Readjusting Learning Methodologies

November 18, 2015 Adrian Leave a comment

I've just finished reading Lara Alcock's book on how to learn about Analysis. Or rather, I've finished reading the first part, and the part on the real number system.

Overall, the book has led me to reconsider my current learning technique. So much so, I've compiled a list of steps to follow depending on whether I read about a new definition, theorem or proof.

In turn, this has made me realise I may benefit from starting my main book on Analysis again from the beginning, but applying these new steps as I go. After all, I am still only at the beginning (kind of), and I don't have any kind of deadline looming over me (which is really nice). Overall it seems like the perfect opportunity to try out some new learning methodologies!

Out of the handful of additions, there are two really big changes for me.

The first being mind maps. As I go through my Analysis, I'll be creating a mind map of concepts, seeing how one builds on another. I'd tried to use mind maps before at university and they'd largely proved completely useless. Here, however, mind maps appear to offer a perfect way to visualise the building of concepts into larger concepts. Here's the beginning of my first mind map!

I'm using draw.io as the tool of choice. It seems flexible enough for what I need it for, it can save as XML, and it supports mathematical notation! (mathjax latex formatting I believe)

The second big change to my learning involves learning by self-explanation. This technique, mentioned in Lara Alcock's book, appears to be one of the key processes involved in truly understanding and appreciating Analysis. You can find out more about self-explanation training from Loughborough University's Mathematics Education Centre website. My difficulty here will lie in concentrating on actually doing self-explanation, rather than just paraphrasing (turns out, it's a very easy trap to fall in). So long as I do it regularly enough, self-explanation will be more likely to come to me naturally.

Expected result: More effective learning and better notes!

Analysis, Text Books

Books For Understanding Books

October 30, 2015 Adrian Leave a comment

I couldn't help myself.

I've bought another book.

The reason I bought it is to help me with the book I'm currently reading... but it's not as bad as it sounds.

The last Analysis query I had, I posted to the OU forums. As usual, my question was answered almost instantly in a concise and understandable way. Awesome!

However, there was another post a few days later suggesting that I read "How To Think About Analysis", another book by Lara Alcock.

Initially I thought it would be a bad idea to drop my current book about Analysis, and pick up a new one, but it turns out that after the first 50 pages, this book can be used as a companion to learning different sub-topics of Analysis. Specifically, after the first 50 pages, the remainder of the book is split into sections about Real Numbers, Sequences, Series, Continuity, Differentiability, and Integrability. Each sub-topic appears to be about 40-ish pages long, and can be read just before or during the actual study of each. Bite-size!

Although I've not quite finished the first section yet, I've already added several new strategies to my set of learning techniques. Despite the fact that the knock-on effect is that the learning process may become slower, the idea is that my knowledge and understanding of the material will become much much deeper.

I'll try and give another review once I've completed the first part, the section on real numbers, and related it all to where I currently am in my larger Analysis text book.

Maths, Proofs

Reading Mathematics

October 25, 2015 Adrian Leave a comment

Still working through my book on Analysis (which I will be until July 2016, so I should probably stop mentioning it...), but I recently came across another proof that I had difficulty understanding. I had to reach out to the mathematics forums in the end, but after getting a reply and working through some further steps that they mentioned, everything fell into place.

I felt really good that I'd finally understood the proof, but I also tried to work out what I could've done to push for that answer myself (so hopefully next time, I won't have to post a question to the forum).

I realised that I might not be reading through the mathematics effectively enough. After reading through Lara Alcock's book I realised how important it is to make effective notes whilst reading through "all the symbols". When reading through proofs I have this nasty habit of reading them like a novel, keeping this story of logic in my head... and then very quickly becoming confused because I didn't see how you could logically progress from one sentence to another.

This sounds really simple, and almost obvious, but I think all it really takes is to sit down with a pen and go through the mathematics of the proof in gritty detail, liberally re-arranging things as you go. It's worth mentioning that this is perhaps quite a different act than just "taking notes".

Being completely confounded by something only to solve the issue entirely on your own is enormously satisfying. The hope is that if I stay mindful and remain aware as to when to write the right kinds of notes, not only will I be able to solve more complicated problems on my own, but also in time these seemingly large logical steps will become second nature.

Anyway, doing independent reading of mathematics is proving, generally, to be really satisfying. Let's see if I can become a more effective reader...

Analysis, Maths, Proofs

Proof of Inequalities by Mathematical Induction

October 2, 2015 Adrian Leave a comment

Still reading though my book in Analysis, I've come across a section on proving inequalities. I'm glad to say that all of this made sense... until I reached a sub-section on proving an inequality by mathematical induction.

As I've written previously, I find that proofs are notoriously unintuitive. In the past however, I have been particularly puzzled by the logical steps involved in proving an inequality by mathematical induction.

To explain my difficulties, let's have a look at the example provided in the book:

Prove that $2^n \geq n^2$ , for $n \geq 4$ .

If we're proving this by mathematical induction, we generally follow these steps:

Let $P(n)$ be the statement $2^n \geq n^2$ .
Show that $P(4)$ is true.
Assume that $P(k)$ is also true for $k \geq 4$ .
Show that $P(k) \Rightarrow P(k+1)$ . Or rather, show that if $P(k)$ is true, then $P(k+1)$ is also true.

Step 4 is the key step here in the proof as it shows that if any number is true, and the next number is also true, then you can apply this rule forever, and your original statement must be true for all numbers!

Anyway, lets jump to step 2. Show that $P(4)$ is true. Well if $n=4$ , then $2^4 = 16$ and $4^2 = 16$ . So $P(4)$ is true! Easy.

Let's look at step 3. Let's ASSUME that $P(k)$ is true for some $k \geq 4$ .

Now, this is the part that caught me by surprise... At this step, the text in the book reads as follows:

"So, we are assuming that $2^k \geq k^2$ . Multiplying this inequality by 2 we get:

$2^{k+1} \geq 2k^2$

,

so it is therefore sufficient to prove that $2k^2 \geq (k+1)^2$ ."

Wait, what? How is it that all of a sudden, all we need to prove is that $2k^2 \geq (k+1)^2$ ? This isn't explained explicitly in the text so I had to close the book and do a bit more thinking.

First thing I had to realise here is that the "Step 4" I've listed above requires a bit more detail... What you're actually trying to do is show that you can progress naturally from $P(k)$ to $P(k+1)$ . ie: We should be able to show that we can progress naturally from:
$2^k \geq k^2$
to:
$2^{k+1} \geq (k+1)^2$ .

Now, if we multiply $2^k \geq k^2$ by 2, as mentioned in the text, we do arrive at:
$2^{k+1} \geq 2k^2$

This is good, as we've managed to get the $2^{k+1}$ we were looking for on the left-hand side of the inequality. But the right-hand side looks nothing like the right-hand side of $P(k+1)$ ie: $(k+1)^2$ .

Here's the key though... It doesn't matter they they're not the same. We only need to see how $2k^2$ and $(k+1)^2$ relate to each other. Look back at Step 3. Part of this assumption is that $k \geq 4$ . Just as a test, let's try $k=4$ :

$2k^2 = 2 \times 4^2 = 32$
and
$(k+1)^2 = (4+1)^2 = 25$

Well this is interesting. It's looking as if $2k^2 \geq (k+1)^2$ . This is exactly what was written in the text!

But to really ram it home, what we really have now is the following:

$2^{k+1} \geq 2k^2 \geq (k+1)^2$

So.... this show us that IF we can prove that last bit $(2k^2 \geq (k+1)^2 )$ is true for all $k \geq 4$ , and not just $k=4$ we have managed to prove that we can get from $P(k)$ to $P(k+1)$ !!! This is exactly why the text in the book said "so it is therefore sufficient to prove that $2k^2 \geq (k+1)^2$ ."

I'm sure in future I'll jump on this immediately and say "oh yes, of course that's all we need to do now", but working through the derivation of why it was sufficient was extremely useful. Long-winded... but useful.

Ah, the learning process...

Analysis, Text Books

Year Four Begins!

September 17, 2015 Adrian Leave a comment

My new learning materials have finally arrived!

This is the amount of reading and exercises that I would normally have to work through over a nine-month period. Entirely doable.

However, as I've mentioned in previous posts, this module is merely to make up the credits for my first stage. Bad news is: it's all quite basic. As a result, I'll also be working through a nice fat book on analysis, to ease me in to my second stage that starts next year. As such, my nine-month work load actually looks a little more like this:

So we'll see how far I get with all that!

Flicking through the first couple of pages of these new materials, I realise that they cover some very basic areas of mathematics indeed. Perhaps even a little more basic than I was expecting. I think I'll have to keep reminding myself that this was my plan from the beginning, and just concentrate on submitting my assignments on time. This work will definitely be taking a back seat...

More importantly, over the past couple of weeks I've also been working through the book on analysis, and although I'm only a few pages in, it's proving to be really valuable; for one, I'm becoming a lot more comfortable with proofs and solution sets of inequalities.

Analysis

$0.\bar{9} = 1.0$

September 7, 2015 Adrian Leave a comment

Lara Alcock wrote in her book "How to Study for a Mathematics Degree" a warning to all mathematics students to be prepared to adjust their intuition.

Very quickly after starting with "A First Course in Mathematical Analysis" by David Brannan (see my earlier post), I have come across something that very much breaks my intuition. It turns out that $0.\bar{9}=1.0$ .

$Let\: x = 0.\bar{9}$

$10x=9.\bar{9}$

$10x = 9 + x$

$10x-x = 9$

$9x = 9$

$x=1=0.\bar{9}$

Simple, and surprising. Although also somewhat disturbing.

The equals sign tends to show that the object on one side is identical to the object on the other. In this case, they look like two completely different objects! We all know what " $1$ " is, and $0.\bar{9}$ looks distinctly different.

Of course, based on Lara Alcock's advice, my first thought was "how on earth do I adjust my intuition to make this feel completely logical?!"

After some thinking, the following helped me a little bit: Before, I considered $0.\bar{9}$ to be a number that was infinitely close to $1$ , due to the infinity of " $9$ "s after the decimal point. I now, however, consider $0.\bar{9}$ to be an infinitely accurate representation of $1$ .

For me, if I consider an approximation to be "infinitely accurate", then that approximation is the object it's trying to approximate. ie: An infinitely accurate approximation of 1 is, essentially, 1.

My opinion of this may change, but this is what's helping me make sense of this for the moment....

Bertrand Russell, Mathematicians, Pop Books

Fermat's Last Theorem

August 31, 2015 Adrian

One of the first "Popular Maths" books I decided to pick up was Fermat's Last Theorem by Simon Singh. In the book, he expertly tells the story of Andrew Wiles and how he (practically) single-handedly solved an age-old mathematics problem. I won't go into any more detail than that because you can read a lot about the story on the web (but really don't, just buy the book).

Part of what I really love about this book is that it goes into a lot of background detail, a lot of history. It builds up your understanding of why mathematics is the way it is today, and exactly why Fermat's Last Theorem was such a prestigious problem to solve.

One section that really sticks in my mind features a certain man called Bertrand Russell. Now, the more I read about modern mathematics the more his name crops up. Right at the beginning of the 20th Century, Bertrand Russell's research in logic appeared to show that mathematics was flawed. Again, I won't write down the details, I'd only end up copying Simon Singh's own words (buy the book!!!).

Although one passage did stand out in particular. Simon Singh mentioned that mathematicians obviously questioned Russell's work, and then goes on to quote Russell's response:

'But,' you might say, 'none of this shakes my belief that 2 and 2 are 4.' You are quite right, except in marginal cases - and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a metre. Two must be two of something, and the proposition '2 and 2 are 4' is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. 'Well at any rate there are four animals,' you might say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. 'Well, then living organisms,' you say. But there are things of which it is doubtful whether they are living or not. You will be driven into say: 'Two entities and two entities are four entities.' When you have told me what you mean by 'entity', we will resume the argument.

A brilliant read (buy the book).

LaTeX

An Introduction To LaTeX

August 24, 2015 Adrian Leave a comment

Assignment writing seems to be a very large part of studying a part-time degree in Mathematics. For my first module I hand-wrote every assignment, but with my handwriting being less than perfect I wasn't entirely happy with the neatness of the end result.

I tried numerous other ways of writing my assignments but the formatting of equations always proved to be a headache. After exploring a couple of avenues (Open Office with equation editor, and Google Docs), I decided to settle on LaTeX.

LaTeX is a formatting (or rather, typesetting) language, much like HTML in a way. For example, the following:

$f(x) = x^2$

is written in LaTeX as follows:

 f(x) = x^2

Which is nice. Of course this is a simple example, and the learning curve involved in producing your first 18-page assignment with it can be rather steep.

What I'd like to do every now and again is post a clever little bit of LaTeX I discover that gets me out of a hole. With any luck, after a handful of posts, LaTeX will become a lot less of a mystery.

My operating system of choice at the moment is Linux Mint. I like it because it's the only version of Linux I've ever installed that just worked straight after installation. No sound problems. No problems with network access. Everything worked.

As such, I started out by trying to work out how to get LaTeX working in Linux Mint. It turns out that to install it, you can run this in a shell:

sudo apt-get install texlive-full

This installation command should work on any Debian-based release of Linux (Ubuntu, Mint, etc...). It will install all the necessary things you need to start writing in LaTeX, including the essential command:

pdflatex

which will do the hard work of compiling your hand-typed LaTeX tex files to beautiful pdfs.

I won't be writing much about Windows, as I don't use it a great deal these days, but if you want an equivalent you can search the net for something called MiKTeX. Any actual LaTeX formatting language I write will of course work on both Linux and Windows.

More LaTeX to come!

Books, Proofs, Text Books

How To Study For A Mathematics Degree

August 22, 2015 Adrian

After my third module, completed in June 2005, I realised that although I enjoyed the module immensely, I still wasn't entirely happy with my ideas behind how to present proofs. Apparently this is a common problem. The shift from normal arithmetic to proofs is a jump that a lot of students find difficult.

I was so bothered by it I decided to post a query on the Open University forums about further reading. Part of my post read:

"...[I] consistently struggle with all kinds of proofs. I either start off completely incorrectly, or have no idea where to even begin."

The response was amazing, everyone (including more advanced students) agreed with me on the difficulty proofs and the shift of mindset required, and I now have a sizeable reading list to work through!

One of the books suggested in the forum was How To Study For A Mathematics Degree by Lara Alcock. I've tagged this post as being about a text book, but only because it's certainly more of a text book than a pop(ular) book. Although pitched at students that have just finished their A-levels and are about to start at university for the first time, I thought I may benefit from picking it up.

I came away with some good notes as to how to adjust my learning slightly, and she talked a lot about the basics of proofs and how to acclimatise yourself to them. One of the parts that I feel was key was the realisation that I can try to construct premises and conclusions in terms of definitions. "Starting with the definitions" seems so obvious in retrospect, but when shown something so unfamiliar it can be difficult to align your brain so as to give yourself a solid starting point. Outlining all the definitions you might need gives you that solid starting point. I'll be keeping all this in mind as I start through my first text book...

Another section I benefited from was a section on how to reading mathematics. Or rather, how to benefit whilst reading mathematics. After spending three years of reading mathematics and feeling I've so far done a pretty good job, but Lara introduced some thoughtful comments on note-taking and comprehension. Again... I'm looking forward to applying this when sifting through my next text book.

As good as the book was, I must say I didn't benefit much from the second part on "Study Skills", but this section was only about 80 pages long. The section covers what "university life" is like, and how best to be a student in such an environment. If you've already been to university it'll be of limited use, but if you're straight out of your A-levels it could make for quite an insightful read.

All in all I feel I took away from it what I wanted to take away; that is a base understand of how to approach proofs in a more structured way. At the very least, I now have a starting point!

Analysis, Books, Maths, Text Books

The Maths Club

August 20, 2015 Adrian

After my first year, I had a look at the clubs and societies you could join as a member of the OU. I thought it would help me feel a little less like I was learning so remotely. Sure enough, there was a mathematics club, M500.

A nice little bonus that you get for being a member is a discount for the bi-annual revision sessions that are held by the mathematics faculty at the OU. So when I started my last module, I signed up for a revision weekend that was arranged to be held about a month before my exam.

It was great meeting other students. With all that remote learning, it was great to be reminded that you weren't alone. We even had nightly trips to the pub! Just like a physical university!!!

Towards the end of the weekend, one of my fellow students, studying the same module as me, mentioned a book that he'd recently ordered. It was written by David Brannan, a man apparently partially responsible for the syllabus of our first double-credit module in our second stage, and wrote the book to reflect the content of that module. My classmate thought he buy it to give him a bit of an edge when he finally started the monster-module. In our class he gave everyone the details. The book is called A First Course in Mathematical Analysis, and at over 450 pages, it looks like one serious introduction!

Finding out about this book couldn't have been any more perfect. It means I can start my easy introductory module this year, and read through my new book on analysis so I'm ahead of the game for when my double-credit module starts in 2016! Perfecto.

…is learning mathematics.