Friday, September 28, 2007

In want of a peaceful title: Hi

The more provocative title has to be 'unfair lectures'. Do I proceed lightly, trying not to step on as many peoples feet as possible, or do I just jump straight in?

Ha, my original title may have raised some eyebrows, but rest assured this may be a somewhat trivial matter. *cue weird look for using the 't' word* This post is going to be messy, but it will basically be a round up of the weeks lectures. I know it has been a week (as the Tweenies constantly remind me), but I have a feeling that Friday's are not going to be enjoyed as much as they used to be, and probably Thursday's too. Thankfully Monday to Wednesday will be nicer.

Today, after Bella and myself bought the book Complex Analysis for a module, I was asked something along the lines, 'Why complex numbers, and why i?' [or something similar]. We were sat on a bus, which I was forced to take (exercise: what is wrong with that statement?) and I tried my best to keep my voice down as I constructed a reply. Ignoring the weird looks, I started with the Natural numbers or the counting numbers. This was all obvious to Bella but I couldn't resist. From the naturals I went to the integers - why because we needed negative numbers. Then in came the rationals, and finally we had the real numbers. All the time saying that we extended the sets for completeness.

The way I used to see it was there is this huge line that is infinitely long. We basically want to be able to give a 'label' to every point on that line. Thus enter the integers, rationals etc. but this doesn't correspond with imaginary numbers. As I learnt the other day, and try to recall, complex numbers are 4-dimensional? The only way I was able to introduce i, was that we wanted to know the solution to the equation x^2=-1. However, even I went onto wonder why 'i' in particular? Why not another letter. Yes, that question is quite silly compared to the 'what's so important about complex numbers?' question. I know that they are used a lot in mechanics and oscillations, but isn't it remarkable that an "imaginary thing" has such a huge role?

I obviously am not well versed in such answers, so I did point out to Bella not to take my word for anything! Nevertheless the conversation was interesting. After the free bus came to a stop, I found myself discussing lectures with Bella.

I told Bella that lecturers are story tellers - they tell us a story. The idea is to communicate this story to us. Different lecturers are different story tellers. Some talk to the audience. They ask the audience for a number. No one replies, so they ask again. Still no one replies, so again they ask until finally someone shouts the number three. These story tellers make you feel part of the story. They write on the board and you copy down what they write with them. All the time being involved with this story telling process. If they pause for a few minutes, and discuss other matters you listen carefully. If something funny is said you laugh; consider it an intermission during the tale. (I later repeated all this to Fizz as we walked to our final lecture today too).

Everyone has their own outlook on things, but I really enjoy going to lectures because of this story telling process. I love writing notes. I love writing notes with the lecturers. I love listening to what the lecturers say when they are NOT writing. I am able to listen, because if the lecturer is not writing, then that means I am not writing as well.

I think my feelings during the holidays were quite clear. I was itching to get back into the action - to get to some solid maths. I was desperate for that challenge. All my modules are challenging indeed, but the ones that tell a brilliant story are Algebraic Structures and Real Analysis. We have had two lectures for them two modules, and it feels like many more. The story has started beautifully, and I am desperate for it to continue in the same way. Where am I going with this? Where indeed.

I don't know whether to apologise for being a weird learner, but it is something that just is. I like pictures and they greatly aid my learning process. The final thing that made the epsilon definition stick last year, was the graph drawn in the example class. The graph of epsilon and any natural number depending on it. I can survive without pictures, for some things can't be viewed in such ways, but they aid me greatly.

Unfortunately, as I continue to state: I am 'poor' at multi-tasking. I believe that it is better to do one thing and do it properly, rather than do a million things, of which a few may be improper. This may be a problem for me later on, but I don't know whether it is possible train one to 'multi-task'. I don't care about the future problem, but about the current one.

If I am copying notes from a prepared acetate, then I am sorry but I am not listening to a word you are saying. Fragments of the important story you are communicating to me, are picked up, but it is infuriatingly nonsensical. As I am 'copying' parts of the story i.e. not 'writing' the story with you, I once again don't take anything that I copy in. The process of this note copying is like being asked to remove all the peas from a pea pod. You just remove them - not caring about a thing in the world and not thinking about what you are actually doing.

Sigh. This is all connected to my inability to copy and listen at the same time. A possible solution is not to copy things down, but just to listen; however that is only half a solution. When I am writing the story with the lecturers, my brain actually tries to acknowledge the story and tries to process it. That is 90% of my learning - it all happens during lectures!!! If you asked me about my lecture at 12pm-1pm I will be able to tell you that we discussed binary operations. We had three 'exotic' examples (which I found cool!) and some natural ones. Today we talked about the identity element in a non-empty set S, and some associativity properties. (a^n= a*a.......*a*a, n times). We did a neat little proof of why the identity has to be unique as well. Ah, we also drew a multiplication table for modulo 3 and noticed that a binary operation is commutative if and only if the table is symmetric about the diagonal. And no - I have not looked at my notes since the lecture.

If you ask me about the real analysis lecture I will dodgily be able to communicate the idea of a neighbourhood, and that of a deleted neighbourhood. When I saw dodgily I mean the lecture was on a Tuesday or Wednesday! However, I am sure that I could tell you about the limit of a function f, as it tends to a, being L iff.....(for all epsilon more than zero, there exists a delta such that: 0 <|x-a|< delta, implies that | f(x) - L | < epsilon). BTW - don't take my word for that since it is probably mumbo jumbo and has a mistake somewhere. I haven't defined the function properly and made other things clear.

The rather dragged out point I am trying to make is that I am really following these two stories continuing. Albeit the second one being slightly above board, I am really looking forward to it. I love these two stories already, namely because of the story tellers. Already they are slowly bringing me back into everything.

I really miss Professor Dold. He wrote a magnificent story using the OHP. Yes, you read that correctly - a story WITH the OHP. What did Prof. D do? Well he wrote on the acetates and we then copied down what he wrote with him. That's fair enough. If he had just stuck the acetate up with everything written on it beforehand then that wouldn't have been very fair in my opinion. I don't mind lecturers using the OHP, however it would be fairer if they wrote on it during lectures, rather than preparing stuff before hand and sticking them up. Maybe diagrams and other things are an exception, but is it fair that we have to try and quickly copy down everything, whilst trying to listen to what is being said and then process it all? (Yeah, I know life's not fair, but when has that stopped people from complaining?)

I know that the Roscoe theatre is awful, and we are doomed with acetates. But I just want to write the story with the lecturers - is that too much to ask? I suppose it is, since I tend to be the only one voicing these opinions. I have a certain way which works for me and I try to achieve that way. More than white boards, I hate the OHP. And I also hate notes being put up on the internet before the lectures. That kind of spoils the story if that makes sense... I know shoot me - I am a lame student who makes a big hoo haa over small things. I can't help it. The two best story tellers in the university and the two best stories that I am going to be told, are being overshadowed. And that by the OHP. Isn't that enough to make any bean angry?

Surprisingly I kept my cool today. I could really feel my blood boiling and causing me internal physical problems! The urge to exclaim loudly was too darn strong, yet I kept it bottled up. What to do beans? I suffered in a similar manner in the first six weeks of the linear algebra lectures, but I think even they were brilliant compared to now. Then the problem had been having to copy everything from the book into my notes. Now the problem is that I am copying things down but not actually taking anything in whatsoever. Please story tellers - can't you write the definitions or notes on the OHP during the lectures WITH US, rather than before hand.

I want to go to lectures and learn and be happy about it. I don't think that is going to be the case for two lectures (on Thursdays and Fridays). A lot of definitions were put on the OHP, and I honestly can't remember any of them. Only fragments like the words: open set, closed set, limit of a point... are coming to mind. And for the other lecture - I only can remember the double integral.

Maybe the OHP is used to save time? I doubt that for we did finish five minutes early, so there was time to spare. I think this is a lose lose situation for me, but I am just going to bite my tongue for another week.

Sometimes I wish I was born 20 or so years ago. All this tecnological advancement is making my hair go grey. (Well if we didn't have OHPs I wouldn't be having this problem, and if we didn't have the Internet then no notes could go online before the lectures! Ha - imagine life with no Internet. I was joking at that suggestion BTW.)

The week did end on a dull note, but it was really great to be back. I now realise why there are soooooo many people on our course this year: we have joint honour students and people re-sitting their second year too. Normally if you came late, you could find a place to sit down - now that is impossible. And for a blind bat like me, I can't really see much the further back I go. Also, before I do shut up I have a tip. I would definitely recommend that you sit near the front of the lecture room. Due to me being early 'but late' at the same time (i.e. I went after ten too but before o'clock!) I had to sit at the back in the mornings lecture. At the front you have to look straight ahead at the lecturer, and so you focus on what is being said. However, at the back the whole room is your distraction and refocusing yourself is a pain. Note to self: aim to get to lectures before ten too if possible! Do you think my story telling analogy is a fair one?


Jake said...

However, even I went onto wonder why 'i' in particular? Why not another letter.

Why 'i'? I guess because the i is standing for 'imaginary'. You could use any letter you wanted but sometimes for important constants it becomes the convention to use a particular letter/symbol etc. so that when reading unfamiliar texts etc. the write doesn't have to explain to much of the notation i.e. convention makes it more apparent what the writer is trying to communicate. e.g. When you see a diagram of a curve representing a function of one variable, the author can refer to the x-axis/y-axis etc. without defining which is which because she knows that the reader will know by convention which she is referring to.

I think there was a slight mistake in the post too. Complex numbers aren't 4-dimensional but are two dimensional. In fact, they are often defined without reference to root(-1) in terms of ordered pairs (a,b) where a,b in R.

I think you may be referring to curves of functions of one complex variable i.e. f: C --> C which are four dimensional as you are mapping a two dimensional plane onto a two dimensional plane much in the same way that curves of functions of one real variable is two dimensional as you are mapping a one dimensional line onto another one dimensional line.

Btw. the story telling analogy is in my humble opinion a very good one. I think the key is motivation. Not so much motivation as in motivating people to study (although that also helps) but more in terms of motivating the theory being discussed. e.g. Why particular things are interesting and why we study certain objects.

e.g. in group theory, one could just start off and give the group axioms and say that we can manipulate them in certain ways and prove various theorems but it is nicer if it is told as a story and we can understand its origins in permutation groups of roots of equations and the relations to symmetry etc. etc.

beans said...

Thanks for the correction - shows how much attention I actually paid! I suppose i does make the most sense, but before university I used to use 'j'. It's quite 'weird' but you can also ask why '2' or why '-1'?

Yes, motivation is the right word - you got the nail on the head. If the lectures can motivate you to study, then that is indeed a bonus.

PS: Sorry for the disjoint reply; it's a Monday!

Anonymous said...

'j' is also common for people with a background in physics or electrical engineering. In those contexts 'i' is often reserved to denote current. There are often differences in some definitions across disciplines. Physicists often define the Fourier transform and its inverse in a slightly different way too (it is all done in a consistent way however). Another example is if you're educated in continental Europe you will probably use Rot instead of Curl in vector calculus. These changes keep things interesting and stop us from assuming uniformity and leaving out definitions ... so ultimately its a good thing ;)

beans said...

They definitely keep things interesting! I like the sound of rot too.

You make a fair point though, that it does teach us not to assume certain things without defining them properly.