I'm not so hopeless after all, mum!

So, I'm sitting here trying to work out if I'm any better at the Entanglement game than a trained monkey, when my phones rings. It's my friend, calling on the way to the airport for a romantic weekend in Bangkok with a new lover. "What are you up to?", she asks (except she's Californian, so she actually says, "Hey! Wotcha doin'?"). "Trying to see if I'm better than a trained monkey at playing a computer game", I don't reply, partly because I'd then have to explain about the monkey.

And I'm pleased to announce that I am indeed better than a trained monkey. Yay, me! My mother will be pleased. Although no doubt she'll find fault somewhere. Still, at least she didn't raise a complete idiot. I've resigned myself to the fact that I'm just not very good at this game though. The best scores every day are all 400+, and I've only passed 300 once. I have no idea how that happened as the score just appeared without my knowing what was going on. Maybe the monkey did it while I was making a cup of tea?

So how did I do (and a chance to insert some images for the first time)? I played 30 games using each approach. The top graph shows the mean score after each game, the bottom one shows the standard deviation. The monkey is red, and I'm blue.

The main points are that my strategy produced better and more consistent results. I also improved with time, whereas the algorithmic approach didn't. Maybe that was because I started to hone my strategy, but this isn't really about me. Just look at the opportunities there are for the classroom!

I'm not expecting 11-year old students to be computing variances, but they can draw bar charts and pie charts, calculate means and ranges, and interpret their data. Surely that has to be a whole lot more fun, and hopefully therefore a whole lot more memorable as they have something enjoyable to hang the ideas on. How about some display work? One class against another? So we've tricked them into doing some work while they play the game, and then again when they analyse the results!

Well, I think I'm done with Entanglement, at least for the time being. It now feels like one of those moments at the end of a Simpson's episode, when Homer and Marge are looking back on the events of the previous twenty or so minutes, and wondering what it all means. Maybe there is no meaning? Maybe it's just a bunch of stuff that happened?

A fine line between work and play

To anyone else, he was just having a long soak in the bath. To Archimedes, it was work.

And so it is that my time today has not been pointlessly spent playing Entanglement, but has been a thoroughly time-efficient exercise in developing teaching ideas and resources.

The absence of sufficient Geometry teaching during my own schooling has left my capacity for spacial understanding hopelessly under-developed, so I am still rubbish at the game (current best score: 348; percentage of scores about 200: <10%). But I've turned this rubbishness to my advantage. By appreciating that my own strategies weren't working, I developed a new one, and it's produced interesting results.

My new strategy is: accept whichever piece is offered, in its original orientation, assuming it won't end the game. If it will, then keep rotating clockwise until it doesn't, and accept that. If they all end the game, the I use the swap button and repeat with the spare piece. If nothing offers me a way of continuing the game, I take the highest scoring move to finish.

How has my new strategy worked? It produced two new scores above 200, including my third and fourth best so far! Which might suggest I'm even more rubbish than I thought, were it not for the fact that it has also produced some of my worst scores, including my only single-figure score.

So how is this work? With two classes about to start studying hypothesis testing I'm sure there's some scope here. I can't imagine them complaining too much if I set them a homework to play this game 20 times and record their scores (of course, it needn't be this game, but I'd want it to have some Mathematics in it). The lesson will then be about seeing if Player A is better than Player B. At the 95% confidence level.

Right, I'm off to see if an algorithmic approach really is better than trying to think about each move. I may publish the results, if they aren't too shameful.

I'm Frank Benford and this is the law

Whilst he may not be quite on the same level as Alan Turing, or even William Gossett, Frank Benford is perhaps one of the unsung heroes of twentieth century Mathematics. True, he didn't help shorten World War II, and he didn't even enable us to compare two sample means taken from populations of unknown variance, but a knowledge of his work could help prevent you from getting caught next time you try to fiddle your tax return governments around the world spot fraudulent accounting.

What Mr Benford spotted was that, in many data sets, the distribution of the leading digits is not rectangular. That is, the digits 1 to 9 won't occur roughly the same number of times, and nor should they be expected to (so don't ask them to as it will only create tension between you). The digit 1 should be expected to appear as the leading digit roughly 30% of the time, and successive digits are decreasingly likely, following a pattern of exponential decay.

This phenomenon is now known as Benford's Law and is, would you believe, actually admissible as evidence in a court of law in the US! At least, that's what Wikipedia says. The same Wikipedia article also says that the law was previously "stated" by Simon Newcomb, which makes me wonder... what constitutes being-statedness? Did he have his work published? Did he merely write it down in his notes? Or did he just mention it in the pub one night while having a few beers with his friends? Although this being the nineteenth century, he was more likely in a tavern quaffing ale with stout companions.

Whoever said Statistics isn't interesting? This is the stuff we should be teaching in school! Which gets me on to the topic of how we teach Statistics, another one for the soap box one day. I really should write these ideas down on a list somewhere so I don't forget. In fact, I'm going to make a note to do just that right now.* Why do we teach it using the techniques as the starting points, rather than the contexts in which they are used? And I'm not just talking about really difficult stuff like formal hypothesis testing, it starts at an early age.

Every school I've ever worked in has had a "module" where I have to teach children how to draw a bar chart. Why? What's the point in ever drawing a bar chart unless you have something interesting to say about it? Unlike the quadratic formula or trigonometric identities, there is no inherent beauty in a bar chart. Try doing an image search for "beautiful bar charts" and see what you get. Are any of those so beautiful you want to hang them from your wall?** Most of the images aren't even bar charts! The blue bird in the hard had is quite cute though.***

My point is, shouldn't all our Statistics teaching be centred of analysing and describing real data sets? Wouldn't it make so much more sense? In the earlier years of secondary, this is really just about making the content more interesting/relevant, but as it gets tougher it should also help comprehension. Rather than learning all about t-Tests and Z-Tests and Chi-Squared Tests and Mann-Whitney Tests and Product-Moment Correlation Coefficients and Stuff, just so that we can answer some pseudo-contextual questions in a book or on an exam paper, why not have students create their own data sets, ask them the right questions, and then show how the techniques can be used to answer them. Simple, no? I may put this blog on hold for a few months while I make a fortune writing a new post-16 Statistics textbook.

Still, Frank Benford, eh? Pretty cool stuff.

* I did actually go so far as to start a list of things I feel the need to get off my chest. Or that may be of interest to some people. Or, if I'm really lucky, both. And the teaching of Statistics is on that list.

** Maybe you wouldn't want the quadratic formula hanging on your wall either. The analogy is far from perfect.

*** I do realise that the Internet is an ever-changing thing and you may not get the blue bird in the hard hat, so here it is. And I'm glad I did this, as that looks like quite an interesting website. I'll add it to my list.

Tangled up in red

The much anticipated day back at work didn't happen. A slight relapse is going to mean a trip to the doctor and some drugs. Hopefully that'll sort me out.

It's an ill wind, and all that, and the extra day off has given me time to look at something that I only found out about relatively recently and think is probably worthy of an enrichment lesson. But more on that later, as it didn't take long before I was once again caught up in Entanglement.

Despite a good few hours of my day wasted invested playing the game, I don't seem to be making much progress. My top score so far is a paltry 348, which does at least put me 78th on the leader board. For today. So far. I'm trying to set a target of scoring at least 200 points, a target I have met. Once. Which is starting to make me question whether the act of setting targets alone is enough to improve performance. Who knew?

The main problem I'm having is which strategy to use. Okay, the main problem is that I'm rubbish, but the main problem that I can do something about is which strategy to use. I've tried several: inside to out, outside to in, prioritising making long chains and trying to attach to them, and prioritising joining up loose ends.* But all of those seem inferior to my current "try something different and see if you can stumble upon a better strategy". Well, I say inferior... the current strategy hasn't actually produced any results yet. But I'm optimistic.

All of this makes me wonder whether we should be taught more Euclidean geometry in school. I remember lots of algebra and statistics, but not many geometric proofs. Which reminds me of an interesting probability proof that uses similar triangles. Oh, if only Blogger would let me include PDF files I could show you! Maybe that's a good thing... you could try it for yourself. Here's the problem:

You are playing a game which involves spinning two coins. If they are both the same, you win; if they are different, your friend wins. If the coins are fair, then so is the game. But what if the coins aren't? For the sake of simplicity, let's say that they have the same bias, and in favour of the same side (hmmm... maybe I should have started by saying there was one coin, which we spin twice). Does the game now favour me or my friend?

There's a fairly simple algebraic proof which the absence of a superscript button on the formatting toolbar prevents me from demonstrating (Wow! Now I know just how Pierre de Fermat felt!) but can you come up with a geometric one? Go on... give it a try!

* This might make more sense if you've played the game.

So much for an early night...

I was just about to go to bed, honest I was! And then I discovered this. Go on, try it... and I bet you won't be able to stop either!*

* Assuming you have the sound muted.

Tarsia Software

I've been off school the past two days with man flu so nothing to report from the world of teaching as I haven't been doing any. Unfortunately I'll be back tomorrow.

On the up side, I have recruited another blogger to the team, who will hopefully be joining us next week. I was going to introduce her, but then thought I should leave that to her, and maybe introduce myself. Or maybe we should introduce each other? Hmmm... such important decisions...

So, today's treat is another free and useful piece of software. It's Formula Tarsia, from Hermitech, a company based in Ukraine. The program allows you to make jigsaw type resources which students then have to put together by matching questions with correct answers.

The download page is here, however it appears that there is only a Windows version. Once you've downloaded the software, you can see what it can do by downloading examples from Craig Barton's website. Bryan Dye's Mathsnet site also threatens to make file-sharing available, however this doesn't seem to be in place yet.

The software supports full WYSIWYG equation editing, and provides you with an answer sheet as well as the jigsaw. The latter will need cutting out, but that's something the students can do. Maybe I should include some screenshots at some point. It would also mean I could try out the "insert image" button and see how it works.

Early night for me now, what with having to work tomorrow...

Geogebra - it's free!

Hi,

This may be a case of preaching to the converted, but if you haven't yet seen the light then you really should download Geogebra software. It's fantastic and free. Possibly the best free software that doesn't carry advertisements a Mathematics teacher could find.

It's essentially a geometry and graphing package, arguably better at the former than Autograph, and better at the latter than Geometer's Sketch Pad. There's a command line user interface which isn't ideal and will take students a bit of getting used to, but the being free it means that all your students can use it, at school and at home.

There a Mac version and a PC version, and if you don't want to install the full program on your machine, then you can run a web applet.

I may write more on these three applications, once I've got the hang of how to use this blogging software.

Hello, and welcome

Hi,

Well, at long last the blogging project is underway, and with student exam leave imminent, who knows, it may even continue!

This idea was born of many conversations with fellow teachers that have left me with a load of wonderful ideas on teaching Mathematics, but no sense of how to order them. So I thought I'd share the process of trying to sort them out and hopefully, along the way, make some changes to some other classrooms around the world.

Okay, so up to this point the post has been written in the first person singular. It's all about me. Except it's not. Or at least I hope it won't be. The plan is to have a team of bloggers, which will hopefully mean that there are more frequent posts than if it's just me, and also it will offer different perspectives. There are a couple of people I have in mind, but if you happen to read this one day, and want to join the team, then please get in touch.

Well, that's an introduction. Now I'd better go and post something useful...

Maths Geek