Wednesday 20 March 2013

Why Are You Just a Teacher (part ii)

Remember back in May when I wrote this post? Give it a quick re-read if you want. I'll wait. Ho hum.

Ready? OK, well I had a similar sort of conversation a few weeks ago. To be honest, I have this conversation fairly often. Every single one of my classes has asked me at some point, why are you just a teacher? I'm used to it now, but something one of my students said a few weeks ago really hit home, and I'm still thinking about it now.

I was talking with one of my A level groups, and we must have been discussing universities or something. The question that is the bane of my life (see title, I can't face typing it again) came up, and as usual I deflected  it with some verbal hand-waving. But then a student, in the most sombre voice you've ever heard an eighteen year-old use, asked me, "Do you feel like you've achieved your dream?"

That's one of the problems with my school: we're so big on promoting following your dreams and believing you can achieve them. The students are so full of the optimism of youth. It's quite sickening really.

I just sort of stood there with my mouth trying to form words that my brain wouldn't supply. I needed to say yes. I needed to for them, because they need to believe that their teacher loves teaching them, and I needed to for me, because the opposite would mean admitting to myself that I have failed.

But I said no. I gave the word an extra two syllables, as if stretching it over three would distribute the weight of its meaning. It came out like this, "...Nuh-oh?-oh..." That's right, the question mark was in the middle of the word. I didn't even know that was possible. Try it.

I think at that point I distracted them with the estimated monetary value algorithm. But I kept thinking about it.

I think a teacher was what I was meant to be. But maybe I'm not. Yeah, I love maths, and I love sharing my passion with others, and I love speaking in front of an audience, and I love teenagers, but is teaching the only job that fits with all that? And what about my other passions: dance, fashion, reading, writing blogs?

Well there's probably not a job that encompasses all of those things. But this notion of a "job" is so old fashioned anyway. These days, you don't have to have a "job". In this day and age, there are people out there making money from just about everything. At the weekend I read a couple of books on this subject: Be a Free Range Human and Screw Work Let's Play. These books are all about escaping the 9-5 and making money from something you enjoy. I would not quit teaching at this stage, I'm far too attached to having a roof over my head (call me materialistic), but there are some great ideas for side projects that can be done alongside your regular job. The idea is that when these become profitable you quit your day job.

I became a little bit inspired. What could I do to achieve my dream whilst still teaching full time?

One of the suggestions was blogging. As much as I enjoy writing this blog, I have to accept that it probably won't ever make any money. I know some people do make money from blogs, with advertising and sponsorships etc, but if they say that sex sells, then a blog about teaching maths is going home at the end of the day with a full briefcase and an empty wallet.

Do you feel like you've achieved your dream?

Emma x x x

Thursday 14 March 2013

Official Outstanding Lesson: Permutations and Combinations

In this post I am going to describe to you the lesson that I taught this week which was graded as Outstanding by an OfSTED inspector.

Obviously I can't guarantee that if you do this lesson you will get a 1, because a lot of it is in the delivery, and in how your students respond. There was a sort of magic spark in my lesson, which I think was there because of the good relationship I have with the group.

My lesson was structured as a series of problems relating to permutations and combinations. I didn't tell my students what perms and combs were until right near the end of the lesson. Not one of my students could have told you the learning objectives, until the very end. But who cares about learning objectives when you're busy doing PROPER MATHS? FYI, the only resources you need for this lesson are a deck of cards and, optionally, some mini whiteboards and pens.

1) The Dramatic Intro

Here is a rough transcript of what I said at the start of the lesson:
"All of you in here are extremely lucky. I am going to show you something no one has ever seen before. *dramatic pause*. This is a once in a lifetime experience, literally. In the whole history of the human race, no one has ever seen this before, and there's a very good chance that no one ever will. I'm not just doing this because we have a special visitor today (nervous glance at inspector) but because I love you all very much [I tend to babble when I get nervous]. Are you ready? *Another dramatic pause*"

Then I took out a deck of cards and shuffled them. Then I fanned them out and showed them all of the cards. Then I just looked at my students, paused, then said, "Amazing, yeah!" In my most excited voice.

There was kind of a confused awkward silence, followed by a chorus of "no"s. I feigned disbelief. Then I explained to them that that particular order of cards has never been seen before by anybody, living or dead, and probably won't be seen again for an extremely long time.

I then explained the maths. I said, let's assume that playing cards have been around for about 500 years, because that's roughly what Wikipedia suggests. Let's assume the population of the world has been a constant 6 billion for the past 500 years. And let's assume that everybody shuffles one deck of 52 cards every second. How many orders of cards would that make? I let them work it out. I gave them mini whiteboards to do their working out on, because pupils are much more likely to start attacking a problem if they're not worried about making mistakes. With MWBs they can erase all evidence of stupidity later. They worked out the answer (I leave that to you as an exercise!) and then I got them to calculate the number of possible orderings (permutations) of 52 cards. Both tables of pupils worked out very quickly that it would involve factorials. Again I will leave the answer to you as an exercise.

[*edited to add* I've just remembered a terrible joke I made. This is pretty cringey, but I'll include it anyway. A student asked the sensible question of what to do about leap years. I said to ignore leap years, and pretend that everyone gets the day off from shuffling cards once every four years. Then I said, "oh no, but that contradicts the song!" So they're all like, "What song?" and I said, "You know, Everyday I'm Shuffling!". Pretty good considering I thought of it on the spot! ]

We had a discussion about the hugeness of the result. The students were pretty amazed, I think. The inspector was too!

2) The iPhone Problem

I moved swiftly on to my next problem. Here is roughly how I introduced the problem:
"You're sitting with your friend, and their iPhone is on the table. They leave it there and get up to go to the toilet. You want access to their phone, because you want to frape them, or whatever [is frape an appropriate  word to use during an observed lesson?] but they have a passcode on it. It's four digits, and you happen to know each digit is different, so there are no repeated numbers. The question is, do you have time to try every possible passcode before your friend comes back from the toilet?

The thing I like about this question is that students have to make certain assumptions in order to answer it. Like how long the friend will spend going to the toilet (actually that ends up being kind of irrelevant once you've worked out the answer), and how long it takes you to type in the four numbers. My two tables ended up with different lengths of time that it would take to try all the possibilities, because they made different assumptions.

You can have a little discussion then about phone security and whether a four-digit passcode is enough. That will take care of your PMSSC. I didn't really go into it that much though.

3) The Baby Chris Problem

"Chris' mum gave Chris five wooden alphabet blocks to play with. She gave him the letters A, H, M, S and T. She left the room to make a cup of tea, and when she came back, baby Chris had spelt out the word MATHS. So Chris' mum is going around saying he's a genius. Do you think her claim is valid?"

The students calculated the number of possibilities and decide whether the probability is low enough to claim he's a genius. The class decided it was just a coincidence.

So then I gave them part two:
"This time Chris' mum gives him all 26 alphabet blocks to play with. She leaves the room, and when she comes back, he has taken those same five blocks and spelt out MATHS. Now do you think her claim is valid?"

This is trickier to calculate, and the first thing many of my students wrote down was 26!. I had to question them a bit to steer them in the right direction. The probability is about 1 in 8 million, so we concluded that he probably is a genius.

At that point my inspector left, but I'll tell you the rest of the lesson anyway.

4) An exam style question

Each of the digits 3, 4, 5, 6 and 7 are written on a separate piece of card. Four of the cards are picked at random and then laid down to make a four digit number.
a)  How many even numbers are possible?
b) How many odd numbers smaller than 5000 are possible?

I had to help them quite a bit with this question, but it's a nice one. FYI, the answers are 48 and 30 respectively. 

I then formalised the idea of permutations slightly by showing them the notation. I didn't give them the formula though, because I find the formula is quite unhelpful. I think it's easier to just think about the question and work it out. Someone said, "Oh! That's what that's for! I always wondered that!" Meaning the key on the calculator.

I only spent a couple of minutes on that, and moved swiftly onto the next problem.

5) The lottery

My students have all memorised by now that the odds of winning the lottery are 1 in 14 million. I suppose I must go on about it a lot! But none of them has ever thought to work out where that number comes from. I think they were quite excited to actually be able to work it out. The key thing I wanted them to notice is that it is a different type of question because in the lottery, the order of the numbers doesn't matter, so it's not the same as finding permutations.

They worked out that they needed to divide by the number of arrangements of 6 numbers. Then it took them a little to work out what that was. The most common guess was 36.

I took their workings out and demonstrated that what they'd done was divide 49P6 by 6!. I said that finding the number of combinations will always be like that. Then I said, "If only there were a key on your calculator which could work it out for you!" At which point they all realised that what they'd just done was work out nCr. They'd used nCr in binomial expansions in core maths but hadn't realised (or forgotten) the connection with combinatorics. They were very relieved that there was a shortcut! But I think it is so important to show them where the formula comes from. I proved the formula using the nPr formula that we'd already derived.

6) The Cricket Team Problem

This was just a question I took from the textbook. A cricket team consists of 6 batsmen, 4 bowlers, and 1 wicket-keeper.
These need to be chosen from a group of 18 cricketers comprising 9 batsmen, 7 bowlers and 2 wicket-keepers.
How many different ways are there of choosing the team?

This question is actually really easy using nCr, but it was useful because they could see how to use nCr rather than working it out fully themselves. This was a good question to end on, because they found it quick and easy, which felt like sort of a reward for having derived the formulae earlier. 

Then I set the homework. I didn't really feel like I needed a plenary, because we'd been plenarying all the way through. 

So there you go, my outstanding lesson on perms and combs. I really enjoyed teaching it, and I think my students enjoyed it too.

If you decide to use this lesson, please let me know in the comments below how it went for you.

Outstanding Emma x x x

Winning OfSTED

So, this week we had a little visit from the inspection fairies. It's funny: we were sort of expecting them last week (just because it was exactly a year since our preliminary inspection), and when they didn't come, we sort of decided they weren't going to come until after Easter. I took my eye off the ball a bit. When the phone call came, it definitely caught me off guard.

Strangely enough, I felt like staff morale actually went up. I think we were all more excited than anything else.   The maths department was definitely brought closer. We had all spent some time over the past few months jointly planning series of lessons, and so it had this sense of it being a team effort.

When the results of our individual lesson observations came back, our spirits got higher and higher. We just kept knocking them out of the park! Historically, our department hasn't really done that well in formal lesson observations, but we were killing it this week!

I was observed last out of all the maths teachers, which wasn't very nice, especially as most had been done the previous day! I will blog about my lesson separately, later. But the headline is this: my lesson was Outstanding!

I think on the whole, the staff knew the final result before it was given. But it still felt so good to hear our principal say those words.

Here are my tips for winning OfSTED:

-Carbohydrates. Bring in some muffins for your colleagues. Chances are, people will have been too nervous to eat breakfast, so they will need something to keep them going. And the great thing about muffins is that they're dual-purpose: comfort food for if you do badly, celebratory food for if you do well.

-Club together. Show each other your lesson plans, and offer some tweaks. I think one of the things that made my lesson so good (well, outstanding) was one of the tiny tweaks that my head of department made less than an hour before my lesson.

-Share your excitement, nerves, and success with your students: I sort of briefed my students before their lesson. I went into their previous lesson and said (slightly manically, I'll admit), "ARE YOU EXCITED!?" which I think got them in the right kind of mood. As soon as I found out my result, the first thing I wanted to do was find my students and tell them.

-Come and work in my department. I have such amazing colleagues. They really made this horrific experience not just bearable, but enjoyable. There are just some people who really know how to lift the mood of a group. And luckily for me, my department is full of them!

This whole year, and this term in particular, have been pretty stressful. The pressure was on us for such a long period, that it was really starting to wear some of us down. But it feels like none of that matters now because it all paid off and we (hopefully) won't be having another "O Day" for six years! Now that's a good feeling!

Happy Pi day everyone!

Emma x x x

Saturday 9 March 2013

When Multiple Wrongs Make a Right: Parrondo's Paradox

If you have a maths degree, I recommend that instead of reading this post, you read this page instead. This is a very interesting article but not very accessible to non-maths grads. I have tried to make my version more simple and easier to understand.

Imagine you are at a pretty dodgy-looking carnival stand and there is a game you can play in the hope of winning some money. The odds of winning are less than half, which means if you play for a little while, you will probably end up broke.

Opposite this stand is another one, just as dodgy-looking, and with the same premise. Again, the odds of winning are less than half, so again, you will end up broke if you keep playing.

Common sense, and your maths teacher, would tell you not to play either of these two games and to go and buy some candy floss instead. But then a mysterious man called Parrondo appears and tells you to play both games. He says that by playing them both in a particular sequence it is likely to earn you a tidy profit.

How can this be possible? Two bad things surely can't make a good thing, can they? These two games are not tied to each other, you can play them independently, so surely one can't affect the other's odds?

Let's take a look.

Let's keep game A nice and simple. Let's say it's a coin toss game, where if it's heads you win a pound, if it's tails you lose a pound. But because it has to be a game you're more likely to lose, let's make it a biased coin. We'll say the probability of winning is, oh, I don't know... 0.495 (I'm pretending to have chosen that randomly, I'm guessing you can tell I didn't!) and hence the probability of losing is 0.505. If you kept playing this game for a while, you will gradually lose your money. You are likely to lose a pound for every 100 games you play. Anyone can see that this is a bad game.

Game B will have to be slightly more complicated, but bear with me. Game B involves tossing a coin as well. But there are two different coins, and the one that you use is determined by the amount of money you have at the time. If your balance is a multiple of three, you flip the bad coin, otherwise you flip the good coin. Both coins are biased, but differently.

The bad coin has a a 0.095 chance of winning £1, a 0.905 chance of losing £1.
The good coin has a 0.745 chance of winning £1, a 0.255 chance of losing £1.

This game is also bad, as the bad coin is soooooo bad. Even though you will only have to use the bad coin a third of the time, the chance of losing is so great that it overrides any wins you get from the good coin. Unfortunately, my knowledge of Markov chains and such like is not good enough to compute the expected win/loss, but running a simulation shows roughly the same results as with game A. Although it's not obvious, take my word for it, this is a bad game.

I can sense that some of you are already putting two and two together here: does playing game A somehow make you less likely to have a balance that is a multiple of three, and hence less likely to use the bad coin on game B? Obviously you can't only do game B when you have a certain balance, as the carnies would see what you are up to and throw you out. The idea is to plan the sequence of games before starting so you don't look suspicious. The best sequence to choose happens to be ABBABBABBABB... Although there are others you could use to win.  But how does this work?

I simulated this game 100 times with 102 goes each time. The average (mean) winnings at the end of 102 goes was £9. The highest was £42, and the lowest was -£12. Out of the 102 simulations, only 10 ended in a loss. I think this is enough evidence to suggest that this combination of games is worth playing.

My simulation spreadsheet can be downloaded here.

Let's think about how this works.

Say your balance is a multiple of 3, and you are about to play game A. If you lose, you're more likely to win next time, because you'll be flipping the good coin, and also the one after.

Now say your balance is 1 more than a multiple of 3, and you play game A. If you lose, you will probably lose the next one, but win the one after.

If your balance was 1 less than a multiple of 3, and you play game A, If you lose, you will probably win the next one, and also the one after.

So, without putting probabilities on it, it just sort of looks right that you'd end up winning.

So there we have it, two wrongs can make a right!

Now I'm off to alternately buy lottery tickets and put 10p coins in one of those slidey things.

Emma x x x

Thursday 7 March 2013

What to Expect When You're Expecting OFSTED

My school is expecting a visit from the observation fairies any day now. By which I mean, they are due to come some time between now and the end of the year. Everyday a large percentage of my colleagues are glancing nervously at the tannoy speaker, just waiting for it to be announced. 

Now take a moment to think: does that sound like a nice working environment to you?

Here's what you can expect when you're expecting OFSTED:

-You will either mark your books every week, desperate not to be caught out, or you will be not marking your books at all, thinking you'll do it in one go the night before they come. 

-You will skip horrible chapters like perpendicular bisector constructions, to make sure you don't get caught in the middle of it. 

-You will "save" good chapters for when they come, which means you end up finishing your scheme of work two months early.

-You will go out and buy some kind of containers for mini whiteboard pens and rubbers, but keep them hidden in the cupboard so they don't get trashed before they arrive.

-You will breathe a sigh of relief every Wednesday afternoon, because you know they won't be coming until next Tuesday at the earliest.

-Everyday you will hear one person say "I don't care when they come, I'm not going to do anything differently". That person will be lying.

-You will choose what activities your students do based on the potential for display work.

-When making plans with friends, you will always add the proviso, "unless we get the call that day". 
e.g. "Will you be my maid of honour at my wedding in April?"
      "Yes, unless we get the call that day". 

But at the end of the day, we know that our school is amazing and we'll be fine. When we do get the call, we'll probably stay up all night preparing, and that will be enough. There's no point worrying about it now.

And besides, I don't care when they come, I'm not going to do anything differently. ;-)

Emma x x x 

Tuesday 5 March 2013

Things My Pilates Teacher Taught Me about Teaching

Every Friday after school there is a staff Pilates class. It is physically incredibly demanding and can be ridiculously painful on Saturday morning (sometimes Sunday too, if you're "lucky"). It is really effective though. Me and my Pilates-mates are getting seriously hench.

I was thinking last Friday, whilst I was struggling to hold my plank: how does Gary, the instructor, manage to convince the class to do unpleasant things like jump squats over and over again, when we're tired and cranky and in pain? In my mind, this is equivalent to getting year 8s to add fractions again and again when they're tired and cranky and causing me pain. Are there any tips I can pick up from Gary?

Teaching Tricks I've Picked up:

1) Have a "number of the day"

 Gary always gets us to choose a number for the week. This is the number of reps of burpees or sets of press ups or the number of seconds we'll be holding our side planks or whatever. We always try to pick a low number (particularly if reverse lunges are involved) but Gary somehow manages to manipulate it.
Gary: What's today's number going to be?
Liz: 4
Me: 3
Gary: OK, 7 it is!
He's an evil genius.
But because we have "chosen" that number, we have some ownership of it, and we can't complain it's too high or too low.

Next time you're setting some questions for your class to do, first set a number of the day. You can discuss the properties of the number first, give some cool facts, then set them that number of questions, or have them work for that number of minutes.

2) Demonstrate many levels

Gary is probably the best differentiator I have ever seen. When he demonstrates a move, he always has at least three different versions. One of the options is always ridiculously hard, which, to be honest, no one in our class is really able to do, but he always shows us it anyway, to show us what we might be able to do in a few lessons' time.

He always makes us do one (or one set) of the lowest difficulty move, then says, stick with it, or if you want, do ---. Then after a few more he cranks it up another level, but always reminds us to stick with what we're doing if it's working for us.

I think this is a great idea to apply to maths lessons. Show your class three types of equation, demonstrate solving each one, and make one of them just above their current capabilities, so that they can see where the learning will lead them next, and give them something to aspire to. Start them off at the same point, then let the high fliers progress.

3) Be your subject's best advertisement

Gary has a typical surfer's body: wiry, lean, and deceptively strong. He does these moves that leave us all open-mouthed. I sit there thinking, I'd love to be able to do that one day.

Do your students ever look at you and think that?

How often do you wow your students with your amazing maths skills? You're the expert, you should be the role model. You should be the person they look at and think, "I'd love to be able to do maths like that one day". Build some showing-off into your lessons next week, and see how your students react.

And if all else fails, do a hand-balance. My favourite is the Crow.

Emma x x x