All posts by Adrian

Maths

Choices, choices.

It's only a few weeks now until my next module starts! The beginning of my fourth year looms...

This module choice proved to be an interesting one. My degree consists of three stages (think of them as proper university academic years), and this module will be the last of my first stage.

The last module choice is actually very broad (some of the modules are foreign languages!), but of all the modules, there is only one that has 100% mathematics content called "MU123". The only other one I would want to take has about 30% mathematics content.

The problem with MU123 is that is an introductory mathematics module. -one that I'm bound to find far too easy. Worried by this, and wondering whether I could just skip straight to the next stage I called the OU support center. They agreed that I would indeed find the module easy, but did suggest something that other students have done in the past. Previous students have taken MU123, AND the first module for the Stage 2 together and done them in the same year. The problem for me there is that the first module of Stage 2 is actually a double credit module, meaning twice the usual workload. So instead of doing one modules worth of work in a year, I'd be attempting to do three!

Faced with the absurd workload, I decided to just sign up for MU123 on it's own, have a nice easy year and get the credits I require to move on to the second stage.

Having said that, my fourth year will not be all plain sailing... more to come in a future post!

Group Theory, Maths, Proofs

Year Three

My third module, completed last June, took me by surprise. As I'd started my statistics In February 2014, and completing in September 2014, I was hoping for a nice break before starting this third module.

The module I needed to take was MS221, rather innocently called "Exploring Mathematics". However, it transpired that the class that would be starting in September 2014 was to be the last ever of MS221. Normally this wouldn't be a problem, but MS221 followed on directly from material presented in my first module, and I was told I had to take this course. If I waited to take the replacement course I would face a ton of catch-up material .

So in summary, due to the slight course re-structure, I ended up not stopping to have a break between my second module and my third. This meant I was studying non-stop from February 2014 to July 2015. Exhausting.

I did manage to take a long holiday at the beginning of 2015, but tried to make sure I was several weeks ahead of my study before doing so.

Although "Exploring Mathematics" sounds rather unassuming, it covered some rather in-depth topics. On the one hand I was presented with proof by mathematical induction (which only proved itself to be baffling) but on the other I was also presented with group theory which describes a really nice formal form of symmetries. Seriously cannot wait to learn more about group theory...

Once July arrived, and I had taken my final exam I was once again a free man. Finally free to enjoy a Summer.

So that brings us up to date! I'm now enjoying my first proper break away from mathematics in 17 months, and the materials for my next module are due to arrive in under a month!

Maths, Statistics

Year Two

After a long break from my first module, I decided to start my next module in February 2014, half-way though the academic year.

My second module was to be Statistics. I was looking forward to this, and wasn't disappointed. The module covered a lot about survey results, and how to conduct surveys effectively and make sense of the data. It also had a biology experiment thrown in there for good measure, the results of which were meant to be used in an assignment.

One of the things that really bothered me though, was their description of statistical variance. To my mind, one measure of variance would be the sum of the difference between the population mean and the sample mean, all divided by the sample count.

In fact, that's not the case. It's not divided by the sample count, it's divided by the sample count minus one. This seemed totally counter-intuitive, and apparently the full explanation was outside of the course material!!! Frustrated I scoured the web looking for an explanation... Eventually I came across a decent youtube vid. Thanks youtube!

To summarise, this video proves that sample variation s^2 is an unbiased estimator of the population variation \sigma^2 as shown below:

    \[<span class="ql-right-eqno"> </span><span class="ql-left-eqno"> </span><img src="http://adrianbell.me/wp-content/ql-cache/quicklatex.com-8309d8e21d5c74b4a21588f648103d5a_l3.png" height="45" width="261" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*} E(s^2) =& E \left( \frac{ \sum_{i=1}^n( x_i - \bar{x} )^2 }{n-1}\right) = \sigma^2 \end{align*}" title="Rendered by QuickLaTeX.com"/>\]

See that pesky "n-1" divisor? Check out the full explanation here.

So after my study starting in February and ending in September, I felt I'd had a decent introduction to statistics.

Although, something that does surprise me is that looking ahead at my possible future module choices... statistics doesn't really crop up again for the rest of my maths degree. With the growing importance of statistics in modern society (let alone mathematics itself!) I would have expected a lot more of those modules on offer. Having said that, there is a separate BSc (hons) Mathematics and Statistics that the OU offers, but this appears to almost be purely statistics and doesn't give much variety. -certainly wouldn't be good for me.

In fact, there are  only two real interesting modules on this Maths and Stats degree course, one of which is the final year module "Mathematical Statistics" (M347) that introduces Markov Chain Monte Carlo, which is an area of interest. I suppose after my ten-year degree, if I'm still desperate for more, I can sign up for it as a stand-alone module. 🙂

 

Maths

Year One

The countdown to starting my latest module has begun!!! With about a month left to go before all my new material arrives I thought I'd sum up the years I've completed thus far.

My first year I spent studying a module the Open University call "Using Mathematics". It was essentially A-level-ish in terms of the content. Although I have never studied A-level in mathematics, I was thankful that a lot of it seemed like revision.

The module covered things like recurrence sequences, vectors and matrices, calculus, and some basic statistics in the form of probability work. Calculus being the most challenging area, as I'd not encountered it before.

This first module was the biggest jump I've experienced so far, as when I started I hadn't done any formal learning for about 12 years. Doing well in this first module acted as a massive confidence boost, acted as a confirmation that I actually knew what I was doing, and had a fighting chance of seeing the whole thing through!

Maths

Starting Late

I really should've started this site sooner, but I only considered how useful it would be to document mathematical learning just the other week.

I'd made post on the Open University forums, and one of the students (who is also a mathematics teacher) gave me a link to her site. It contained all kinds of observations on common mistakes and examples to circumvent them.  So good it was, it inspired me to make my own site!

As it stands I've studied three modules already. The first of which was more like an A-Level-like introduction to mathematics. As I don't have an A-Level in mathematics, this was actually rather useful.

Second year came a module on statistics, and the third year, followed directly on from the first year introductory module.

This most recent third year was particularly good, as it introduced complex numbers (of which I've never formally been taught), and group theory which included some brand new mathematical concepts to get my head around.

So with three years in the bag, I'm now looking ahead at another seven (at least). I'll be writing about my learning process, the great parts of learning and the bad. I'll be writing about mathematical techniques and even about various tools and books I discover to help me along.

Oh, and I'll also be writing lots of pretty maths:

    \[\pi\int_{0}^{\frac{\pi}{3}} \tan^{2}x\: dx - \pi\int_{-\frac{\pi}{6}}^{0} \tan^{2}x\: dx\]