How to Deal with Being Inadequate

It's been a while since I've written a diary-style blog post. Recently I've been sticking to maths and the odd bit of teaching and learning. I decided after finishing my NQT year that I would no longer discuss my CPD publicly, for the sake of my readers more than anything: reading somebody's moans can't be interesting. But then again I suppose most people wouldn't find absement interesting either.

But this week I felt the need to write something that is somewhat more personal. It's been a bad week for this QTpi. I've experienced something that I knew would probably come eventually, although I'd hoped it never would: I have had a lesson observation graded "inadequate".

In case you didn't know, I'm somebody who likes to do well. I liked getting A*s at school. I didn't like getting anything else. So it bugs me that I'm no longer a straight A student. I have just received my first ever U. I don't know how to respond. This does not compute. Syn error. 404 page not found.

I don't want to talk about what went wrong in the lesson, or anything like that. It's personal, and besides, it wouldn't be interesting. But I do want to talk about what happened after I received my "Inadequate".

I felt sad. Annoyed. Frustrated. Victimised. Like a failure. And after 10 minutes of all that (I'm a fast thinker), I started to just feel glum. What an amazing word glum is! It sounds exactly like the feeling. Glum. It's almost impossible to say glum without slumping your shoulders and sticking out your bottom lip.

This glumness, unfortunately, has lasted all week. It has hovered around me like a bad smell. I walked into lessons thinking "what's the point? They're obviously not going to learn anything anyway". And this made me so sad. I love my students. I want them to learn. I'm letting them down.


Things people tell you when you get an inadequate:
-It's not you who's inadequate, it was the lesson.
-It was only one lesson out of thousands of good ones.
-It only went badly because you were nervous about being observed.
-It happens to everyone.

At any other point in my life I would have said that those four things are all true. However, in my current state of glum, they all seem like the kind of cruel lies that parents tell their children (I ate my crusts religiously for years and my hair remained as linear as ever).

Taking these cruel lies one by one:
-You are what you teach, hence I am inadequate.
-All of my lessons are exactly like that. Hence I am always inadequate.
-I never get nervous. Literally never. Unless fire is involved.
-It does not happen to everyone. I can name someone it has never happened to. She is an outstanding teacher and her hair is nice and curly too. I bet she never had to eat crusts. I hate her.

As you can see I have reverted to my fourteen-year-old self.

So if these nice things my colleagues have said (I do appreciate it guys!) have failed to make me feel better, what will? The answer is this: time. That's it. I just have to get back on the horse, and stay on it for as long as it takes to forget I ever fell off.

Maybe one day I will be able to look back at this observation and learn something from it. At the moment I don't even want to read the feedback sheet. I'm not in the mood for learning from my mistakes. I'm in the mood for focusing on what I'm good at. Like integration by parts, and matching my shoes to my dress. My colour co-ordinations and calculus will always be outstanding.

Have you ever been told you're an inadequate teacher? How did you cope?

Emma x x x

A Moral Dilemma: Teach Them Less So They Learn More?

At a *theoretical* school last *theoretical* Monday in a *theoretical* sixth form teaching meeting, somebody *could have* mentioned an idea for raising achievement in A level students. *Theoretically*, this could have caused some disagreement between staff.

This is all theoretical, of course.

The idea was (I mean, might have been) this: if you have a group of students who are unlikely to achieve a high grade in their AS or A level, and are aiming to scrape a pass, why not teach them less of the subject matter, and instead focus on just a few topics? So in A level maths, for example, teach them the calculus chapters, but nothing else. If the students got most of the marks allocated to those sections, they should get an E.

This idea caused a lot of debate. Once we'd got past the cries of "that wouldn't work in this subject!" we got to  some more interesting discussions: if it were possible, would it be morally right to do it?

Let me present some arguments for and against (I'm going to stay neutral, in case any theoretical colleagues are reading this):

For:

We do it at GCSE

In GCSE maths, you will probably have a few groups who are all sitting the higher tier paper. Let's say sets 1, 2, and 3. Well set 1 are obviously going to have to learn everything. With set two, you might choose to leave some of the circle theorem proofs out, and focus mainly on the grade A topics instead. With set 3, whose target grades might be a C or B, you would probably leave out all the A* topics completely, and only look at a few of the A topics. You would want to use your teaching time to make sure they fully understand the grade C and B stuff.

This makes a lot of sense, and few people would disagree with this. On a larger scale, you wouldn't teach your bottom set kids the A* stuff. That's why we have tiers in maths!

It benefits the students

Getting the grade E in their AS or A level would make a big difference to the student. OK, maybe their understanding of maths will be insufficient to actually help them in any way in the future, but at least they have something else to put on their CV. And these students obviously wouldn't be planning on studying maths at Uni anyway, so who cares? Who uses infinite series in real life anyway?

Besides, if pupils fully understand two topics, isn't that better than not really understanding six topics?

It decreases the school's number of fail grades

Enough said.

Against:

It's not fair on the students

Telling a student: "you're not going to learn that, there's no point because you won't get it", is a pretty demoralising thing to tell a student. Especially if the rest of the class are being taught it.

Also, limiting their knowledge in this way automatically stops them from progressing in the subject. If you teach them a limited number of topics at AS level, there is absolutely no way that student could progress to A2, even if they end up with a grade D in the end. Putting a cap on student progress will surely be a self-fulfilling prophecy? And who are we to assume that this student who appears to be on track for a U won't turn it around in the last month by working their socks off?

It's not fair on universities

How do you think secondary school teachers would feel if their year 7 pupils came to them with level 4s in their KS2 maths SATs, but had never learnt about, say, decimals. These students would be placed in a middle ability set with "true" level 4 students, and will have absolutely no idea when it comes to anything involving decimals. This is the problem that universities are having. They call it grade inflation. Students are coming in with the same grades every year, but the students' knowledge is getting weaker and weaker year after year. We do not want to be the cause of this! This applies even if the subject they are studying at university is completely unrelated to maths.

It's "teaching to the test" which everyone knows is a BAD thing

Learning A level maths should not be about the grade you receive. It should be about the journey of discovery, of honing your skills, and developing your way of thinking mathematically. It is not a means to an end.

And I might as well say it: OfSTED do not like this sort of thing. They explicitly criticise teaching in a way that allows students to pass exams without sufficient understanding. You do not want OfSTED to catch you doing this.



What do you think about this idea? Do you think it's morally wrong to restrict the amount of content you teach certain students, in order to teach them a select few topics really really well?

Do you have any more arguments for me to add to my lists above? Please leave a comment!

Emma x x x

The Area Under A Distance-Time Graph

Teaching Mechanics

I love teaching A level maths, but I must admit to feeling a bit disappointed when I found out I would be teaching the Mechanics module (M1 from MEI, fact fans) because... Should I really be saying this? OK, I'll just say it: I find it boring. No, that's not the whole truth, I'm making excuses now. I have to admit: I don't always get it.

I did Mechanics 1 at A-level (also using the MEI exam board) and that's the full extent of my knowledge. No Mechanics 2, none at Uni, I didn't do Physics A level or even separate Physics GCSE. I've never really been a "scientist", preferring to see myself as a creative type. I was always better at English and French than at science.

Anyway, this year (2012) I've been team-teaching M1, sort of. I've basically been watching someone else teach it. And suddenly I'm learning things I never really understood the first time round. I'm starting to get it!

The Area Under a Distance Time Graph

But then I went and spoiled it all by doing something stupid like Googling "What's the area under a distance-time graph?"

I would hope that those reading this are aware of the concepts illustrated below:

When you draw a velocity-time graph, for example, the area underneath the curve gives you the displacement, and the gradient of the curve gives you the acceleration.

The other teacher of the class posed the question to the class "What's the area underneath a displacement-time graph?". My mind was immediately blown. With a bit of jotting down (the diagram above), I could quite easily work out that the units had to be ms (metre seconds, a bit like kilowatt hours), so I knew it was something to do with distance multiplied by time.

I knew that what I was looking for would be the blank in this sentence: "displacement is the rate at which BLANK changes". What could fit? Nothing seemed intuitive.

Absement

That was where that much-relied on search engine came in. After a bit of research (Googling), I found that the word I was looking for was "absement" (oh of course, you all exclaim) and that I was by no means the only geek in the world wondering what it was.

Absement is a port-manteau of the words absent and displacement (knowing which makes it no easier to understand) and there are not many results on Google for it (the eighth result is an English to Urdu translation page. Erm, cheers for that).

I will attempt to explain absement using an example. For another example, visit Wearcam.

You live 2km from school. You walk to school in 30 minutes, stay there for six hours, then return home, also taking 30 minutes.

A displacement-time graph would look like this:

Now let's consider the area under this graph. It would be given by integrating the curve above. You can sketch this curve easily: in the first section, the gradient is constant and positive, so the corresponding bit of our new graph would be increasingly increasing (curving upwards). The middle bit has zero gradient so our new graph will have a constant positive gradient. The last bit is the opposite of the first bit, so our new graph's last section will have a decreasingly increasing bit.

My sketch:

The next thing to do is calculate the numbers on the vertical axis.

Your absement at any point is given by the (average) distance you are from home multiplied by the time that you're there for.

So when you've just arrived at school, your absement for that point is 1000m (your average distance from home) multiplied by 1800. So that's 1 800 000 ms. Up to that point, your absement has been increasingly increasing. When you're halfway to school, for example, your average displacement was 500m, your time is 900, so your absement is 450 000ms (note that this is not half of 1 800 00ms). At the end of your second part (when you've just finished school), your displacement has been constant at 2000m, your time has been 6 hours which is 21600 seconds, so your absement at that point is 43 200 000 ms. You can calculate your absement at many different points so that you can plot a nice, smooth curve. It looks a bit like a cumulative frequency curve. I'll leave this to you as an exercise.

What's the point of absement?

Well if you were on a spaceship and you had some kind of mobile communication device, you might imagine that the further from Earth you are, the more power the device uses. Therefore you might measure its battery usage in metre seconds.



Emma x x x

Praise and Rewards: Overrated?

How often do you praise your students? I bet most of you would answer "not enough", because we're constantly being told that we should be praising and critiquing in the ratio of 4:1 (which I'm sure is completely arbitrary. Where are the calculations?) and that we should be constantly reinforcing good behaviours and boosting confidence.

But really? Really? The hype over praise has reached such a ridiculous level that we're now told we have to seek out things that students are doing and praise them for it. It's called "catching them being good" but it should really be called "searching frantically for something they do which isn't completely moronic". 

I sound quite bitter don't I? I'll dial it back a bit. I'm no educational expert, and I'm by no means an experienced teacher, but I still feel quite strongly that the advice we have been given about praise, and more so about rewards, is bad.

I have a confession to make. My name is Emma and I'm an over-praiser. I will meet every child's contribution to the lesson with a "well done!" or an "excellent answer!" or, more often than not, "that's not correct, but I'm impressed by your creativity!" I actually found myself praising a student for remembering to draw his margin with a pencil today. And this was a middle ability set. I hate the fact that I do this. It's a habit I got into during my training year, before I was confident enough to challenge any of the advice I'd been given. I'd spend hours trying my hardest to think of three "stars" for little Johnny whilst trying to cram all of my hundreds of "wishes" into one line. Result: Johnny walks away thinking he's done a good piece of work. It doesn't matter to him that one of the stars was "You wrote the date! (Smiley face)"

My revelation came during my NQT year. I was talking to a particularly inspiring teacher, and he admitted to me that he never gave out "points" for good work, good behaviour etc. He said he had never even logged into the online points system. I felt really smug then, because I had painstakingly given out every single point in my account every week since the start of the year. My smugness didn't last long, however. He told me he didn't really agree with the points system. He said that students shouldn't be behaving well and doing good work so that they can earn points, they should be doing it for their own self-satisfaction. This really struck a chord with me.

Giving out points encourages extrinsic motivation, where people act to gain external rewards and avoid external punishments. Intrinsic motivation is where people act for their own satisfaction. Studies have shown that using extrinsic motivation to get people to do a certain activity can lead people to see that activity as "work", a job for which they are paid. I have heard that it is common advice to tell parents not to reward their children for reading, because if they do, they will stop reading just for the fun of it. 

We all know that Pavlov et al have shown that using praise and rewards allows us to control behaviour, but I personally would rather teach students who are in control of their own behaviour. There's something quite sad really about a dog who salivates when a bell sounds. Would you like to teach a bunch of robots who immediately start working the second you say "VIVO miles"? Are they the kind of people we want in society? There's no one around to reward adults for not dropping litter on the floor, not sticking chewing gum under cinema seats and wearing appropriate clothing in public (I'm talking to you, large blonde lady from the number 5 bus).

And when you throw praise and rewards around willy-nilly, you devalue it. I don't think any of my kids ever really experience the feeling of pride puffing up in your chest. I'm sure my comments wash over most of them, a lot of the time. I watched the teacher mentioned above teach an A level lesson the other day. He asked the class how they thought speed cameras worked. After some answers and some discussion, a boy contributed an answer that I thought was pretty good (embarrassingly, I didn't know the correct answer). The teacher, without saying anything to the boy, explained the boys answer to the rest of the class, rewording it a bit so they would all understand. He then finished by pausing for a second, and then simply saying "that's exactly how they work". He didn't add "well done", he didn't even smile. He just said that, in a slow, low voice. But even from where I was standing, the pride that boy was feeling was palpable.

I must admit, I stopped giving out my points a while ago, and since then I haven't noticed a single bit of difference in my classes' behaviour, effort or achievement. My next step is to stop being so generous with my praise. I'm thinking I might aim to give out one really good bit of praise per lesson. 

Think about yourself as a student. What bit of praise really affected you? What moment of pride do you still remember? For me, it's the time my year nine history teacher (also the head teacher) wrote that my end of year project is the best he had seen for as long as he could remember. And he gave me a pen (my schoolmates will remember how scarcely these pens were given out). Embarrassing admission of the week: I still have the pen. It's in my "special box". 

Am I on my own here or do you agree? Comment below!

Emma x x x 

My Mathematical Jewellery Collection

Today I'm going to a barbecue hosted by one of my colleagues. The theme is maths. We will be doing maths puzzles and games, and there is a prize for the most mathematical outfit. On reading this I exclaimed "What shall I wear?!" Not, as you might think, because I don't own any mathematical clothing. In fact it was because I own so much, that choosing was going to be difficult.

Two of my absolute favourite things in the world are maths and fashion. When you combine the two, you get mathematical fashion. And I LOVE it. I have a few mathsy clothing items, but I'm not going to show you those in this post. This post is dedicated to perhaps my favourite aspect of fashion: jewellery.

At my academy I'm known by both staff and students for having a massive collection of bold and bright rings and necklaces. A subset of this collection I will be showing to you today.

First up, my absolutely amazing, custom-made SOHCAHTOA necklace;

I bought this necklace from one of my favourite jewellery shops, Tatty Devine. As well as stocking quirky and unique perspex jewellery, they make name necklaces (like Carrie's from SATC). Seeing this feature gave me the genius idea of turning my favourite maths word into a necklace. I thought long and hard about what word (containing nine letters or fewer) to choose. I settled on SOHCAHTOA because it's so recognisable, not to mention useful. I've had several strangers see it and proudly exclain "I remember that! It means..." which is so nice to hear!

I think my favourite thing about this necklace is my students' reactions when I tell them I had it custom-made. I think to them it just confirms my status as geekiest teacher ever.

Onto my next necklace: The Infinity necklace:

I bought this necklace on Amazon after a lot of searching. I had decided I wanted a necklace in the shape of the infinity symbol, and then set about trying to find one, which is the complete opposite of how I normally shop, where I see something I never knew I wanted and then buy it immediately. This necklace is sterling silver and contains an actual real diamond, which sadly my poor eyesight inhibits me from seeing. Amazon assures me it is there, though.

Necklace number three: the Rubik's Cube:

Everyone loves my Rubik's cube necklace. I have been nearly strangled countless times as students grab it as  I walk away. Nobody can resist turning it a bit, which is why it is a jumbled mess. I am actually rubbish at solving them, although I keep telling myself I will learn the algorithms soon. My friend Emma appeared to learn overnight when she got one for her birthday, but knowing her she probably spent hours and hours learning it obsessively.

Oh, I almost forgot to tell you where this amazing necklace is from! I got it from Folksy, a brilliant website where random people sell all kinds of handmade and vintage stuff. I have bought so many things from there. The necklace was from a seller called Mary Quite Contrary.


And now, a bracelet:

I bought this bracelet quite recently from Folksy, from a seller called I heart my art. It was the only one of its kind, which makes me feel special. Check out her other items though. I like how the bracelet contains "e" because most of my students don't know about the exponential constant, so they'll ask questions about it. Any excuse to talk about e and logs! Although I bet most of them just assume it stands for Emma.

Next, my Origami necklace:

Boy was this hard to photograph! This little stunner really is origami: it was made from a single square of silver and folded into the iconic crane. I saw a very similar necklace on Anthropologie, one of the most beautiful shops in the world, but it cost way too much so I went looking elsewhere. I found this on Etsy, which is basically the American version of Folksy. I bought it from the hugely talented AllegroArts who handmakes all of these stunning origami pieces. This exact necklace can be found here.

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So that's my mathematical jewellery collection as it stands at the time of writing. I fully expect it to be twice the size this time next year!

I will leave you with one last photo: a little cutie I just had to buy from Ryman's as I was picking up my back-to-school stationery:

I just can't resist cute geekiness.

Emma x x x

Playing to Lose

(I was contemplating calling this post "Every Loser Wins", but I hate that song).

As all of you should know by now, I'm a winner. I win things. So when my friend Stacy challenged me to a game of Noughts and Crosses today, I was prepared to win. Until she said this:

"The winner is the person who loses. The aim is to NOT get three in a row".

As you can imagine, I was flummoxed. Winning is easy, losing is difficult. She told me to go first, which, now that I think about it, was a sneaky way of increasing my chance of winning - and hence losing. I obviously avoided the middle square. I figured the corner squares were also too good to use. So I opted for a side square. That decision was pretty easy. The rest was not so trivial. Stacy lost the game, and hence won. Apparently I am such a winner that even when I'm trying to lose, I win.

Please please please grab your partner/your child/your flatmate/the guy next to you on the bus and challenge them to lose a game of Noughts and Crosses. It's the only way you can really think about the strategy involved.

Have fun!

Emma x x x

Why We Actually Won The Olympics

Before this year, I'd never watched so much as one minute of an Olympic event. Not even China's opening ceremony last time (I have a phobia of fireworks). I am an incredibly competitive person, and the idea of watching a massive competition into which I'm not even entered, really holds no interest to me. Who cares what happens? The bottom line is, I'm not coming home with a shiny medal around my neck.

But then I went to spend a week in the house of a family who were interested in the Olympics, and (this is important) actually had television (we stopped buying our TV licence several years ago as our favourite shows only air in Japan anyway). I didn't have much else to do that week, no longer being a fifteen minute walk from a city centre (and hence, shops) so I found myself watching the Olympics A LOT. And I found myself enjoying it. I finally got it: sure, I wasn't going to win anything, but my country would, and in a way, me and my country are one and the same. I suddenly understood what Nick Hornby was on about in Fever Pitch (OK, I never actually read it, because, I enjoyed About A Boy and all, but seriously, a book entirely about football?) and I felt that I was Team GB. It was a great feeling.

And then we didn't win.

Those of you who know me well know that I don't particularly enjoy not winning. Please note this only applies to competitions I deem worthy: me always failing to win at Laser Quest, Mario Kart, and anything that involves Geography does not bother me in the slightest. Because they're stupid. But the Olympics was something I wanted to win, something I thought I was really good at (by me, I mean Team GB of course). I wasn't going to let stupid countries like China and the USA beat me. It's not fair: they already have crazy cheap designer clothes and fifty flavours of Pop Tarts, why do they have to have this too?!?

Naturally I did what I always do when I am told I have not won. I set about moving the goalposts, scouring the rule book, and  examining the data to try and find a way that means I actually won. But whichever way you count the medals, we were not higher than third place.

Rank by Gold	Country	Gold	Silver	Bronze	Total
1	United States of America	46	29	29	104
2	People's Republic of China	38	27	23	88
3	Great Britain	29	17	19	65

There was only one thing I could do: "To the math cave!!"

I'm sure that anyone who is even slightly mathematically minded automatically saw a problem with the scoring system. I don't mean the fact that it's all done on the number of Golds and not on a 5-3-1 system, because we still wouldn't win then. I'm sure that when most people looked at the scoreboard, their first thought was "Well of course those two countries are winning, they're absolutely massive". China and the USA are well known as having huge populations, along with India, Brazil and Indonesia (interesting point: this is probably the one geographical fact that I actually know without having to look it up. Maths teachers from my Academy will know why). Is it really fair that we are being compared to countries with 5 or even 21 times more people than us? (I had to look that up). If we randomly divided the USA into five pieces, and selected the best athletes from each piece, would they still be the best? And if we cut China into 21 pieces? I don't think so.

So, here comes the maths: I have created a table of the competing countries, and their number of golds per 10 million people. This is like doing population density (urgh, there is way too much geography in this post) but instead it's gold density. Except out of the number of people, not the area of land. Hmm. OK.

 

Country	Golds	Population (10 millions to 3sf)	G/10mill (2 dp)
USA	46	31.5	1.46
China	38	135	0.28
Great Britain	29	6.23	4.65

As you can see, we're actually 3 times better than the USA, and loads better than China. So, we actually should have w... Wait a minute. What?!?! Looking further down the list I can see some pretty disturbing numbers. I'd better give you the whole table. This is it ordered by golds:

 

Country	Golds	Population (10 millions to 3sf)	G/10mill (2 dp)
USA	46	31.5	1.46
China	38	135	0.28
Great Britain	29	6.23	4.65
Russia	24	14.3	1.68
North Korea	13	2.46	5.28
France	11	6.54	1.68
Germany	11	8.19	1.34
Hungary	8	0.996	8.03
Italy	8	6.08	1.32
Kazakhstan	7	1.68	4.17
Australia	7	2.27	3.08
Japan	7	12.8	0.55
New Zealand	6	0.443	13.54
Netherlands	6	1.68	3.57
Ukraine	6	4.56	1.32
Cuba	5	1.12	4.46
Jamaica	4	0.271	14.76
Czech Republic	4	1.05	3.81
South Korea	4	5	0.80
Iran	4	7.51	0.53
Croatia	3	0.429	6.99
Spain	3	4.62	0.65
South Africa	3	5.06	0.59
Ethiopia	3	8.43	0.36
Brazil	3	19.2	0.16
Lithuania	2	0.319	6.27
Norway	2	0.503	3.98
Denmark	2	0.558	3.58
Switzerland	2	0.795	2.52
Azerbaijan	2	0.924	2.16
Belarus	2	0.946	2.11
Romania	2	1.9	1.05
Poland	2	3.85	0.52
Kenya	2	4.27	0.47
Turkey	2	7.47	0.27
Grenada	1	0.0105	95.24
Bahamas	1	0.035	28.57
Trinidad and Tobego	1	0.132	7.58
Slovenia	1	0.206	4.85
Latvia	1	0.207	4.83
Georgia	1	0.45	2.22
Ireland	1	0.459	2.18
Serbia	1	0.712	1.40
Dominican Republic	1	0.945	1.06
Sweden	1	0.951	1.05
Tunisia	1	1.07	0.93
Venezuela	1	2.72	0.37
Uzbekistan	1	2.91	0.34
Uganda	1	3.29	0.30
Canada	1	3.49	0.29
Algeria	1	3.71	0.27
Argentina	1	4.01	0.25
Colombia	1	4.67	0.21
Mexico	1	11.2	0.09

And this is it ordered by gold density:

 

Country	Golds	Population (10 millions to 3sf)	G/10mill (2 dp)
Grenada	1	0.0105	95.24
Bahamas	1	0.035	28.57
Jamaica	4	0.271	14.76
New Zealand	6	0.443	13.54
Hungary	8	0.996	8.03
Trinidad and Tobego	1	0.132	7.58
Croatia	3	0.429	6.99
Lithuania	2	0.319	6.27
North Korea	13	2.46	5.28
Slovenia	1	0.206	4.85
Latvia	1	0.207	4.83
Great Britain	29	6.23	4.65
Cuba	5	1.12	4.46
Kazakhstan	7	1.68	4.17
Norway	2	0.503	3.98
Czech Republic	4	1.05	3.81
Denmark	2	0.558	3.58
Netherlands	6	1.68	3.57
Australia	7	2.27	3.08
Switzerland	2	0.795	2.52
Georgia	1	0.45	2.22
Ireland	1	0.459	2.18
Azerbaijan	2	0.924	2.16
Belarus	2	0.946	2.11
France	11	6.54	1.68
Russia	24	14.3	1.68
USA	46	31.5	1.46
Serbia	1	0.712	1.40
Germany	11	8.19	1.34
Italy	8	6.08	1.32
Ukraine	6	4.56	1.32
Dominican Republic	1	0.945	1.06
Romania	2	1.9	1.05
Sweden	1	0.951	1.05
Tunisia	1	1.07	0.93
South Korea	4	5	0.80
Spain	3	4.62	0.65
South Africa	3	5.06	0.59
Japan	7	12.8	0.55
Iran	4	7.51	0.53
Poland	2	3.85	0.52
Kenya	2	4.27	0.47
Venezuela	1	2.72	0.37
Ethiopia	3	8.43	0.36
Uzbekistan	1	2.91	0.34
Uganda	1	3.29	0.30
Canada	1	3.49	0.29
China	38	135	0.28
Algeria	1	3.71	0.27
Turkey	2	7.47	0.27
Argentina	1	4.01	0.25
Colombia	1	4.67	0.21
Brazil	3	19.2	0.16
Mexico	1	11.2	0.09

So Grenada (a country I'm not even sure I've heard of) has stormed the league table with a whopping 95, making our 4.7 look pretty pathetic. We don't even make the top ten! Random places like Lithuania and Slovenia (who?) have done well, and places that I'm sure only exist for people to take holidays in have managed to come second and third!

I was really hoping that maths would prove we won the Olympics. I suppose this is why I should work out all the data BEFORE writing a blog post. It's not my fault: how was I supposed to know that our little Island actually has a lot of people on it? I felt so sure we were one of the smallest countries. I mean, we only have two types of Skittles over here! Big countries have at least five (including the sour ones, mmmm)! I am outraged.

And before anyone mentions it: the failure of this post has absolutely nothing to do with my lack of geographical knowledge.

The Point of This Post

This post has been a cluster-fudge of poor formatting and lazy researching, so you probably want the payoff now. Well here it is: whatever country you're in, think about doing the following activity at the start of the new term. Announce to the class: "I don't know what you've heard on TV, but Grenada actually won the Olympics". They'll be like, what? Show some pictures of the beautiful country of Grenada, and pictures of, uh...  *googles* nutmeg and mace and the uh... Grenada dove. Then, with or without giving them any extra information or instructions, but giving them access to the internet, get them to find out why Grenada actually won. I'm sure that intelligent pupils will be able to work it out, and you can give them lots of hints if need be. Extension: get them to prove that another country actually won the Olympics (e.g. by looking at the number of golds per GDP or something).

If you're worried the kids will google and find this blog and get the answer from here, fear not. As if any teenager is going to read through this long, dull piece! I'm surprised you've got this far, to be frank.

My Final Conclusion

That last bout of googling I did to find out the main exports and national bird of Grenada revealed something interesting to me: Grenada is a Commonwealth country. Do you know what that means? Yep, it's owned by Queen Elizabeth. Who? Yep, the Queen of England. So technically, technically, we won the Olympics. I knew it!!!

Emma x x x

Why Is It Called a Quadratic?

Today I had a really weird experience. I taught a group of students different from any other that I've taught.

Adults.

It was a lot of fun, and something I'd love to do regularly. Adults are just so... reasonable. The best thing about the session is that I got some really interesting questions. Like this one: why is it called a quadratic equation when the highest power is 2, not 4? I'm ashamed to admit I couldn't answer off the top of my head. I had to have a long think about it (and then I googled it, natch).

The reasoning is this: quad in this case doesn't really mean four, it means square. The formula for the area of any quadrilateral will be a quadratic equation, and for every quadratic equation, there exists a quadrilateral whose area is described by it. Similarly, a polynomial of order three is called a cubic not a tresic or something, because it is the formula for the volume of a cube or cuboid.

The pattern breaks down here, because there was no Latin word for hypercube (a four dimensional cube) and they probably weren't too bothered about finding the hyper-volume of one either, so they just called a polynomial of order four a quartic (as in four) and five a quintic, six a sextic, etc.

I know this question has been plaguing you your entire life, so I'm glad to have cleared this up for you! Now go out an enjoy the sunshine whilst it lasts!

Emma x x x

True or False PART II

The other day I asked you all a true or false question:

I brought this up at school today and it created quite a bit of discussion.


True or False:

If x^a = x^b  then a = b. 

Let me know your answer in the comments below!

I'll do a full post on this once I have some answers.

Emma x x x 

 Thank you to everyone who replied. As well as asking on my blog, I asked my students, my colleagues, and some non-mathematical people, including my dad. There are a few interesting things to discuss regarding this question. 


The "logic" of the question


The question I posed contained no existential or universal quantifiers. However, some people reading the question assumed that there were implied quantifiers which completely changed the meaning of the question. For example:


If x^a = x^b  for all values of x then a = b. Is this statement true or false? I think it's true.


The other thing some people confused was the logic of the phrase "if... then...". I would normally use an implication symbol instead but a) I don't know how to type that (my html is an fml) and b) I thought that it's likely not everyone would know its meaning. What surprised me was that even phrasing it this way, there were some people who misunderstood. I received some responses along the lines of "it can't be false because it works for [example]". They didn't understand that for this statement to be false only one example has to fail. This sort of logic isn't taught until A level maths. 


The types of number allowed in the question


 It is obvious that if we restrict x, a and b to natural numbers excluding 0 and 1, the statement is true. A lot of people, seeing the letters a and b, assumed they had to be integers. Just for fun, I asked the question to different people using different letters. When I used y and z, they were less likely to assume they were integers than when I used n and m. 

When x = 0 or 1, a and b can be anything*, rendering my question trivial. So to make things more fun, I told some people that they weren't allowed to use zero or one. This means to find a counterexample you have to go into complex numbers.

 Complex numbers and e

The question had first occurred to me when I was teaching a Further Pure 2 lesson, aand we had to solve the equation e³ = e^z which prompted one of my students to say, surely the answer's just z = 3? I had already gone through the solution on the board and we had come to the answer which by the way is z = 3 + 2kπj (for integer k). My student didn't believe this could be true, and I even started doubting it myself. I couldn't see how z could be anything other than 3. e is just a number, after all. It's kind of amazing when you think about it. That is what is so awesome about complex numbers, they always manage to surprise you. The reason why it works is because you can write

e^3+2kπj  as   e³  x e^2kπj   and it can be shown that e^2kπj = 1.

When I told somebody that this was the type of counterexample I was looking for, they protested, saying that I couldn't use e because e is a function. I replied that e wsan't a function, it was a number, 2.7182818... a bit like pi. They still weren't happy with this, and changed their track. x stands for a variable, but e isn't a variable, it's a constant. I found this very hard to explain: x is a variable and e is one instance of that variable. That particular instance doesn't work, so the statement is false. The person still wasn't convinced.

So, a simple question, but it generated a lot of discussion. It highlighted to me how many people (including maths teachers) have misconceptions with the whole idea of proof and logic and  numbers. 

Wow, this was a pretty heavy post for the last day of the school year! 

Fellow teachers, enjoy your freedom!

Emma x x x

PS If you're wondering why I call it j instead of i, it's because at my school we follow the MEI A level programme, and their links with engineering means it makes sense to use the letter preferred by those in industry. I also followed this syllabus when I studied A level myself, so j is what I grew up with. It was so weird at uni where everyone called it i! 

PPS I now have a contents page! Check it out!

*OK, not completely true, depending on how you define zero to the power of zero.

True or False...?

I brought this up at school today and it created quite a bit of discussion.

True or False:

If x^a = x^b  then a = b. 

Let me know your answer in the comments below!

I'll do a full post on this once I have some answers.

Emma x x x