Friday 30 September 2011

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.

Wednesday 14 September 2011

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

Sunday 11 September 2011

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