Mathematics in Literature

Lewis Carroll's oeuvre, Alice's Adventures In Wonderland is pretty much my favourite book of all time. Why? Because it's FULL OF MATHS!

Lewis Carroll is the nom de plume of Charles Dodgson, who was a maths lecturer at Oxford. He was actually more famous for being a children's photographer, but his maths was pretty good too. He was really obsessed with Euclid's Elements and wrote textbooks to go alongside it. He loved logic too, which is what you see most of in his books.

"if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic".

Tweedledee says that. It's one of my favourite quotes from the book. The best logic puzzles appear in his other books, like The Hunting of the Snark.

Here's another favourite quote (it's just as Alice is falling down the rabbit hole):

"I'll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is - oh dear! I shall never get to twenty at this rate!"

Alice's wrong calculations can be explained by using different base systems:

in base 18, 4*5= 20 = 18^1 + 2 = 12

in base 21, 4*6= 24 = 21^1 + 3 = 13

if you carried on the pattern, you'd get:

base 24: 4*7 = 14

base 27: 4*8 = 15

...

base 39: 4 *12 = 19

base 42: 4*13= 42^1 + 10 = 1X (where X is the symbol for 10 in base 42)

So Alice is right, she'll never get to 20!

Interestingly, it breaks down at base 42. Lewis Carroll seems to have a slight obsession with the number 42, it appears everywhere in all of his books. It has been suggested that Douglas Adams used the number 42 in the Hitchhiker's Guide to the Galaxy because of this. Adams named his chapters of the radio show "fits" which is also what Carroll did, suggesting he's a fan.

My favourite maths thing that Dodgson did though, was his work on voting systems. This is really interesting, it's got some great maths in it, and if you tackled it in a lesson you would be hitting the moral, social and ethical aspects of maths.

So if anyone asks you to come up with ideas for a maths/English connected curriculum day, suggest studying some of Lewis Carroll's work. I love it when literature and maths combine, as they are my two favourite things. Another good mathsy book is Mark Haddon's The Curious Incident of the Dog in the Nighttime, which I love, but the maths in that is very separate from the story, and I don't think it would be good to use in class.

One of my favourite authors of all time is Louis Sachar, who wrote Holes, which is very commonly studied at secondary schools. He also wrote some great maths books, the Sideways Arithmetic from Wayside School series, so if you have a year 8 class studying Holes, it would be cool to do some puzzles from that to run alongside it.

Finally, another of my favourite authors of all time is Koji Suzuki, the Japanese author of Ring, which was made popular by the abomination that is the film The Ring. There are two sequals, Spiral and Loop, and all three books are outstanding on their own or as a series. They include some brilliant codebreaking linked to genetic codes, and some discussion of meta-mathematics which is just plain awesome. I'm not saying bring them into your lesson, but suggest them to sixth formers who like reading and maybe need showing the coolness of maths. And you should read them yourself because they're sooo good!

Emma x x x

PS if you want to find out more about the mathematical life of Lewis Carroll, I recommend Robin Wilson's Lewis Carroll in Numberland.

Are Equivalent Fractions Really Equivalent?

We teach pupils that 6/7 is the same as 12/14 is the same as 120000/14000 etc. But is it really?

This thought struck me in a faculty meeting this afternoon. The head of maths said that the maths department had the highest proportion of U grades at A level in the school. He then said, "But we did have a much higher intake than other subjects". Everyone immediately said things like "ah, yes" and "true, that needs to be taken into account" and "well there you go". So the message I was hearing loud and clear was: if 7 people out of 42 got a U grade, that's not as bad as 1 person out of 6, and hence the two fractions aren't equivalent.

When you see adverts for shampoo or whatever on TV, it might say 70% of women agree, and then it always says at the bottom, 109 out of 156 women agreed. Now if that figure was 10900 out of 15600, would you be more likely to believe that it's a good shampoo? I would. We all know that the bigger the sample, the more reliable the findings. So the two fractions aren't really equivalent.

Of course equivalence is an undisputed term. It means they have the same value. The thing is, the word "value" can have multiple meanings. The mathematical meaning, which is obvious, and the other one, the one that refers to how much value we give to something, as in, how much we think something is worth. We think 1/6 is worth more than 7/42 when it comes to U grades. We think 10900/15600 is worth more than 109/156 when it's to do with data collection. Mathematically, of course they're the same.

Happy hump day everyone!
Emma x x x

Getting Pupils into a Seating Plan : Activity

This week I wanted to put one of my new classes into a seating plan, but I didn't want to do it at the start of the lesson. I had two lessons with them in one day, which made this work really well, but it would still work if you only had one lesson.

Before the Lesson
I planned my seating plan (alphabetical but adjusted so it was boy-girl).
I numbered the seating plan. I have three rows of ten so I numbered it 1-10 three times.
I wrote a slip of paper for each pupil, saying group 1/2/3 and then their number 1-10.

 During the Lesson
I told the class we were going to have a competition. I told them that as I call out the register I was going to give each person a piece of paper with a group number and a secret number that they weren't allowed to tell anyone. I did this (it took quite a while) and then I told the class what we were going to do:

• You are going to get into three groups. Group one at the back, group two at the side, and group three at the front.
• In your group, you have to line up so that you are in number order. The first group to do this are the winners. 
• You are not allowed to speak or mouth. You are not allowed to write, or hold up fingers to show numbers. You are not allowed to use the slip of paper I gave you.

I then got choruses of "that's impossible!" etc. But they were really keen to have a go.
When they'd finished, I told them to remember what number they were.

Next Lesson
I told the pupils as they came in that group one will sit on the front row, group two on the middle row, and three on the back. Number 1 will sit by the window, and 10 by the door. Voila, no confusion and no fuss. I can also use these numbers in the future for other activities.

Why I Thought This Was Good

I hadn't actually thought of any techniques the pupils could use to get themselves in order. I wanted to see what they would come up with. I was so impressed, all three groups came up with different ideas, and they were all brilliant. I think this exercise gets the pupils to remember that numbers are concepts, not words or  symbols. One pupil picked up a stack of textbooks behind her, and took her number from the pile. The others in her group then followed suit. This emphasises that numbers only really make sense as the size (or measure) of a set (or object). Another group used a calculator. Because they assumed that just typing in their number was disallowed, they typed in calculations that would equal their number. The last group got in order by indicating how high or low their number was by holding their hands a certain width apart. This was nice because it emphasises that numbers are about comparing and about relative size.

It was also good to do with a new class because you can spot instantly who the leaders are. In each group, one or two people took charge from the word go.

It was also kinaesthetic. Need I say more?

I hope you all had a good week. For me it was incredibly long and tiring. I need the whole weekend to recover and hence will get no lesson planning done. It's going to be another exhusting week!

Emma x x x

A Nerdy Day Out: Maze Algorithms

Every August bank holiday, me and my family take a visit to the Wistow Corn Maze in Leicestershire. It's a huge maze made, humorously, of maize. The design changes every year, and this year (spoiler alert) it was designed like a bee.

Photo used with kind  permission from the owner of the maze

Now it might not surprise you to hear that my family are all big geeks. Neither of my parents did maths A level, but they're both a product of an experimental system called "New Maths" which was where the curriculum suddenly completely changed in the 1970s and 6 year olds were learning set theory. My parents both know some basic group theory, and claim to have been taught Boolean algebra in year seven. My parents seem to know *everything*. They're the cleverest people I know.

One of my brothers did Physics at uni so is a slightly different kind of geek from me. The other one pretends not to be a geek with his cool metal band and all that, but he so is.

What happens when a family of geeks takes a trip to a maze? The first time we went, we took one step inside the maze, and instantly agreed we'd follow the left-hand rule algorithm. We stuck unfalteringly to this path, and looked upon families who were randomly meandering (or worse, reading the map!) with equal parts pity and disdain.

There are 12 boards hidden around the maze with pictures on, and the aim is to find these 12 boards and draw the pictures, then convert the pictures into letters at the end, to spell out a word. Naturally, my family takes this part very seriously.

Let's have some maths, shall we?

Maze Algorithms

The Left (or Right) - Hand Rule (Wall Follower Algorithm)

As long as the walls of the maze are simply connected (every wall is touching the outside wall, so there are no islands), then placing your left (or right) hand on the wall and never letting go as you walk will mean you will always be able to find an exit, or if there isn't one, the entrance. This works because (topology alert) the maze can be deformed (stretched and squished) into a circle, and obviously this rule works on a circle.

This rule fails when there are bits that aren't connected.

Pledge Algorithm

This is the only other algorithm I know (wow, how well-researched is this post!) and it works when the wall-follower algorithm doesn't (on disconnected mazes). This is how you do it: you choose a random direction and walk that way. When you hit an obstacle (a wall), you do the right-hand rule with it and count how many times (or what angle) you turn around it. When you have turned a full 360 degrees (or 0 degrees, or 720, or, ... basically when the angle sum is zero) and you are facing the direction you were when you started (not necessarily the same thing) then you stop doing the right-hand rule and continue to move in the original direction (before you stopped at the obstacle).

So what did my family do this year? Well as usual we started with the left-hand rule. My "cool" borther protested that it would make the maze less fun. But his protests lacked conviction and were easily batted away. He knows deep down that he is one of us. After about twenty minutes, we ended up back at the entrance. Darn. Follow my family's route on the map if you like.

This is the map you're given when you come in. Excuse the crumpledness.

Wistow maze is always a disjoint maze, with an inside track and an outside track. It has two bridges at opposite ends (shown on the map) for you to get from the outside to the inside and vice versa. But as you can see (if you can be bothered), following the left-hand rule means walking past the bridge in the NE but not walking across it. This is why our algorithm failed us. So we had to back track (easily done using the right-hand rule) to the bridge and go across it. Then we resumed the left hand rule until we got to the Mid Point, which has picnic tables and is a good place to stop for a Werther's Original and/or a bag of crisps.

From the Mid Point, there are 6 paths you can take. One is labelled the "quick exit" (the West path) and we came from one of them (the NW path) so we had 4 more to explore. We took the North path because that's what the left-hand rule dictates.

The twelve boards we were trying to find are not usually marked on the map, but this time they were (they're the black crosses, not the numbers), probably to speed people up as it was a busy day. We allowed ourselves to diverge from our algorithm in order to get to a nearby board, but we returned to the original path *immediately*. The left-hand rule is not an algorithm that guarantees to cover every path in the maze. Is there an algorithm that does? Obviously it depends on the maze. In graph theory, I know that you can cover each edge in a graph exactly once as long as the graph has an even number of odd nodes. I'm not sure how to translate that into mazes at the moment. I need to have a think!


School Trip???

Wistow Maze welcomes school trips and even has a lesson pack that you can email to get. However, it's aimed at key stage 1 and 2, and won't have anything to do with maze algorithms. I personally think they are missing a trick there!

I think you could do a really cool school trip with higher level GCSE and A level pupils. When I did D1 there were questions on the exam (MEI) about maze algorithms, so it would be a nice accompaniment to that module. If your school is in Leicestershire or neighbouring counties, definitely consider taking a group there. The maze is open mid July until the end of September, so it would be a good end/start of term treat.


Have you ever been to a maze? Did you geek out about it or did you walk it like a normal person?

Emma x x x

Advice to New PGCE Students

This time last year I was feeling really excited about the start of my PGCE. The PGCE, for those of you who don't know, is the Post-Graduate Certificate of Education, which is a one-year course you do to get Qualified Teacher Status (QTS) in England and Wales. There are other ways to get QTS (like the GTP) but I think the PGCE is the most common. Our neighbours in Scotland do a similar course, but they call it the PGDE, because they just had to be different.

If you're about to start your PGCE, you might be wondering what to expect. My university gave me hardly any information about what the course was actually like before I started. Luckily, you guys have me to fill you in.

Expect to...

• Have loads of fun in the first few weeks. This will probably be university-based and, if it's anything like my experience, you will feel like you're back in seconday school again. I loved it, especially as I made lots of new friends (hi, if you're reading this!).
• Be asked to do lots of reading. My advice is this: don't do it. It's pointless. You learn nothing about teaching from books. You only learn from hearing other people's stories, or from being in a classroom yourself. If you get picked on during a seminar, use an anecdote from your own school experience as a pupil, rather than something from the book. The teacher will never know ;).
• Have lots of paperwork. When you get given a piece of paper, decide imediately which of the following categories it comes under: "important" (as in, to do with you getting your qualification), "seems important but is actually unnecessary" (the details could be typed into your phone so that you can throw the paper away, the information on it is somewhere on the university's website), "useful" (an idea for an activity, the name of a good book for your essay, a website to check out),  "they say it's important but honestly I'm never going to look at it again" (an article, a PowerPoint handout). As soon as you know what category it comes under, you will know what to do with it: file it, make a note of it then recycle it, ditto, recycle it immediately. **Edited to add** Start collecting articles and things that could be used in the masters essays right from the start. As Liz pointed out in a comment below, the "post-reading" that was set us was often very good for using in the masters essays.
• Be placed in a school far, far away from where you live. If you have a car, this means not too much inconvenience but a sudden huge rise in your petrol costs. For pedestrians like myself, it means catching the bus at 6:20am, and falling asleep on the bus on the way home. My advice: buy a bus pass, and if you're in the midlands, use Traveline's brilliant website.
• Spend about five hours planning your first lesson, only to have it turn out a complete disaster. Hey, it's a rite of passage!

Our university sessions involved a disproportionate amount of origami.

The task was to make something that will hold five ping-pong balls. Our group made origami bunnies.

I think my most important piece of advice is to make some friends within your subject and organise a Friday afternoon pub trip every week. It's great to be able to let off steam about how annoying the course is and how terribly your first lessons have gone. Without this, I don't know how I would have survived!


I think that's enough for the first half term (also known as the calm before the storm).Good luck to all new student teachers and enjoy it whilst you can. The NQT  (Newly Qualified Teacher) year is apparently a lot harder (gulp).

Emma x x x

Explaining the Fourth Dimension

Has a kid ever asked you what the fourth dimension is? What it looks like? Whether it could possibly exist?

No, no kid has ever asked me either. But I really wish one would one day.

Anyway, I was reading this book by one of my favourite authors, the hilarious Scottish crime writer Christopher Brookmyre. The book is called Pandaemonium and for goodness' sake don't recommend it to any of your pupils. The "c" word appears frequently, there's really really gross violence and gore, and it's just generally very offensive. If it was released as an audiobook, it would be read by Frankie Boyle.

I loved every page.

Anyway, I'll try and get to the point. There's this bit in it where CB explains the concept of a fourth dimension really nicely. I loved it so much I highlighted it and wrote a little note (I LOVE my new Kindle!!). I'll try and paraphrase it here:

Imagine there are some ants crawling across the duvet on your bed. They are only aware of two dimensions: walking forwards and walking sideways. There is no up or down for them. So if you picked one of the ants up and suspended it in the air, all the other ants would think the ant had vanished. They would have no idea where it went. And then if you put the ant back down a few centimetres from where you picked it up, the ants would all think it had teleported.

Maybe there's a higher dimension out there that we can't comprehend. Maybe there's something out there that could pick one of us up and put us down somewhere else, and it would look like we've teleported. If we could access this fourth dimension, think of the possibilities: we could perform surgery without breaking the skin.

Back to the ants on the duvet: imagine picking up two opposite corners of the duvet and bringing them together. The ants are still only moving in two dimensions, but now they can walk off one edge of the duvet and return at the opposite end, walking in the opposite direction. This involves moving their world around in three dimensions, but keeping it as a two dimensional world, and without them noticing. Could something similar happen to us? Could we walk off one "end" of the universe and end up on the other side? Is dying walking off the end? Does God live in the fourth dimension?

Well, I found it interesting.

Emma x x x

Khet - The Laser Game

Today I played Khet for the first time (and won, natch). It's a strategy board game similar to chess, which involves moving pieces with mirrored edges around the board, and firing a laser so that it bounces off your pieces and hits your opponent's.

Obviously, there's a lot of maths to be had here. Firstly, bouncing lasers off mirrors requires some understanding of angles. The mirrored edges are at 45 degrees to the grid of the board, so the laser always hits the mirrors at this angle. As the angle of incidence must equal the angle of reflection, the laser comes off the mirror at 45 degrees, which is 90 degrees from where it was hit. So the laser beam always moves in L shapes between mirrored surfaces.

The game also involves problem-solving, as you have to consider where to move your pieces in order to block your Pharoah (did I mention all the pieces are Egyptian-themed?) and to take out your opponent's pieces. It's similar to chess in that you have to be thinking several steps ahead at each point.

The game was a Mensa Select Award winner, so clearly it's a game that can involve a lot of intellectual stimulation. But it's easy for anyone to understand enough to at least have a go.

It would be impractical to have everyone playing this in a maths lesson, but you could give pupils a situation on a worksheet and ask them what their next move would be, or ask them where the laser would point if it was fired then. Like how in newspapers they often have a picture of a chessboard and you have to decide what your next move would be.

Khet would be great for an afterschool or lunchtime maths club. It takes about 10-30 minutes to play (according to Wikipedia) so you could easily fit in a game. Obviously the school would have to buy a few sets first. It is available in many places online, including Amazon.

To watch a video demonstration of Khet, click here.

It's starting to get somewhat annoying how I can't play a new game without thinking: how could this be used in a maths lesson? Does anyone else suffer from this?

As it's the summer holidays, my posts will be fewer and further between for the next few weeks. I am aiming to blog once a week. I'm also going to be doing some re-blogging, which means I'm going to re-post some of my favourite entries from my old, defunct blog into this shiny new blog, so that they're not lost forever.

So for now I'm going to set you some homework and leave you for a week. Your homework is to watch the latest series of Futurama. There's loads of cool maths slipped in. For example: one writer created a new maths theorem just so he could use it in the show. Click here for more details.

Emma x x x