will serve as my excuses, this post will deal with the real reasons.

my bad

At times I have worked hard, but too often I’ve been coasting on this course. True, I have a fair[-ish] grasp of the subject-matter, but I haven’t done anything the like level of work needed to ensure that I can ‘diddle the symbols’. I’ve been lazy. And I’ve been particularly lazy over the last month, I have done virtually nothing.

This caught me out during the exam—too often I had to look up my handbook because I wasn’t sure what I needed to do, which should have been automatic.

For example, there was a question in part II on diagonalizing a matrix [more-or-less], even in severe time-trouble I should have been able to ace this. I didn’t even get it finished and I’m not sure that what I did will attract many marks. I knew that this question was likely to come up, I intended to do it but I had only practiced it once. Under stress I fell apart.

There were too many questions like that.

Today’s paper, while not easy, was everything that one could have desired. It should have been catnip for me, even a wee bit of work during this last week, or a less panicked response during the final hour should have ensured me a grade 2. Wasn’t to be. My fault.

so

I must admit that as I rushed my way back to work I was tempted to just give up, to cry, or to move over to a computing only degree. However, I’m perfectly capable of making the same mistakes in a computing exam and while I’d like a good degree that’s not exactly why I’m doing this. No, I have to stick-in and learn from my mistakes.

When I dropped out of University, first time, it was for exactly the same reasons that I muffed this exam—an inability to work, when work was needed. I had thought that things had changed—because they’d seemed to; somehow, just by getting older, things were different. Not so, I’m still making the same mistakes that I made as a youth.

Just after a bad exam is not a good time to be making decisions, I know that. So I won’t. But I have made one: I’m not going to cop out on my planned road just because I’m a lazy idiot.

I can stop being lazy and the OU can elevate me from idiot.

Bad days we shall have, and today has been a bad day. The good days will come again. And you will hear of their arrival here first.

I’ve written this as-I-felt-it, because that’s what I should be doing. I won’t edit.

I’ll be doing my course review in the next couple of days, once I’ve recovered from my despair. ;-)

2011–10–07

aaargh

For once this disaster isn’t my fault. I did have a revision plan but it’s been totally buggered up by events. Well, I suppose that I’m partly to blame, judge for yourselves…

One of my minions has been off sick for the last six weeks, leaving two to do the work of three, not good. Worse my other minion, as my headteacher once said, “…does the work of two men, unfortunately those men are Laurel and Hardy.” I worked over forty hours of overtime last month, I’ve been on back-shift two weeks out of three and I’ve had just five rest days [days when I’m not at work] in the last two months. Not exactly great for a boy trying to revise.

I’ve been feeling too drained to open a maths book for weeks now. I scraped through the last two TMAs, just. I’ve been telling myself that it would all be OK, I’d just rest for the-time-being and throw hours at it in the last week. I had two days holiday booked, so including the weekend I had three whole days off before the exam.

You’ll have guessed what happened—the much-promised assured cover will not occur [I should have expected this], the minion on the sick failed to return as he’d aggravated his injury by sitting on his arse and my other minion went on yet another of his periodic binges. Result: no days off. Worse, I have to work stupid hours in the run up to the exam, I even have to go back into work after the exam.

Swear word, just swear word.

So I’m going to fail and I’m getting my excuses in early? Not so!

Last night I forced myself to sit down and do some maths. I found that although I’m rusty and slow, I grokked, mostly, the things I’m supposed to have learnt. So despite my knackeredness, my unpreparedness and my aaarghedness I still feel that a result is achievable. So after I finish work tomorrow morning I’m going to spend a day-and-a-half getting back up to speed. I will have to take some desperate measures—three days off the drink. This will hurt, but has to be done.

My hell is a typical OU hell; to be expected, at the very worst moment circumstances will conspire. My recipe: when you can work, work hard, when you can’t work don’t flagellate yourself and never say that you can’t do the course because of your circumstances, you always can. It might be impossible, but don’t let yourself know that. I may get a crappy mark for this course, but I don’t intend to, and it won’t stop me putting myself in this position again.

Reading this you may think, ‘rubbish’ but it’s not. At some point in your OU career things will go wrong at exactly the wrong point: don’t lose your optimism, desire and confidence.

I’m going to stop now, I’m beginning to think that I’m sounding like David Cameron—saying something that I don’t believe in the hope that what I believe is wrong.

2011–09–28

october comes

And with it new courses and new bloggers. The OU blogosphere is awash with eager people posting about their new courses. I don’t have a new course so I’m somewhat jealous—but not too much.

This course has taught me a lot—and not just about maths—it’s time for an OU break and some pondering upon my progress. I need a rest.

The first question, to ponder, is: am I going to finish my degree? There have been times this year when I’ve felt like giving up. Fortunately I don’t have to worry about this question for a while yet, I’ve signed up for another sixty points starting in February, so I’m locked in for the foreseeable future.

The next question is where am I heading with computing? I’m not doing a computing-type course next year and there will be [at most] only one the following year. Should I convert to a maths degree? Again this is something that I don’t have to deal with yet.

The third question is: am I up to maths? I’ll answer that yes, for the moment. I’ve been tried but I don’t think that I’ve been found wanting. At some stage my mind will fail, but not for a while yet.

The actual question is: why all these questions?

I’ve enjoyed this course, I’ve learned a lot, but at times I didn’t seem to be having much fun. The main reason for said sorrows has been my work situation, which will probably only get worse. I’m not alone in that, should I let it put me off?

It’s enrolment for night-school this week, and when we had moments of reprieve the Community Education worker and I were bemoaning the blows that are being visited on us. At some point I said, “they cut all the things that make life worthwhile” [in this context I meant adult education classes]. The penny dropped.

In many senses the OU makes my life worthwhile; I shouldn’t expect it to make my life fun.

2011–09–27

oh, dear

My marked TMA plopped through my letter-box this morning. My worst mark for this course so far; couple this with the fact that still no revision has been done and we have a potential problem Houston. Well maybe.

There’s still a couple of weeks to go until the exam, I know, roughly, where I’m weak [I’d rather know where I was strong] and in the real exam I’ll be running on a potent mixture of adrenaline and red bull rather than the numbing cider. Things may still turn out for the best.

My problem is that I’ve never really got revision—it’s a delayed-gratification thing I think. I’m happy when I’m having fun doing the initial learning, I’m happy doing a TMA, but I dislike having to work at something that I don’t see at the present moment. I don’t like working at something when the reward isn’t immediate.

I try various tricks to get round this laziness [for in great part this is what it is], none are very successful. The obvious [revision] trick—doing past papers—I never bother with. For one thing I only ever have the specimen paper that we get in the box [I’m too poor/tight to buy any more], for another I know that I’d just cheat.

It’s probably too late, for this course, for any rational revision plan—so I’m just going to practice some of the questions that I’m fairly sure will come up and read my handbook to see where it is that I don’t understand it. That may be enough.

Oh, and I’m going to start my next course early [you may have to be in-system].

2011–09–19

last TMA away

Now it’s full-on revision time. Ah, revision time, I’ve never really got revision. I’m happy[ish] with TMAs, I have the thing in my hand and I’m working to a deadline—I know what I have to do and when I have to do it by. Revision is different; OK, I know when I have to finish it by, but I’m never sure of the nature of the what-it-is-that-I have-to-do. Hence there are many opportunities for procrastination and sloth.

The wise man knows himself it is said, but as Marx [or Confucius] might have said, the point is to change yourself for the better. That ain’t so easy, and I won’t be doing it.

From the revision TMA and an inspection of the specimen examination paper that we got in the brown box, I’ve a fair idea of where I’m weak. I should revise that? Well of course I should!

But I know that I won’t, that’s just the way it is. At least I can still spot, most, of my own lies.

In the past I’ve come up with many a [flawed/not followed] revision plan, a plan which was consistently ignored/flouted. I know me; I’ll never follow one, I might as well face facts. So for this exam I’ve come up with a better plan—not to have a plan. Insane but brilliant.

I’m going to try and do a bit of maths every day. I don’t have to do any maths, but if I do then I can choose what I want to do. I’m going to rely on worry to get me moving.

This [stupid] plan sits comfortably with my so-called philosophy—that I’d rather understand a subject than pass an exam in it. The problem, with exams, is that they are about an understanding—they’re about you proving to an examiner that you can do certain things on a certain day. They’re not about having a clear understanding of the fundamentals of what we’ve been studying, exams can never be that—they are the Polaroid of learning.

I may not be wise, but I do know when I’m spouting tosh. And the above is tosh.

I’ve always had the ability to rationalize my, patently wrong, actions/decisions after-the-fact, it’s one of the reasons that I’m so comfortable in my own skull [few regrets], now it seems that I can rationalize wrongness before I even do it. Progress?

2011–09–08

why?

I had intended this post to be about my revision woes and my lack of mental-energy. But this type of post [my meat & drink] is becoming boring, even to me. I suspect that I could churn such posts out to order even if I only had a nose/keyboard interface. So I decided to talk about something else.

I’m in TMA/revision avoidance mode at the moment, so yesterday I picked up one of my ‘bought’ maths books—doing maths but not the maths that I’m supposed to—and I came across I name that I knew: Paul Erdos, which lead me to this—PDF [W Gower].

I was struck by the wonder [and utility] of Edros’s life and by the fact that I could grok [parts of] Gower’s argument. The first may always have been in me, but before this year Gower would have just been gibberish. [And most of it still is.]

This is why, after all, I got started on this humped-road of ours—the desperate urge to know. I now know that I’ll never know. But I am better-in-my-head. Take today.

Due to an idiocy of one of my minions I had to do a big split-shift; up at 04:30, home at 22:00. I did have a break and on my walk-back to work from this break I proved to myself that all prime numbers must be of the form 6n ± 1, [n > 3]. A year ago I wouldn’t have been able to do that. I’m getting better, I’m becoming who I want to be.

What, you want my proof? Well here it is…

  1. Let n, k ∈ Z+, positive integers.
  2. Even numbers are of the form 2k and odd numbers are of the form 2k + 1
  3. What we are trying to show here is that there can’t be any primes that don’t meet our condition. That means that we have to consider— 0, even numbers and odd numbers, and show that they just don’t work, unless they are 1 or are equivalent.

That should be enough.

update [2011–09–04]

Whoops. Neither for the first, nor for the last time a gross error is knocking at my door and demanding redress. I woke up this morning with the realization that I’d proved nothing. So here is the real proof that I worked out on today’s walk to work.

  1. Take a prime number, p—by definition it has no factors except itself and 1, so it cannot be divisible by either 2 or 3 [Unless the number happens to be 2 or 3, which is why we exclude them]
  2. Thus both p + 1 and p - 1 are divisible by 2. [That’s the way numbers go…odd, even, odd, even…]
  3. One of p - 1 or p + 1 must be divisible by 3, again because that’s the way that numbers work— 1, 2, 3, 4, 5, 6, 7… [Can’t be p from 1 above.]
  4. Thus either p - 1 or p + 1 must be divisible by both 2 and 3, and hence divisible by 6
  5. Thus any prime number must be of the form 6n ± 1, [n > 3]

Learning maths sometimes feels like a game of snakes and ladders.

2011–09–03

re-vision

I should have been having maths-fun at a tutorial this morning, instead I was stuck at work moving boxes and updating the school’s web site. The reason that I mention this is that as I was twiddling with various [site] bits & peices I was struck by how bad people are at putting themselves outside their box. I should explain that shouldn’t I?

I notice that people often assume their own knowledge in others—KU, obviously Knowledge and Understanding. To you maybe, but not to your average reader methinks. This is a common problem—misjudging, actually that’s wrong—the word is targetting—your audience. Your audience becomes fixed; you say what they want to hear. But if you are forced to produce cant for your peers, you lose the ability to communicate with your non-peers. Marxists have this problem. You develop an esoteric short-hand that is meaningless to others. What has this to do with me and maths?

It’s now revision time and I have a dilemma. I’ve winged my way through this course so far, by which I mean that I’ve worked the course books to produce answers to things that I don’t understand. Should I continue to do this?

I’m stuck in a similar situation to the teachers who contribute to my web site—should I concentrate on reproducing what, I know, that the examiners want to see, or should I try to be able to explain this stuff to my mum?

Put like that the answer should be obvious, but it isn’t.

2011–08–30

deranged

I read my last post this morning, I’m pretty sure that I know what I meant to say, I’m almost certain that nobody else did. Happens.

First off let me try to explain what it I was that I was trying to say:

  • Sometimes you come back to something that you didn’t like/understand the first time and it doesn’t seem so bad/hard
  • You can often get good marks without understanding the material
  • Are the above something that you can/should rely on?

What I didn’t mean to say was that I detest Taylor Polynomials, it just came across that way.

The sub-text was that I’m beginning to suspect that I may be odd. Well, that’s kind of a given, but I mean odd as a mathematician. In fact I may not be a mathematician at all.

I came to maths from programming and I came to programming by mistake. There’s something about the way that I think that isn’t quite, right for maths. I’d love to spin this into a positive, “I’m visual”, or something, but I know that this is a nonsense.

What I have noticed, and what I knew at school, is that I have difficulty in putting the maths that I know to use when it is presented to me in a way that I’m unfamiliar with. I don’t have this problem with computing, the pattern is usually obvious. What does that say about me as a programmer? About me as a mathematician?

If I evaluate myself as a programmer: I guess, that my strengths are that I don’t over-complicate and I have a strong sense of when I’ve gone wrong. I don’t seem to have that with maths; I miss things out, I include the unnecessary, I think that my bad maths is good. This may just be down to practice—I forget that I’ve probably spent more time at serious programming than I have at non-rote maths.

I’ll get through this course but I have some serious thinking to do—if I’m going to go ahead with maths I might need to change the way that I tackle it. Actually I might need to change the way that I think about it.

2011–08–28

taylor hell

Or, when analysis meets Taylor polynomials—the ultimate storm for the neil? Strangely enough not.

When Taylor polynomials and I first encountered one-another I wasn’t impressed, in fact I was a wee bit hostile. So I wasn’t looking forward to another encounter. But oddly it hasn’t been too bad.

None of which means anything unless we look at the bigger picture [the meta]; how do we learn? And how do we learn better?

By which I mean: whatever you study there will be bits that you don’t like much [Javax.swing springs to mind], how do you cope? How do you ensure that it goes into the meatware?

For this course I’ve been cheating big-time. You can do a TMA without any understanding of the course-materials at all. I know this to be true because I’m amongst the guilty. That isn’t good, an M208 Nuremburg trial would have me shot.

Well let’s hold on a wee minute there. Could it be that if I come back to this in a year’s time and I’ll [just suddenly] understand? I don’t think so, but I do.

Learning isn’t about knowledge, it’s about a mind-set, it’s about a hunger, it’s about a skill-set.

It’s something you want.

2011–08–26

hard[y]

I’ve been struggling away at the penultimate TMA for the last week or so. I’m nearly there, but the shadow of behind still looms—it’s only three weeks until the final TMA is due and then it’s revision time.

So what with, TMA hell, revision worry, money angst and work woes, I’ve rather enjoyed having the soporific of Hardy’s pure maths book to transport me to the land of nod. Which got me thinking—actually that’s a lie it was one of Nilo’s posts—does an OU course prepare you for reading a maths book? From my own experience the answer is clearly no.

That’s not the full story however.

I’m going to use my course-mates as examples here, the ones who post in the fora anyway, I’m not going to name names. We seem to fall into three categories [to me]:

  • Those who clearly grasp what they are doing and who are interested in the subtleties
  • Those who have issues
  • Those who are struggling a wee bit because the course books leap in places

Obviously you can fall into more than one category for a single topic and into a different category for another topic. [Typical mathematician’s get-out!]

The first group probably wouldn’t have any difficulty with Hardy, the second and third groups assuredly will.

Let’s look at the third group, I’m in there often. There are a lot of posts of the type, “…how did we get from here to there…?” This never bothers me much: I know the answer and I like the challenge of getting to it—it’s a maths I’m comfortable with. I can see that for others it might be daunting, but I’d suggest that you try, even if it disrupts the flow of what you are trying to learn.

Now to the second group. I suppose that most of us belong in this group—without the OU texts as an in we’d have little chance of grokking Hardy raw. I get bits of it, and in places it helps to have something put in a different fashion, but if I only had Hardy…?

Most maths books are written, either for other mathematicians, or for the brilliant student, or as course notes to be taught-to. They aren’t meant for mere mortals. OU texts are designed to be stand-alone.

And that’s surely the point! I doubt that many of my cohort are going to be professional mathematicians; we just want to learn. Should we be excluded from mathematics because maths books are written for an élite [or to be taught from] because we’re average or we aren’t able to pursue our maths in a traditional way? The answer is clearly no.

There’s going to be an update here because an interesting discussion [that will take me a while to understand] has started on one of the threads about how maths should be taught.

2011–08–14

history of maths

I’ve done little maths the week, I’ve been busy at work and, therefore, too tired for rigour. [I missed yesterday’s tutorial because I had to work.] The maths that I have done has proved useful, that maths was history. I read A Concise History of Maths by Struik.

I have several, personal, issues with analysis, the main one was why? Reading about Newton’s discovery and exposition of the calculus was something of an eye-opener [We’ll leave aside the Newton/Leibniz controversy]. I began to see why we needed analysis. It’s too easy to assume that something that seems obvious is true. Or to put it another way, can you continuously make something smaller until it goes away. At what point does it disappear? We’re back to Zeno’s paradox; something that our common-sense knows isn’t true but is very difficult to prove wrong. And it’s by no means clear that moving from point A to point B is the same as tracing the graph of a function with a pointing device. Are there gaps?

In short I realized the need for this rigour; which led me to question my own.

My problem is that I don’t get the level of the rigour that I need to answer the questions that are handed to me. The normally excellent unit texts have let me down here—sometimes they go into great detail, sometimes they seem, to me, to skimp on some issues.

This may be my fault, assuredly it is in fact—I haven’t paid enough attention to the proofs. The way that you prove things [in maths] is directly related to how you use the results of said proofs.

So, for the neil, back to analysis block one to grok the proofs. We’ll see if eating(pudding) = ∃pudding works for me and analysis.

2011–08–05

counting the pennies and checking my sanity

From a thread in the M257 course forum I’ve realized that I’m going to have to do M256. The question is can I do it this year? Should I do it this year? Can I afford to do it this year? [This year would mean academic year, ie 2012.]

Let’s look at the second question first. I’ve battered on at length here about it being inadvisable to do more than sixty points in one year. If I do M256 I’ll have to do it alongside Geometry and Groups and Topology. Two third-level courses. Now, whilst I don’t foresee too much trouble with Groups I’m guessing that Topology is going to fry my brain. The workload would seem extreme. Perhaps.

All these courses start in February so there will be huge TMA collisions and exam hells. But…I think that I can manage this—probably hubris on my part. We’ll come back to this.

The problem is that bankers’ greed and trams have messed up my world—it’s an almost certain surety that I’m either going to be made redundant or privitised into, an even greater, poverty—I can’t predict how I’ll be fixed this time next year. And I want to leave myself with financial wriggle-room.

In the end. Can I? Yes. Can I afford it? No but I could give up the fags, drink less and skip on inessentials like rent and food. If I want to do this I can.

But should I? Gloss it how you will this is blatent idiocy. [If I do commit to this I’ll do it drunk; I’ll then be able to absolve myself of some of the responsibility.]

The stupid part of me [neil = {stupid, sensible, risible} : risible > stupid > sensible} a well-ordered set] is saying something along the lines of, I’ll be doing computing anyway, the sensible part of me is saying, this is waay too much.

I’m either going to do something stupid or regret that I didn’t do something stupid.

Fortunately, being a maths/computer student I see a way to re-phrase the question so that the answer that I want makes sense.

2011–08–01

analysis, again, alas?

After an enjoyable weekend spent messing around with group theory it’s Monday and back down-to-earth and analysis. The next month isn’t going to contain much maths-fun for me methinks. But… A weekend’s messing with maths has confirmed that my last post was on-the-money—I can’t just read the units, I need to mess around on my own; I need to diddle symbols and draw shapes. I need to bring this insight to my analysis.

This means yet another re-visit to the first analysis block to tackle it in a different fashion. I haven’t done too well so far, I haven’t forgotten everything but I sense that my understanding, over time, is a null sequence.

After my analysis TMA I re-visited the block, but it was in the same stilted style that I first tackled it. This meant that I didn’t really gain any new understanding and will cost me marks on my group theory TMA as I got short of time. [Although something interesting came out of this—that’s what I’ve been messing at over the weekend, and from a long discussion in the forums, and others’ insights, I think I see a way of creating an algorithm for colouring problems. If I do I’ll post it here.]

From just a quick glance at unit one of the second analysis block I can see that I haven’t grasped some essential concepts—in particular I’m unclear of the methods of proof that are required. This means that I probably need to spend a week on just catching up.

I could do the unit without a full understanding, but that was one of the lessons that I took from this weekend—because I was comfortable with the concepts and techniques I could think while I was doing it and I enjoyed myself.

To me analysis looks uncongenial, but how many times in your life have you disliked something that you came to love on your first encounter?

2011–07–27

limit

For some time and for some posts I’ve been struggling to understand my maths—what works for me? Today I think I’ve achieved a breakthrough. Or perhaps not.

There are going to be some ‘mathy’ terms thrown around here, I’ll try to keep them to a minimum and still write something that’s relevant to non-maths students. But, essentially this post is aimed at those who are studying maths. [And I’m going to deliberately obfuscate what I was doing as this involved a TMA]

I was working with a function, f, a homomorphism, that mapped a group, (C, +) onto itself, or a subgroup thereof—the image of f. I couldn’t picture what the function did geometrically, I couldn’t see the kernal or the image. I did know that it was one-to-one, but was it onto? Stumped, I started playing around with a general term [a + bi], as soon as I did I began to get an idea of what was going on—that’s what I’ve been missing! I have to mess around with examples, that’s what I do when I’m programming, that’s how I understand.

For too long I’ve been avoiding getting, ‘down-and-dirty’ with the symbols; but that is what I need to do.

It’s too easy to pretend that you can think in a conceptual manner—to assume that you can lie in your pit pondering—if you are René Descartes you might be able to get away with this. For me this isn’t a workable plan.

In some ways this is a bit gutting—if all all can do is an axiomatic manipulation of symbols then there are truths I’ll never reach. But I knew this anyway, and at least I now know my lowest-upper-bound.

2011–07–25

nibbling at the edges of understanding

Sometimes it’s better to reflect than to strive. That’s what I’m telling myself anyway.

I didn’t make much progress on my TMA this weekend, none wouldn’t be too bad an approximation. That doesn’t mean that I did nothing—I thought about stuff.

True, few books were opened, none were studied and little paper was scrawled upon—but maths-understanding often lurks in the parts of the brain that we don’t have always-access to…am I tickling for trout in my own head?

Perhaps. I often think that I’m just stupid, but then I realize that I don’t care. Stupid people have a voice and if I’m that stupid I won’t grok your scorn.

Tonight I’ve been mashing symbols, working the books, in short—trying to complete my TMA. I’m feeling wrong. Why? Because I’d rather lie in my pit watching the sun play upon descending dust pondering the infinite? Or do one of the many other things that I put off? There’s a middle way here.

The middle way is thinking, me thinking, about the partitions of my mind; the boy that does the TMAs, the boy who paints soldiers, the boy who loves lego, the boy who lies in bed watching the dust settle in the dawn light. They are the same boy, but different.

Somewhere, sometime I heard a song [the the] that captures my mood exactly. Last night I had to fix my wife’s computer, I had to use a few unusual tools—she asked me how much I would cost if I weren’t her husband. I don’t know, I only know that I would do it for anyone who asked, without price.

I need to think about why I understand computers when I don’t really care about them, and why I have trouble with maths and me, when I really do.

2011–07–22

coming together?

I’m two-thirds of the way through this course and I’m beginning to feel that I may be beginning to get it. Is this a good thing? And what is it?

I ask the questions because I’m not sure that I’ve felt this way about a course before. There have been times when things have suddenly come together, where I’ve felt on the verge of a breakthrough or when I realize that with a bit more work I will understand—I’ve never had the sense of glimpsing a deeper unity in the materials before.

I’m sure that this is mostly down to the nature of the subject itself—of course maths has a deep interconnectedness. But there seems to be something else going on, something to do with me. I think I may have been lying to myself about me.

If you’d asked me three months ago where I felt my [mathematical] strengths lay, I’d have said spatial awareness [I can usually see symmetries], pattern recognition [I can often see that something is true] and a sense of what ‘tool’ to use. Now I’m not so sure that this is true.

One of the hardest things in this life is to understand what ‘kind’ of person you are—you probably minimize your strengths and pretend that your weaknesses aren’t, because you want to be a type. This is a trap that’s easy to fall into. [I’ve noticed a huge rise in the number of people who are Autistic since Rain Man made it cool.] But you are what you are, you have to work with what you are given. Don’t stick labels on yourself. I think I’ve been guilty of that.

I said that I didn’t like analysis because I’m sloppy—I am, but I ain’t always so; I can be accurate when I have to be. I don’t like analysis because I don’t really ‘get’ it. There’s no reason but sloth for me not understanding it, if I work at it I will, I’m making excuses for myself.

The coming together that I opened this with is just that—I can’t pick and choose, I have to understand everything, I can’t let the fact that something doesn’t particularly grab me stop me learning it. I can’t salve my ego by pretending that I’m intrinsically unable to grasp some concepts, or do some things, because I’m special.

Supposedly I’m a touch dyslexic—I certainly had a lot of problems learning to read and I have a lot of left/right issues—I’m not so sure. I don’t use that as an excuse not to try to spell things properly here [and I’m a programmer, I have to spell correctly when I’m doing that]. I need to take that attitude to maths.

I had to edit the first paragraph of this, I realized that I’d fallen into the, solving the if and not the only if trap that I’m prone to falling into. Always think upon Deep Thought, having an answer isn’t much good unless you know the question.

2011–07–04

sunny

Yesterday was very sunny, very hot and I had a rare day off. So I did a bit of gardening and then sat on the front-step, nursing a cider, pretending to do some Group Theory. Nice relaxing day.

Last night I had a wee ponder upon my past and future progress, both life and OU-wise. The following conclusions were reached:

  • The Linear Algebra block could have been done better
  • The first Analysis block was a car-crash
  • More than sixty points a year is too many if you want to be served fun along with your learning
  • I haven’t spent enough time programming this year
  • I may have made an error choosing to do concurrent third-level maths courses next year [albeit a forced error].
  • Robotics & the meaning of life is back on the programme; the October/February gap is beginning to worry me
  • Having to work [hard] for the money to pay for my OU courses is a pain

Some of these problems will have to be pushed down-the-line for future solution. What does need to be decided upon now is how to tackle M208.

on-course?

Technically yes—I’m studying the right unit book, the TMAs have all been not-too-badly done and I feel confident with Groups. All to the good? Alas not.

TMAs are less than a quarter of the battle—the war will be won or lost in the exam hall. At present I will fail there, and that won’t change unless I work much harder.

I don’t know how this works for others, but for me TMAs are, or should be a means for you to test your understanding. You shouldn’t have to rely too much upon the supplied texts if you are doing well. I am, therefore I’m in trouble.

Obviously since this is July and the exam is in October this trouble isn’t terminal trouble. But small troubles are like small snowballs—you have to stop them before they become big ones. So my plan?

For now, to forge ahead as quickly as possible. Analysis, in particular, isn’t something that I want to work at—so I need to have time available when I do fancy doing it.

There’s no avoiding it—I’m going to have to force myself to work when I want to play.

2011–07–02

better…

For the first time this year I’ve got my baseball boots on—summer has started. I had a tutorial this morning, lovely Groups. Getting to my tutorial is always fun; It goes in stages:

  1. I walk along the canal to Tolcross[ish], very bucolic—ducks, swallows, reeds, swans, dogs and the bikers who seem to be trying to end my life.
  2. I get to Tolcross[ish], the city, and head for the Grassmarket, via the West port, the second-hand-bookshop region of the town. Here I buy my old maths books, but I don’t tarry
  3. Then there’s the Grassmarket, a seething mass of tourists and drunks even at nine-thirty on a Saturday morning. Full of light and pubs
  4. The Cowgate—a shaded urban-canyon where strange pale-people stagger around looking around for the night-club they think they’ve just left
  5. You swing up towards the High Street where the world streams past
  6. You do the maths

Afterwards we go to The Task and have a drink and a catch-up chat.

After a slightly torrid-time personally I feel healed, and ravenous for maths.

2011–06–29

analysing my analysis

It’s activity week at my school this week. Half the school is abroad, somewhere, others are concentrating on a single activity—golf, cooking, film-making…whatever.

Then there are those on the, ‘daily activity programme’. One of these activities is chess. I can’t/won’t ever walk past a room where people are playing chess without at least a wee peek. My look revealed that there were those there that couldn’t play chess at all. [Why?] I picked upon one wee girl to teach.

You teach chess by showing people how each of the pieces moves and set up the start position, right? Wrong. Here’s what you do:

  1. Set up four pawns each side, the winner is the one who gets a pawn to the eighth rank first
  2. Make sure they are comfortable with the pawns; and make sure that en passant comes up somewhere
  3. Keep adding pawns to their side until they can beat you, when you are trying
  4. Introduce the King, the idea of check and play the same game
  5. Three above
  6. Introduce the Queen, the concept of promotion and checkmate
  7. Three above et al…

I’ve, probably, taught hundreds kids to play chess, or about it, and I can assure you that this is the best way for their future progress. Chess from the start-position is just a nightmare, if you haven’t grasped what it is you are trying to do. First introduce the patterns, “if I’m here, I can get there by doing that, and I can win from there…”

Anyhoo, I was going through my usual routine, the wee girl was doing well, she showed some flair [she grasped the concept of a passed-pawn and an outside-passed-pawn in a way that some quite-good players don’t]. But she had a distressing tendency to push her King up the board when she should have been gobbling pawns. Why? Because she was doing exactly what I’d told her to do—get a piece to the other side! I hadn’t made it clear that the game-rules hadn’t changed [1 above].

She was right, I was wrong, and that’s my problem when it comes to analysis.

Spot the other dodgy assumptions for yourselves.

My faultz

I got my latest TMA back today, my tutor said, something along the lines of…

Accuracy is everything in analysis…you [me—neil] often miss out parts of the argument.

Exactly what I’d done with the wee girl and the chess. I assume; you can’t do that with analysis. Or with maths in general, but particularly with analysis.

My wife was in heaven when I read this out to her. Once, I drew her a map, a map which sent her astray. I missed out a couple of streets, shouldn’t have mattered—it was obvious. She got lost, asked for help, displayed the map, the map was rubbished by a stranger. My wife has a keen intelligence, especially when it comes to my failings, she’s never ceased to bring this up.

So, I have a problem.

What to do?

I suppose I could just ignore it and carry on. We’ve another analysis block—I could catch up there? Catch up in what sense? Stopping being sloppy? More of the same won’t help there methinks.

To stick with the chess metaphor, where we started—you can be aware of the positions that you don’t like and try to avoid them, but in the end you are better fixing your flaws. Otherwise others will head for your weakness. So we must master the basics, like I teach chess.

I’m going to go back to analysis, to teach myself a discipline that I don’t have.

2011–06–17

overload[2]

Well M257 exam done-and-dusted and discussed elsewhere, now it’s time to get back to analysis. Not before time, I still have the sequences and continuity units to read. By Monday, for my TMA is due in by next Thursday. Again, a time-crunch time for the neil.

analysis

Seems a bit of an oddity. This Thursday I bumped into Graham, [an M208 course-mate mate] when I was buying the ridiculous quantities of red bull that I have to take into an exam. He’d missed the last tutorial [He’d been at a philosophy day-school in London] he asked me whether the general-thought of the tutor-group was that analysis was as simple as it looked. I replied in the affirmative. I now suspect that I was wrong.

Not because I’ve since stumbled upon anything that I’ve personally found difficult, or because of any blinding revelation but because [and this often happens] the slow osmosis of not thinking about something lead me to realize that my understanding was shallow. Zeno’s paradoxes are subtle, it took the cream of the nineteenth century’s mathematicians to prove the bleeding obvious fact that they’re false. That should give us all pause. If it took that much effort and that much talent…

Mathematics is all about, [I’m beginning to think], crafting tools that you can rely upon in outer-space, in a black-hole or an n-dimensional non-Euclidean space. We spend much time on the foundations so that when we’re eighty storeys above the we don’t have to worry about tool-wobble.

That’s going to be hard for me.

2011–05–28

overload

OU/life overload is a subject that’s forever churning in the back of my head. It’s come to the fore in the last couple of weeks for a few reasons—

  • I had a couple of TMAs that were unpleasantly close together, and hence were skimped and possibly botched.
  • Next Saturday I have back-to-back tutorials. Fortunately there are in the same building but the fag/red bull break will have to be swift
  • I’m finding computer revision/maths TMA juggling a wee bit taxing
  • My mate Chris seems to have given up on one of his maths courses

I think that Chris was mad to even attempt 120 points in one year, but I’m full of admiration for his courage in deciding to ditch a course. I’m not sure I would be able to do that if I was in a similar position.

So how many courses/points can reasonably be attempted in one year? The answer will differ depending on the person, the courses and the available study time. I think all of us are impatient to get our degrees, but we have to be sensible. If you go to a brick university the study schedule is made for you, we have the luxury of cocking it up. Here are some factors to consider—

  • Why are you doing this? Do you want the qualification or the knowledge? Do too much and you might get the former but will you get the later?
  • Are your circumstances in a state of flux? I’ve lost count of the number of students who have been kibboshed because their lives have altered
  • Do you want to have fun? If so you need to leave some time for having it
  • How good a degree do you want? More work == less marks
  • How much stress can you cope with?

I want the knowledge and the fun, I can live without the stress and the marks and my life is pretty settled. [I hope!]

I’m doing 80 points this year, for me this was too much. Despite the 80 points being October/February staggered I’ve missed out on a lot of things. I started personal activities for both courses which I’ve shamefully neglected, I haven’t been as active in the fora as I like to be and I’m working the TMAs because I don’t properly grok the subjects.

Next year I’m signed up for 60 points of concurrent third level maths. Which I’m now beginning to think is a mistake. It might be but it’s a mistake that I was forced into because the courses are the final presentations—if I don’t do them now I’ll never get to do them.

For me the knowledge comes before the fun.

2011–05–21

how hard is linear algebra?

This is a hard question in itself. I found this block testing, I’m not sure that I found it hard. My reading of the fora suggests that other students are definitely in the, ‘this was hard’ camp.

To my mind linear algebra is about an abstraction—Vector Spaces [or just spaces]. We’ve dealt with abstractions before—Groups [yes, I know—maths is all one big abstraction but…]. Indeed many of the techniques that we use in Linear Algebra should be familiar to us from our work with Groups. I think that the problems people had with this block were down to the fact that we went far, fast and there wasn’t a clear sense of why.

Somewhere in the unit texts it was stated that, Vector Spaces are one of the great unifying concepts of Pure Mathematics. Get that? When the word great is used we can take it that they mean hugely important. So, very important and hard them. Important yes, hard I’m not so sure of.

I think that the unit texts let us down slightly here. Not because they were bad but because they didn’t make it clear what it was that we were doing in the way that the Group blocks of M221 and M208 did. Consider their structure:

Unit 1
A sedate review of vectors
Unit 2
Messing around with matrices and simultaneous equations
Unit 3
Vector Spaces
Unit 4
More matrices and Linear Transformations
Unit 5
Eigen Spaces and quatrics

At no point, working my way through the units, did I feel that I wasn’t understanding what I was currently doing, what I was having trouble with was seeing how it all fitted together.

Now that the TMA dust has settled and I have a wee bit of ponder time I can see that this all had to be covered. And perhaps covered in this particular order, but I lacked a unifying concept—I felt that I was jumping around from topic to subject. We had spanning sets, bases, orthogonal bases, orthonormal bases; then matrices, linear transformations, more matrices and then boom—we were rotating quatrics. It was hard to take in that we were working up to one example of where this all came together.

Writing this has helped—the reason Groups seemed easier was that it was all about Groups, for Linear Algebra we need to put diversity together. I think that I may have learned something that I knew but didn’t know—in maths everything is connected. A technique, a theorem, a conjecture, a lemma in one area may have important implications in what seems, at first blush, an entirely unrelated field—the Riemann Conjecture anyone?

So the unit texts were right and neil was wrong—who would’ve predicted that?

2011–05–19

the evening after

I decided to have a look at my latest TMA effort today, in fact I got out my red pen and decided to mark it.

When I used to play chess there were people who wrote down their move before they played it, [this is no longer legal under FIDE rules]. I never did this, I thought that I didn’t need to—I was wrong then, and I’m wrong again—I should have checked my effort before I posted it. What do you know? It’s full of stupid slips, slips that I caught in the fifteen minutes it took to read it back.

Read your submissions, twice, before you submit folks.

2011–05–18

phew!

That was close. I managed to get my TMA away a wee bit ahead of time, still, too close.

Because I spent more time than I’d budgeted for on the final computer-course TMA I was seriously short of time for this one. That isn’t the complete story however, mistakes were made all the way down-the-line:

  1. I was lulled by the sedate pace of units one and two into thinking this was going to be easy
  2. When I discovered that vector spaces were going to be something that we need to completely understand, I didn’t alter my study–plan; up
  3. I wasted too much time on frippery—Doctor Who, Eurovision and drinking
  4. The do-the-TMA as-I-go-along algorithm broke down

The Linear Algebra block has been a shocker for me—you really needed to understand [with your hands & head] what has gone before to tackle what comes after. Because I was rushed I missed out stuff like orthonormal bases, they didn’t seem all that important. Fool that I am.

All my stupidities—I envision my stupidities as birds that are black—returned to roost for this one.

When it came to the TMA I was handicapped by the way that I had chosen to tackle the units and by my utter ignorance. I don’t suppose that I can do much about my ignorance but I can choose the way that I do my TMAs. I can come back to this block later. [Will, not can. neil]

I’m still unsure about the lessons that I should learn from this mess, but I can confidently predict that I will have to learn them again.

2011–04–27

two months of misery?

I’ve got a lot on my plate in the next couple of months. For M257—a TMA and the exam; for M208—two TMAs. Things far easier to write down than to do.

I’ve also got a life, of sorts, to deal with. I have messy workies in both my home and school, which has meant that no OU work has, or will be, done for a while. Something has to give; alas that has to be M208, for now.

My last couple of posts were about, and I hope that this came across, the fact that you can achieve good marks while storing up future woes. Often you can ‘work the symbols’ without fully understanding what you are doing—this is something to be avoided. Mostly.

Ach! That weasel-word: mostly. Sometimes the wrong way is exactly the way that you should go.

dealing with the pressure

An important learning skill is knowing when you are out-of-your-depth. Such moments come in a myriad of forms—usually work-load or comprehension related. At such moments you need to assess your resources and make a sensible plan. Most of us just panic.

I’ve been lucky, I started off down my current study-road with the web-apps—six intensive twelve week courses that taught me many things about my learning processes. The most important of which was the concept of context-switching—the futility of trying to do more than one thing at once. Do the most urgent job first.

You can always return to what you’ve skimped upon at a later date. [Missed would be better, but in most cases you have to skimp something I’m afraid.]

This is what I intend to do with M208—skimp. Not optimal, and not something that I’m happy about, but you have to be brutally realistic sometimes—trying for too much may get you nothing.

how far am I behind?

Not much actually, but it’s an important not much. For a long time now I’ve had the slight worry that I’ve been coasting along with this course because of previous work. That slight worry is beginning to firm-up into a certainty. I’m beyond, ‘the fields that I know’ and need to tread carefully. By which I mean I need to work harder for my understandings. Which takes time, and can’t skimped forever. But perhaps for the next TMA, at least, it may be.

2011–04–25

vector spaces

Mostly I gibber about me here. I gave up trying to be a knowledge-portal long ago [once-upon-a-time this site pedalled dodgy JavaScript]. All you now get is me and my flaws, rather than a wrongness about something that I don’t understand. But sometimes we need to talk about what I’m learning I’m afraid. So, Linear Algebra.

With the above warning we should be fairly safe to talk about this.

Today I spent in the garden digging earth and watching my wife plant crops, cider was drunk and soon so was I. But seated in the sunshine with a pint of the orange necter in my paw, I had an insight—I need to redo a unit. Even if it leaves me behind the road map.

Vector spaces. I think that I understand them, or rather I thought that I did until I dipped into the fora where people were asking the questions that I would have loved to have been clever enough to ask. Everyone can learn, the difficult bit is to use what you know. I don’t think that I know and I can’t ask any sensible questions. I’m effectivly ignorant.

Perhaps I should spend less time drinking cider in the sun and more time at the paper-and-pencil-face? Nah!

The watching-point is when you’re absorbing the knowledge and not asking the questions, that’s when you need to do whatever you do when you have this problem.

Me?—I intend to eat the beans that I planted, sometime, get sober and do this block again.

2011–04–18

roadmark

The other day, whilst making a token wife-appeasing effort at tidying up my ‘stuff’, I came across one of my old notebooks and one of my MST121 TMAs. The notebook I’ll talk about elsewhere, but the TMA was interesting. [It was the whole-course TMA04.]

This bit may seem like a non-sequitur but hopefully it’ll all come back together at some point…

I got back my TMA02 result the other day—my cunning, ‘do the TMA as I work through the units’ plan seemed to work. Although this was Group Theory—something that I’ve previously studied on my lonesome and is a part of maths that I particularly like. The first real test will be the Linear Algebra block—something totally new; although, already, I think that I like.

There’s another, potential, problem with this method—cherry-picking the knowledge to get a good TMA result and not developing a proper grokking [the complete understanding]. I was already uneasy about this, but chatting with Chris F. at the last tutorial turned an unease into a real worry—am I skimping on understanding for marks? [Chris’s latest post also resonates, although it might not be exactly apt for what I’m trying to say here.]

Into my internal debate dropped my old TMA—which was, and I suppose still is, crap—I’d do it better now. What does that mean? I’ve improved? Yes, I’m a much-much better mathematician than I was in them-there days; looking at it now I found places where Ford, my tutor, wasn’t hard enough. I’d have been harsh in the ultimate. I now work smarter, harder and longer—if I didn’t I’d be a wee bit devastated. We do this to improve ourselves; if we don’t, what’s the point? I’m beginning to acquire some of the needed-skills. Perhaps, more importantly I’m beginning to dimly discern what the needed-skills are—

  • A broad understanding of the subject being tackled
  • An ability to commmunicate that understanding
  • An ice-hard rigour that won’t melt in the sun
  • A firm grasp of the techniques of proof

None of which I have, so, although, I’m going to keep up my current TMA algorithm I’ll be watching me closely.

2011–04–02

choices

This week it has dawned upon me that my next choice of courses may commit me to a path. Thus far I’ve been doing the compulsory courses, now it’s me-choice-time. Which is the problem that I’ve been swithering over—who am I?

I have two problems: computing and maths. Actually I have three problems, I have to decide whether to focus mainly on my maths, or mainly on my computing.

Because of the way things work, come October I will have two-hundred points—an even number that is odd. I have the choice between six thirty-pointers or five thirty-pointers and a ten-pointer. [I could wait until the new sixty point Further Pure Maths becomes available, but I’m avoiding that option.] I’d choose to do the six thirty-pointers if money wasn’t a worry and I wasn’t faced with an OU hiatus between October 2011 and February 2012. I can’t be having that—so I need to do a ten-pointer come October.

The ten-point maths option is The Story of Maths, which appeals, but… I think I’ll go with Robotics and the Meaning of Life. I have my big box of Lego and the subject appeals anyway. All of which suggests that Natural and Artificial Intelligence should appear on my curriculum at some point. We’ll see.

So, three level three maths courses, or two level three maths courses? The sticking point is Software Development with Java, which I would have to do if I wanted to do Software Engineering with Objects. This is a decision that I’d like to put off.

Maths I enjoy, though I’m probably a better programmer than I am a mathematician. But I’ve realized that I’m never going to create that ‘dream application’; I’m never going to do it for real. Sad, but one must be realistic. So, although it would be nice to know, I’m never going to need to be intimate with design patterns and UML. There aren’t too many computer courses on offer that tick all my boxes.

So I’m going to do something wild. I’m going to do Groups and Geometry and Topology next year. There will be no computer course.

The decision has been a difficult one—no computing! I’ve been heavily influenced by my mate Chris F. As he wisely said in a comment to one of my posts, We’re not here for an easy ride. Always listen to your mates—I’m doing maths because of what someone once said to me.

Sometimes the only right course is the wrong one, off to sign up before I change my mind!

2011–03–30

TMA away

In a most unusual way—I handed in at the tutorial. Which provoked some, deserved, derisive comment from the group. Although nobody actually punched me, which I was slightly disappointed about. Why, and how, was my TMA done so early?

The how is that I changed the way that I work—I did the TMA as I worked my way through the units. Normally I do the units then I do the TMA. The why is that I’m feeling that…what am I feeling?

Hard question. This course doesn’t exactly worry me—but it does. A sentence that needs explaining I suppose. As soon as I saw M208 units, in fact before I saw then, just lifting the box informed me that I was dealing with weightiness. I knew that it wasn’t enough to say my usual, “I’ll be more organized”, this time I really was going to have to be more organized. So I’ve been watching me carefully, I’ve not been much pleased with what I saw—I wasn’t learning well. I was disorganized.

While I didn’t exactly fail the first TMA, I didn’t do as well as I should have. We’d covered most of the material during MS221 and the new stuff shouldn’t have been too hard to grasp. But I was weak in several areas—there were bits that I didn’t ‘get’, and I knew that I didn’t really get them. But instead of working at my weaknesses I tried to blag my way through. Maths is all about rigour and understanding, there aren’t any shortcuts—you have to know and be able to explain. You can’t pick just one. I was trying to explain without fully grokking.

So I came up with the do-the-TMA-as-I-go-along plan. With block one I felt like I’d sleep-walked through the units, “I’ll get it eventually” I said to myself. I didn’t. I wasn’t focused enough, I didn’t work at understanding enough, I was lazy.

Doing the TMA as I go along [with the intention of handing it in at the tutorial] seemed to make me work…not exactly harder, but better. It stopped me rushing past potential problems—I had to fix things there-and-then. I couldn’t leave it with the intention of coming back later—it had to be fixed before I could go forwards. The questions had to be done, and not done by cribbing the examples, which I’ve done in the past, to my shame.

I feel that it worked, but this was Group Theory and I haven’t got the marks back yet. In the mean time I’ll try the same methodology for the Linear Algebra block [matrix-diddling, mostly]. That will be the real test—can’t be doing with matrix-diddling.

2011–03–11

hitting the books…

Is what I’ve been doing this week. Mostly GTA, but I did have a few side-dabbles into Halmos’ Naive Set Theory and Patterson & Rurtherfords’ Elementary Abstract Algebra—a boy needs some down-time! Well, down-time may not be the right [hyphenated] word. Both books are just right for me—slightly too hard for my present knowledge, and therefore exactly what I should be reading—if you’re reading what you already know and understand you’re reading wrong. If you’re reading what seems like gibberish, same thing applies.

Things have got hard this year. It’s not that mathematics has suddenly got more difficult for me, it’s that we, now, have to be a wee bit more careful about how we write our undersatnding down. I should explain that: what we once got off with will not, now, cut-the mustard. We need to be flawlessly exact. And I’m sloppy. So I’ve changed my approach.

As I’ve been working my way through the course-books I’ve been doing the TMA. Has this helped? Well we’ll see, won’t we.

The point is to try different approaches—if we don’t try something different we’ll never know. There’s a trouble with different, there’s a trouble with ideas—they’re often rubbish and wrong. My different and wrong especially.

Still, this whole thing, this journey of ours, this learning, is an exercise in putting our head onto a table and hoping that others won’t cut it off. So I’ll just bringe forward, a panzer of stupidity, in the hope that ‘this stuff: goes into my head

It’s a gamble on the best estimate, but if we don’t want to roll the dice…?

Why are we here, in the trenches of learning.

I have two thoughts reading this back [strange, but I do], why am I wasting time with group theory when I could be learning set theory? What, in the name of Hades are Euclidian Rings?

2011–03–05

stop, start

Has been my maths progress on M208 so far. Every so often I do a lot of maths, perhaps eight-to-ten hour’s worth. And then nothing for a week or so.

I’m not sure that this hurts me, it’s not as if I stop thinking about maths, it’s just that I’m not scratching symbols upon paper or posting in the fora.

When I started back on the maths with MST121 I had to work pretty hard, MS221 was easier—I was up-to-speed with maths and I felt in control. This course… I feel comfortable, perhaps a bad place to be. I submitted my first TMA this week [well, the second part of same] so we’ll see how I’m doing when that comes back.

I’m going to struggle to explain this, so I apologize to the people who I’m going to use as exemplars if I get them wrong. There are different types of mathematicians, and they work in different ways. Here are our candidates:—

  1. Nilo—Top 50 Maths Blogs!!
  2. Chris
  3. Me

There are other [undergrad.] mathematicians that I could have included (Phil W. G1, Marcin T. et al)—I restricted myself to those who have web sites that I know of || those whose methodologies fascinate me.

Nilo is interested in everything maths, his blog is the first site that I open every day. I often don’t understand it, but I always enjoy it. He’s a great man for tools. I suspect that he's a very visual mathematician.

Chris, who I’ve met, is just frighteningly brilliant. A philosopher, musician, mathematician and an all-round giant, is interested in the practical application of maths. Often I don’t understand him either. He understand how maths works. A man who is comfortable with symbols?

Me. I have a feeling for this stuff, and because I come from a programming background I often see the algorithm before I see the principle. And because I’m a programmer I know when to go away—those essential fag-breaks that allow the knowledge to seep into the head. Hence the stop/start. Don’t know about me.

So, what am I trying to say? Nothing really, I suppose—how you tackle your maths is down to you. But be aware of what-type of mathematician you are underneath.

2011–02–18

back…

With an appetite. I was a wee bit worried that I’d lost interest in maths—no, that isn’t the right word—focus may be a better one, still isn’t right. Whatever the word is, it may not exist in English, doesn’t matter—my maths head is firmly back on.

The catalyst for this sea-change? The fora. I’d been avoiding them, but I decided to dip my toes the other day, and, what-do-you-know I got hooked by bigger fish.

Over my OU career [in the sense of rush uncontrollably] I’ve been involved in many fora, none of them as good as the M208 one. People, wonderful people, there, love maths, and that’s infectious.

I, again, have the maths disease. Good.

2011–01–29

first day of the course—first tutorial

I was shocked by the shocking state of my maths—I actually didn’t spot that f(x) = -2x + 1 has a negative gradient, I sketched it in entirely the wrong way, I knew that when I was fourteen. To make matters worse I spoke up and exposed my ignorance. Tut.

Still, thus is the way of tutorials—you can sit back and appear knowledgable, or you can be stupid, ask, and have a chance at, us all, actually learning something that we may not had learned otherwise. Because I’m me I’m always willing to make a fool of myself, even if it doesn’t help even me. What would help if I wan’t being so utterly asinine. It would also be nice if, just once, I asked an intelligent question.

So, now that I’ve made an idiot of myself am I going to crawl off into a corner and itch myself to death? Hah! If you think that you don’t know me. Making an idiot of yourself isn’t terminal—it’s an essential. If you don’t feel stupid you can’t learn. What on earth does that mean? I’m not sure, but I think that it’s essential that you have to realize that, essentailly, at bottom, you’re a twat.

But it’s important to realize what type of twat you are. Slowly I’m beginning to get in touch with my inner twatitude. That’s one of the side-effects of the OU—you become more self-aware: I’m smart, but I ain’t that smart and I jump to ill-judged conclusions that I find hard to shake-off. Other flaws are waiting to be exposed.

people

Being an OU student is often…I was going to say lonely, but that isn’t right—we have the fora, so I’m going to say disconnected. [Although that isn’t the right word either.] Tutorials are an important part of ‘our journey’—they’re the place where you get the chance to meet people who are going through what you’re going through when you’re going through it.

Yesterday it was a real joy to meet a lot of old friends and to make some new ones. [It helps the bonding process that we go to the pub afterwords.]

In particular, yesterday, I met a fascinating guy—a fellow smoker, who works in a similar-type, low-end, job of-the-type that I do. He lives near me, so we walked home together. [At this point I should mention that he is doing a philosophy MA]. We walked home through the Meadows, chatting, waving and wind-milling our arms, flapping our mouths and generally being talking-mime-artists. I worry about what we looked like. Two people discussing whether maths is real, abstraction, Gödel and the consequences of his work on philosophy and if there are any real truths just have to look odd. I suspect that we did.

We didn’t argue, we discussed, but the one thing that we did strongly agree upon was the importance of the OU in our lives. Without it we would have had no access to higher education. Both of us skirt with poverty in order to enrich our minds. Without it we wouldn’t have met each other—we would have little-or-no access to education. Without it we’d be…you tick the box

Without the context of a tutorial we would have never shared what was in our minds—we must have met before, he works in a shop that I visit often—but there was no way for us to interface. This trail of ours will have savage lows [ even the greats stutter sometime.]. But, but, while there are lows, the highs are savage too—striding through the Meadows, with the world all open, and large, and green, and grey, with the wind scaring your face and discussing the fundamentals of the world, with a mate, that makes it all worthwhile.

I’m blessed.

2011–01–27

not doing maths numbers #12 & #47

I’ve been ill and other excuses but that ain’t good enough—everybody must do maths. Well at least I do. The fora have been open for days, not dipped into [properly]; the books have been here since the eve of last year, not looked at [much]; all my mates are posting and I’m not riposting [at all]. So what am I doing?

Mostly lying in my bed sweating, hurting and doing M257 stuff.

This has alway happened to me—the two-course flip-flop; I partition my time in a pseudo-random fashion. Just just-now I think I need to focus my focus on the computer course. But there is a nag at the back of my mind—perhaps maths could help me? After all essentially what I’m doing are affine translations. But I can’t see the join.

But I see the potential join and I have a tutorial on Saturday. The sun will come up, the world will go on, but in the end…

Everybody must get mathed!

2011–01–07

getting to know you M208

It always ends up thus—me squatted on the floor messing around with cut-out bits of paper. Yup, Group Theory and Symmetry. This time it’s a wee bit worse as we were supplied with, what’s the word, nets for some Platonic solids. You know the sort of thing, don’t you? So glue, knife, cutting mat and multitudinous uses of the word ‘bugger’ will feature here, at casa anderson, soon.

After no thought I decided to just tackle this course as I always do—a fast, shallow, reconnoitre of the terrain. So I’m already getting into the Group Theory. Hence the bits of paper—we always start with symmetry when it comes to Group Theory. But…

the introductory block

Was interesting. Mostly it was about stuff that we covered during MS221, there was new material. Enough new material to be getting along with, enough to get my brain re-engaged with maths. Rather too much re-engaged—with the fervour of the learner I began posting tosh [that would be a different type of tosh] on my computer course-forum. Enough to make my mornings slightly more hellish.

I keep a book, usually of an improving nature, by my bed at all times, so that when I wake I can, immediately, read it, rather than think about all of the wrong things that I’ve done yesterday. Until after my breakfast.

Still, I suppose that I should appraise you of what’s involved. The books are labeled thus:

  • Real functions and graphs
  • Mathematical Language
  • Number systems

None of which precisely describe what’s ‘in the tin’. Let’s take Number systems:

  • Real Numbers–ok
  • Complex Numbers–ok
  • Modular arithmetic–well…I suppose so
  • Equivalence relarions–fuck off

Then what do I know? The best thing about learning stuff is that you become aware of your ignorance. I shall always be stupid and I glory in that. But I’ll retain still the right to nit-pick at you.

Karma—snow is falling. Heavy snow.

2011–01–03

ratcheting it back up again

Over the past couple of days I’ve been trying to get myself back into the maths-groove. Shockingly I’ve allowed my maths-brain to atrophy rather badly, I haven’t done any maths since October, and it shows. Worse, perhaps, is that my hands seem to have forgotten their business.

Mathematics, like mechanics, or programming, or writing, isn’t something that you can learn just by reading about it—you have to do it. And when you do some thinking-things your hands learn alongside you. You’ll then rely on them to ignite the mind—you sharpen the pencil, you shuffle the paper, you moisten the throat, you apply pencil to paper and the mind kicks in—the hands have started their moves.

Well in my perfect world perhaps. Still I’m a firm believer in the value of externals when learning. By which I mean the props that we use to support our learning. For me, for maths, this is mostly a pencil and a lot of paper—to teach my hands. Others use other tools—visit the OU blogs and you’ll find a hundred variations on a hundred techniques. But not everybody can use tools—how does Steven Hawking study? [If he has to.]

Well, maybe he doesn’t use tools/props now, but I’m willing to bet that he did. Perhaps he now uses ‘mental’ tools, tools that us, mere mortals, are unaware of. As we grow, the plastic bucket-and-spade that we, once, took to the seaside may no longer suffice; the sharper, bigger tools that were once denied us may be what we now need.

David Bronstein, the nearly world chess champion famously used to analyse using the display board, that’s because he was one of the best chess players ever, not a good reason for us to do the same. And in this case I know he didn’t start off this way—you have to get very-very good at chess before people care to start following you on a display board. This couldn’t have been his teenage behaviour.

One of the great moments of my life was playing a chess-congress in Livingstone in the early nineties. Such things are such things—things you do to play chess. But when Dad and I arrived there was a buzz—Bronstein was here, we were in the same room as a ‘Great’. He was a truly gracious man who spent many hours going through others’ games. Personally I didn’t learn anything—Reimann drops into your maths club? Talk well-over my head.

So we must always be looking at our toolbox—proverbially if all you have is a hammer… Perhaps it’s time to be looking at mine?

2010–12–31

finally…

After journey worthy of a Homeric ode the books arrived. So, now that I’ve got them, what do I think?

Yesterday we were entertaining the in-laws so I didn’t have a chance for an in-depth perusal—all I can really say is that there seemed to be a lot of them—I strained my back lifting the box. [There were videos and audios too, but I, almost, never use these so we’ll ignore them.] The one thing that I did notice was that there seem to be two blocks devoted to group theory, good. Nay, great.

I like group theory [my maths hero is Galois] but I only ever managed to get so far with it—at heart I’m lazy. No, that isn’t quite right, but without some kind of outside pressure I can’t/won’t/don’t focus. I get distracted, I wander off.

One of the hardest things in this life of ours is to be honest about ourselves. Indeed should I live to be a hundred I’ll still find myself telling myself lies about myself. But I am beginning to become honest about the way that I learn—the OU does that to you. [Take this beautiful piece of honesty—one of the most thought-provoking things that I’ve read in a long while.] I now know that I have to trick myself into learning. Think about that. Bit odd isn’t it? Trick yourself! Can you imagine what it’s like to live inside my head?

But then we are all odd in our own unique way. One of the great joys of this online-learning life of ours is that we get opportunities to inspect the oddnessess of others.

I’m chaotic, others are different and they fascinate me. Let’s take, for example, Chris and Nilo, they’re both have distinct styles of learning and I’m a wee bit in awe of both.

For the sake of a point I’m going to eschew subtlety here—but you’re grown-ups, you can ignore my abstraction of truth and make up your own minds.

For Chris it’s the book. I have a collection of ‘old’ maths books but for me they’re never the starting point—they’re merely a source of exercises. I couldn’t take a maths book and work my way through it. Chris seems to do this without effort.

For Nilo it’s the tools. [This is a gross calumny, Nilo is one of the most probing (that’s not quite right—universal?) mathematicians that I know of, but point!] He submits PDFs for his TMAs. That’s dedication. He watches videos, he has maths applications, he approaches maths from every angle that he can. I can barely bring myself to use Wikipedia.

They’re different for me, I’d like to be like them—but I’m not. But we all share the same fountain of joy: maths.

That’s the lovely thing.

2010–12–22

the books came…then…went

They almost got here—so close…but so far.

Those of you who follow this tosh will, at this point, be expecting an acerbic rant about the incompetence of delivery drivers—not going to happen. Why not? Well because this ‘OU journey of mine’ has made me a better person. Actually not so, but I have learnt some things.

My house was built after the other houses in my street and is numbered strangely. What, I think, happened was that the driver had lost the ability to see my house. He’d made an assumption about the way the road was numbered, one that was normally reasonable, it was just that it was wrong in this particular case. At which point, to his mind, my house didn’t exist; because it wasn’t where it should be. This happens to me all the time—I create a mental-picture out of who’s frame I cannot break.

I started this ‘OU journey of mine’ with the intention of learning how to, properly, do what I did for fun—computing; to be able to understand what other people said/wrote about said subject and to stretch my flabby mind. The mind has been stretched—I just hadn’t noticed how far. Most of that progress has been down to the rigour imposed upon me by mathematics.

One might think that creating programmes/applications would impose the same discipline—doesn’t. You can write arse-code that works, all-too easily, alas.

During one of my early maths tutorials someone asked a rather searching question [although I didn’t realize the searching nature of it at the time] of the tutor. I was surprised at what happened, his face went blank for about sixty seconds before he gave an answer. I now see that he wasn’t searching for an answer—he was searching for a way to explain it to us! He saw what we didn’t—we weren’t ready for the real answer. That’s how I want to be.

I may never get there, for now the road itself is enough. There have been a couple of times this year where I gave a considered response where once there would have been flim-flam bullshit. I’m prouder of that than of anything else I’ve done this year. More of those moments are more important than the letters that I might get after my name. I suspect that it will be easier to get a degree than to be who I want to be.

Kay [in system] would say ad astra, but for me it’s always going to be per ardua. To tell the truth I’m looking forward to the per ardua.

Hello M208!