Wednesday, 13 November 2013

Confession: I Hate Drawing Graphs

Shocker! Emma, the girl who loves maths, hates drawing graphs. This includes: cumulative frequency curves, histograms, graphs of parametric equations, roots of unity, and those log ones from C2 chapter 10.

Students are always surprised to find out there's an aspect of maths I find boring. They exclaim, "but you're a maths teacher!" Well, the nicer ones do. The not-so-nice ones exclaim, "but you're a massive nerd!"

I have a huge problem with this. Think of your music teacher from school. They loved music, didn't they? Do you think they loved all types of music? Classical, rap, indie, ska, dubstep, One Direction? Probably not. You wouldn't be surprised if you found out there was a genre of music they hated.

Think about an art teacher. They love art. But in their house, would they hang impressionists, surrealists, er... etc... on their walls? Would they love every medium, every era, every subject? Of course not. If they did, you would probably think less of their love of art. "Proper" art lovers usually know what they like, and are dismissive of, or even angered by, artwork they don't like.

So why shouldn't mathematicians be the same? And more importantly, why shouldn't our students see this? I let my students know when we're studying a bit of maths I'm not interested in. Maybe this puts them off it. But who cares? Of course, when my favourite bits come up, I make that known too. I also encourage students to have favourites in maths. I like my students to feel they have a specialism. It makes them feel good when that topic comes up. I want them to see mathematics as being similar to music and art: subjective.

Do you admit to students that there are things in maths you don't enjoy?

Emma x x x

Thursday, 10 October 2013

When Will I Use Algebra in Real Life?

There is one question my students ask me that I hate far more than all others. Far more than "Why are you just a teacher?" and "Do you have a boyfriend? What's his name? What's his job..." etc.

That question is: "When will I use simultaneous equations/ the laws of indices/ completing the square in real life?"

A good maths teacher, of course, would have a bank of answers to such questions. "Satellite dishes are in the shape of a parabola!" etc. Urgh.

Even though I do happen to know a few applications of algebraic principles to "real life" (picture me making air quotes with my fingers), I never tell these to my students. I refuse. 

First of all - real life? I'm sorry, are my maths lessons not real? When you enter room 204, are you entering some kind of alternate universe? Is a  maths lesson merely a state of mind? Some kind of lucid dream that looks real and feels real, but can't possibly be real because instead of English the teacher is speaking in an alpha-numeric jumble?

Secondly, where did students (and, for that matter, teachers) get the absurd idea that everything one learns has to have some kind of practical "use"? Can't we simply enjoy learning maths for its own sake? Does everything we do have to have a useful purpose? What kind of depressing life would that be?

Maths is beautiful. It is deep and interesting. It is a language. It is an art. It is a self-contained world with its own rules, patterns, and mysteries. So don't try to spoil my beautiful maths with your ugly "applications". 

I will leave you with some profound quotes from mathematician G H Hardy: 

"Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics". 

“Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.”

"I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

Of course G H Hardy was wrong with that last one - his work was actually widely applied to genetics and thermodynamics. But the point is, that wasn't why he did it. And he didn't need these applications as motivation for producing this work.

Do you agree with me or are you someone who sees maths as a "tool" with which to get things done?

Emma x x x

Friday, 4 October 2013

7 Habits to Get Your Year 7s into

(Substitute "year sevens" with "seventh graders" if you're American, "S1s" if you're Scottish, or "first years" if you're posh/old/a wizard).

Year 7s are so cute, aren't they? So eager to learn, so willing to please. Some are actually shorter than me, which is nice. By Christmas they'll have had their growth spurts and be taller than me. That's why I like this term. (FYI if you want a mental picture of me, I'm 150cm tall, 50kg, and look a bit like Garth from Wayne's World but slightly more feminine).

The thing with year 7s is that they are lovely little blank slates. They're a fresh batch of play-doh just waiting to be moulded. At my academy, we keep our classes throughout their school career. So it's important that you get your year 7s into good habits early on, to make your life easier later.

Here are the good habits I'd like to get my year 7s into:

1) Leaving the answer as a fraction

To some students, an answer of 3/5 doesn't look finished - because you haven't actually carried out the division. They would much rather put 0.6 because it looks like a proper answer. We need to stamp this out! Fractions are infinitely superior to decimals. The use of fractions should be encouraged from day one. Don't you just hate A level students who convert all their fractions to decimals? Think ahead, teachers!

2) Lining up the equals signs

Some teachers are very anal about this and I admit I'm not really one of them. But it does make algebra look a lot more beautiful when there is neat line of =s down the page.

3) Drawing margins

Why do maths exercise books not have margins pre-printed like all the other exercise books?! It drives me mad having to remind students to draw margins every day. It's amazing how some of them still forget - even my top set year 11s! We need to try some Pavlovian conditioning to get them to automatically reach for a ruler and pencil as soon as they open their books.

4) Drawing diagrams

To solve any geometrical problem, the first step should be to draw a diagram. This is something good mathematicians do automatically. I don't know which is the cause and which is the effect, but it's worth getting our students into this habit.

5) Resilience

Perhaps the most important characteristic of a mathematician is resilience. Try something. If it doesn't work, try something else. Don't tippex out your first attempt. Don't sit there with a blank page because you're scared of writing something that's wrong. If we can instil this attitude into our youngest students, they will grow up to be good mathematicians, whatever their attainment level.

6) Using a calculator properly

Calculators are great. I can honestly say I haven't done bus-stop division with pen and paper for a good 10 years. Because I own a calculator. My computer has a calculator. My phone has a calculator. Even my tape measure has a built-in calculator. Don't diss calculators. However, some students become instantly stupid as soon as they pick one up. They don't question whatever answer it spits out. Students need to be taught to estimate the answer first to check if it's roughly right. Also, calculators are really sophisticated these days, and for example you can type in the entire quadratic formula in one go without pressing equals in between or using complicated nested brackets. Teach students how to do this. Teach them about the magical s<=>d key. Explain how the fraction key works. Get them to make frequent use of the "ans" key. And most importantly, get them to buy their own and bring it in every lesson. But make sure it's a Casio! (Sorry Sharp, but you make my life so difficult. Please stop making calculators.)

7) Taking pride in their exercise book

That's "jotter" if you're Scottish. Or "notebook" if you're American. (Or "parchment" if you're a wizard).
When an exercise book gets filled up, students are supposed to keep it. What do most students do? Throw it away. How awful is this? The problem is that many students see their exercise book as the place where they do work, not the place where they write down things to help them understand. Also some books are just horribly messy! I find that if your work is neat, you take more pride in your book, and hence you put more effort into your work. I know presentation is not about learning and hence presentation-focused comments is considered ineffective marking, but I think good presentation does lead to better learning.

Those are my personal picks. Are there any you would like to add?

Emma x x x

Monday, 23 September 2013

What is x?

How do you introduce "x" to your students?

I'm guessing you start teaching informal algebra in this kind of way:

10 + ? = 20

Where the question mark obviously represents the mysterious-sounding "unknown". Except it's not really unknown, because it's obviously 10.

Eventually you replace the question marks (or empty boxes) with letters. Not just the letter x, obviously, but you have to admit x is a popular one.

In this number sentence, what is x? I don't mean what is it's value, I mean, what is it?

2x + 4 = 12

It's an unknown. It is a number that definitely exists and has one particular value which at this very moment is unknown to us but in a matter of seconds will be completely known. Two quick steps and we and x will be on first-name terms.

In this number sentence, what is x?

2x + 4 = y

Suddenly, x is no longer an "unknown". It is a "variable". Meaning, its identity is still a secret, but it's not one specific number, it could be any (any any?) number in the world.

2x + 4 = y and y = 5.

Now, suddenly, although x is a variable, it has been forced to stop varying, and simply be unknown.

Can you see how the dual nature of x (or any letter really) could be very confusing for students? If students think of a letter as representing one particular number (even if they realise that number can change on a daily basis), this might hinder them when it comes to studying linear graphs, or functions.

Maybe we should try to introduce x as a variable instead of an unknown. Think about how you could do this, perhaps with your brand new, untainted year sevens with their clean-blanket-of-snow brains.

Let me know how you get on,

Emma x x x

Sunday, 1 September 2013

What is a Regular Quadrilateral?

Have you missed me? It's been over a whole glorious month since I last posted. I've been lying on my sofa reading nineties teen romance novels and leaving the house only to visit Ikea for free tea and to pretend I live there. 

But anyway, I'm back and have I got a mathematical ponderance for you!

First, grab a piece of paper and a pen (or open a notepad file, for the more evolved among you) and write down (type) the definition of regular (as in, a regular polygon). 

Done? OK. Hands up who wrote this:

"All the sides are the same length".

And hands up who wrote this:

"All the sides are the same length and all the angles are equal".

Did any of you just write this:

"All the angles are equal".?

A more important question, perhaps, for teachers, is this: what do you tell your students?

Now another little exercise for you. Is this statement always, sometimes, or never true:

"If a polygon has all equal sides then all of the angles must be the same size".

Let's think for a moment. We know it is true for triangles, although you might not be able to prove it, or even justify it, beyond the fact that your Year Five teacher told you it was true during a particularly soul-crushing numeracy hour. We have also always assumed it was true for "big polygons" like octagons, decagons, etc. But is it true?

There is one type of polygon for which the statement is definitely not true. Quadrilaterals. Sorry if the title spoiled this major reveal for you. A square is equilateral and equiangular. However, a rhombus is equilateral but the angles are not all the same size. 

OK, here's another one for you:

"If a polygon has all equal angles, then all of its sides must be the same length". Always, sometimes or never true?

Again, think about triangles first, and then think about hexagons etc. Can you prove your conjectures?

Again, the statement is clearly false for quadrilaterals. A rectangle is equiangular but not equilateral. 

So what is a regular quadrilateral? Is it a square, a rhombus, or a rectangle?

My initial thought (because I admit, I didn't actually know the correct answer), was that a rectangle is regular. I thought this because geometry mostly comes from Greece, and in Greece, they're mostly bothered about angles. Hence the word polygon: "poly" meaning many and "agon" meaning angles. So a hexagon is literally a shape with six angles. In the UK, we're more likely to say a hexagon is a shape with six sides. So I thought a regular polygon would mean a shape with regular angles. 

However a quick tussle with my favourite search engine revealed that a regular polygon must be both equilateral and equiangular. 

I'm ashamed to admit I think I might have taught students that regular just means equilateral. Or, even worse, I think I might have even implied that equilateral shapes were always equiangular! It's funny how such a simple little definition can be messed up because you think you understand it perfectly (after all, I was taught it in year five). Maths teachers like me need to make sure we are completely clear about these things. Maths doesn't leave much room for error, and these definitions need to be water-tight. 

On that note, have a good first day at school!

Emma x x x 

PS I am of course talking about convex regular polygons. Non-convex (star) polygons are of course a whole other kettle of fish! 

Thursday, 4 July 2013

Is Zero a Factor of Zero?

Inspiration for this post came from an unlikely source: a bottom set year seven class and a non-specialist maths teacher. He was teaching them factors, and was asked by a student, "is zero a factor of zero?"

This question would have just about made me explode with excitement. I think my colleague's reaction was a bit different. He is a humanities teacher by trade, so naturally he was very good at fobbing them off with an answer that sounded impressive without actually answering the question.

This colleague has obviously been working with me for too long now, because after this lesson he came and asked me the same question, because he was actually interested in the answer. I even caught him reading my post Is Zero a Square Number?  at lunchtime. He's one of us now.

So, is zero a factor of zero? Well, as is often the case in maths, the answer is: it is if you want it to be.

It all depends on how you define a factor. Here are some possible ways.
For n and m natural numbers, n is a factor of m if:
1) n divides m with no remainder.
2) n x p = m, where p is a natural number.

[Side note: obviously natural numbers have negative factors too, and factors can be defined on integers rather than just the natural numbers, but negative factors aren't interesting, they're just the same as the positive ones but with a minus sign.]

Let's look at the two definitions:

1) From this definition, for zero to be a factor of zero, zero would have to divide zero with no remainder. What is zero divided by zero? That's my all-time favourite maths debate (and you all know I love to maths debate). Here are three possible answers:

-Anything divided by zero is infinity, therefore the answer is infinity.
-Zero divided by anything is zero, therefore the answer is zero.
-Anything divided by itself is one, therefore the answer is one.

In fact the answer could be anything you want it to be:
0 x pi = 0
Therefore 0/0 = pi.

So we say the question (and therefore the answer) is undefined. Or "MA ERROR" on your old Casio.

So by this definition, zero is not a factor of zero, in fact it can't be a factor of anything. However, every other number must be a factor of zero. Zero divided by anything other than zero is zero, which is a whole number with no remainder.

2) I think we'd all agree there exists a p such that 0 x p = 0. There are infinitely many such p! So by this definition, zero is a factor of zero.

So what is the answer? Well I'm going to solve this mathematical mystery the way mathematicians solve most of the really puzzling mathematical mysteries. I'm going to use the magic words: "by convention".

By convention, zero is not a factor of itself.


Emma x x x

Tuesday, 2 July 2013

The Terrible Twos

This week marks my two year anniversary as a qualified teacher, and also the two year anniversary of NQTpi. Woo!

As a third-year teacher, I look forward to:
-A slight pay rise (the last automatic one I'll have. Cheers for that, government).
-The authority that comes with the phrase "I used to teach your brother" (of course this is far less impressive than "I used to teach your father", but I've got a good few years until that one I hope).
-Possibly having a TLR (teaching and learning responsibility). I am interviewing for this next week.

Now that the "terrible twos" are behind me and I enter my third year, I thought I'd talk about the terrible twos that appear in mathematics. That is, the things in maths that are always taught together, but perhaps shouldn't be.

Word association test (please join in at home):

Area and ...

HCF and ...

Differentiation and ...

Volume and....

Here's what I think you said: perimeter, LCM, integration, and surface area. Am I right? If you didn't, then I'm guessing you're not a maths teacher.

These things are always taught in pairs. And these are all things that get confused.

My year 9 class are not completely stupid. But every single time they are asked to find the area of a shape, most of them give me the perimeter instead. Why?! I think it is fairly obvious that the word "area" means the amount of space inside the shape. I don't see how this can be confused with the length of the border. But students always get these confused.

Area and perimeter are always taught at the same time. I have heard many maths teachers say that they shouldn't be. They are two entirely different concepts, after all. If we taught them separately, would this confusion be avoided?

Similarly with HCF and LCM. My top-set students always get these confused. I think it's because they think that the HCF must be higher than the LCM, because of the name.

For me, the really interesting one is integration and differentiation. Obviously these are opposites. They're inverse operations, according to the Fundamental Theorem of Calculus. But when you think about what they actually do, they don't seem to be that linked at all. Finding the gradient and finding the area don't seem that similar. I think what many maths teachers do is teach differenriation, then teach un-differentiation, and announce that this is called integration, and then teach the application of integration to finding areas. I believe it should be the other way round: teach integration in its own right, and then discover that, holy sh*t, it's the opposite of differentiating! By the way, if your students swear in maths lessons it's a sign that you're doing something right.

Where do you stand on the area/perimeter: together or apart debate?

And congrats to all PGCE/GTP/PGDE teachers that have just qualified! Enjoy your NQT year!
And also congrats to all NQTs who have just passed their probation year! Enjoy your terrible twos!

Emma x x x

Friday, 28 June 2013

Introduction to Mechanics: Lesson Plan and a Speed Riddle

First, the riddle:

You are driving along a 2-mile long bridge. You drive the first mile at an average speed of 30mph. You want your average speed for the whole bridge to be 60mph. What speed do you need to drive at for the second mile?
This is how I began my mechanics taster session with next year's AS students. Have you worked it out yet?

The first  answer I got from the class was 90mph. Is that what you think the answer is? If you do, you're wrong. Sorry.

(30+90)/2 = 60, but this is not the average. You would spend longer driving at 30mph than at 90mph, and this average does not take that into account.

Hint Number One:

What is the definition of "average speed"? Total distance covered divided by total time taken.

At this point the calculators started to come out. Pens and paper had yet to make an appearance. I then started to get some bizarre answers like 2mph. This is an example of why a calculator without pen and paper is a dangerous thing.

Hint Number Two:

To average 60mph, how long should the entire journey take?

Speed = distance/time. So 60 = 2/t. So t = 2/60 hours. This would be 2 minutes. Have you worked out the answer yet?

Hint Number Three:

How long have they been travelling so far?

Speed = distance/time. So 30 = 1/t. So t = 1/30 hours which is... 2 minutes.

So it is impossible.

Surprised? I was. It still don't really see how it can be impossible. Surely if you go fast enough you can catch up? It seems wrong somehow.

Now, onto the rest of the lesson.

I borrowed from the science prep room a mechanical weighing scale. The kind you have in your bathroom. Side note: what is it about science prep room technicians that makes them so formidable? I have never returned a borrowed item so promptly!

Anyway, I put the scales on the floor and stood on them. I asked the class to look at the number displayed. I then pointed out I was wearing heavy clothes and block heels and a big watch and I'd just eaten lunch. Then I asked the class what would happen to the number if I put my hands on the back of the chair in front whilst standing on them. They correctly told me the scales would say I've lost weight. Then I asked if there was a way for me to make the scales think I've gained weight. That's a more interesting question. I'll leave you to think about that.

Then I asked them to consider a person weighing themself whilst in a lift. What would happen to your weight when the lift is going up? The class was split almost exactly in half on this. Some thought you would gain weight because the lift is pushing the scales into your feet. Some said you would lose weight because the lift is pushing your feet off the scales. They were also divided over what happens when the lift is going down.

Mechanics is a very sciency bit of maths, and when there's a debate in science there's only one way to settle it: an experiment! So I simply declared, "To the lift!" The students were surprised and I think a little bit excited. We all went to the lift and took it in turns to go in groups of four up to the third floor then down to the ground floor, then back.

We returned to the classroom and discussed our findings and tried to explain them.I won't tell you the result of our experiment, but if you ever get the opportunity to try it out, please do!

Next I held up a tennis ball and a basket ball (borrowed from the PE department, who are a lot less scary than the lab techs) and told them we were going to do another experiment. I asked them which ball they thought would hit the floor first if we dropped them at the same time from the same height.

Again, the class was divided. We discussed mass, surface area, rigidness, air resistance... It was a good discussion. And then we gleefully left the classroom to do our experiment. Half of us went to the top floor (the third floor, or the fourth floor if you're American), and the rest went down to the bottom. Our school is kind of open so that from the top floor you can see all the way to the bottom if you lean over the balcony on the inside. This made it ideal. (The building won an award recently for its awesome architecture).

There were a few students in the corridor working on the computers or printing stuff so we drew a bit of an audience. And when the two balls hit the ground (at the same time? Well, that would be telling...) we definitely drew some attention to ourselves! It was loud.

So we went back to the classroom and discussed our findings. We talked about how the experiment was kind of rubbish because there were too many variables. So I told them we were going to watch a better experiment where these variables were controlled. That's when I showed them this clip from Brainiac. Sorry about the Arabic subtitles.

That concluded the lesson. I think it was a great way to introduce the mechanics module and give them a bit of a taster. Hopefully it has made them excited to start their A level maths in September!

Emma x x x

Monday, 24 June 2013

Happy Palindrome Day to Me

Today I am 8888 days old.

Enough said.

Emma x x x x x x x x

Thursday, 20 June 2013

A New Way to Teach Dividing Fractions

How do you divide a fraction by another fraction?

For example, how would you do something like this:

My guess is you would flip the second fraction upside down and then multiply like so:

If you are a maths teacher, is this how you teach students?

Do you think your students understand why this method works? And, be honest, do you understand why it works?

Well today in the maths office at my academy one of my colleagues showed us a new method he'd thought of.

It works like this:

I think this is a little bit more intuitive.

My colleague got the idea from one of his year sevens who had answered this question without showing any working out:

The answer is quite obviously three. How many quarters are there in three quarters? Three, duh. But I am quite certain most of my A* students would perform the technique of flipping and timesing without even thinking.  My colleague was impressed that this student had used some common sense. He wondered whether the same idea could be applied to fractions with different denominators. It is a little bit less obvious that 21/28 divided by 20/28 is 21/20, but it's not entirely unbelievable. Whereas the "trick" of flipping and timesing can look a little bit like magic to some students.

I haven't tried teaching this method so I can't comment on its effectiveness yet. But as a mathematician it appeals to me. It's quite neat. And in case you were wondering, yes this works with algebraic fractions too.

If you're going to be teaching fractions soon, why not try this out? If you do, please let me know how it goes.

Do you think this is a good method?

Emma x x x

Wednesday, 19 June 2013

A Mathematician's Lament

Sorry it has been so long since I last wrote. Exam season is a very busy and stressful time for teachers as well as students! (Perhaps even more so). Now that the wonder of "release time" is upon us, I should be able to write more.

I stumbled upon a piece of writing called A Mathematician's Lament. I think it is one of the most beautiful pieces of writing I have ever read. As I was reading it, I found myself screaming (inside my head, I don't want to make a scene - I'm British remember) "that's exactly what I think!". It's like Paul Lockhart looked inside my head, stole my thoughts, and wrote them down but in a nicer order and with better punctuation.

I urge you to read it if you can spare the time. At least read page one. For those of you who really can't be bothered, here is a TL;DR summary for you:

*Mathematics is an art.

*Mathematics should be taught the same way that music and art are taught.

*Maths is not about acquiring skills that you might apply "one day".

*Much like music lovers have certain types of music they like and certain types they dislike, mathematicians can dislike some types of maths. It is a matter of taste.

*Maths is about imagination, you do not deal with real things because real things are inaccurate and messy; maths is simple and beautiful.

My favourite quote:

"there is nothing as dreamy and poetic, nothing as radical,
subversive, and psychedelic, as mathematics."

I have shared this with my department and I am considering sharing it with my students as well. I think it is especially important to share with year 11s who are looking to take A Level maths. Those students who have thought they were good at maths all these years because they got all the answers right might not realise that rote learning won't get them very far in A level.

Read the whole thing and then tell me what you think in the comments below.

Emma x x x

Thursday, 25 April 2013

The Top Ten Stupidest Teaching Tips

There is a lot of advice out there for teachers. As an NQT, every experienced teacher you meet will probably start their first conversation with you by imparting some "words of wisdom". Often this begins with, "I don't know what they've taught you on your PGCE, but in my experience [insert gratuitous advice here]".

I have been given a lot of bad advice is my short time as a teacher. Some of it was from my PGCE course, some from books, and some from the soapbox in the corner of the staff room.

So, here is my list of the top ten stupidest teaching tips (in no particular order):

"Don't smile until Christmas"

This is probably the most oft-quoted teaching tip. But does anyone actually do this? I think it's terrible advice!

I love maths. I want my students to love maths. Can you share your enthusiasm for a subject without smiling?  If I explain a topic in maths completely straight-faced, doesn't that imply to my students that I don't find that topic interesting/exciting/inspiring? And they'll be thinking, even Miss finds this boring and maths is her favourite subject!

Also, the first term of teaching a new class is when you're trying to make relationships. Is it possible to make a relationship with someone without smiling?

Of course you want to lay down the law and show them who's boss, but without coming across as cold, boring, and unrelatable.

"As your students leave the room, stand by the door and tell each of them you love them".

You probably think I've made this one up. But no, I read this in Rocket up Your Class! which I actually thought was a great book until I read this particular titbit.

However, being the diligent student that I am, after reading this advice, I decided to try it out. I don't think I can really call it a success. The first student out the room was definitely uncomfortable, the last person out the room only got a very lacklustre "Uvyoo" (you try saying "I love you" thirty times in thirty seconds!) The students didn't even find it funny, they just thought I was weird.

The main problem I have with this is that to be quite honest, I don't "love" my students. They're perfectly nice people (some of them) and all, but my love is reserved for immediate family, pets past and present, and chocolate-covered popcorn. Sorry kids.

"Positives and negatives should be given out in the ratio of 4:1"

I've already written about my scepticism of this. I know teachers who are highly effective and have excellent relationships with students who very rarely use praise, and never use rewards.

I hate over-praising. It just devalues the whole currency. I'd rather be praised meaningfully once in my whole seven years with a teacher, than shallowly every lesson.

"Never take work home with you"

If it were possible to not mentally take work home with you, I might have thought this was good advice.

In terms of physical work: marking, lesson planning, reports, various admin (far too much of it, considering teachers aren't technically supposed to have to do admin)... the way I see it there are two choices.

Go home lugging your books on your back (or in the boot, if, unlike me, you have a car), get home at 4:30, make yourself a cup of tea by filling the kettle from the kitchen sink, not from the communal toilet's sink (the closest water source to the maths office), using milk that's not perpetually "on the turn", in your favourite mug, not whatever vessel you can find that hasn't been appropriated by the faculty slob. You can immediately change into your jammies, put on your guilty-pleasure music (a certain adorable quintuple springs to mind), and mark/plan/do stuff whilst stuffing your face with roasted chickpeas (too smelly to eat in the office).


Stay at school, do your work, get home at 7pm, and smugly say to your spouse/kids/tamagotchi: "I don't take work home with me. Check out my work-life balance!"

"Don't do more than one of the same type of question"

This advice was actually given to me by an OfSTED inspector. Despite my new-found respect for HMIs, after they showed excellent judgement in awarding me the coveted title of "outstanding teacher", I still feel I have to disagree with this particular tip.

I talked at length about this in this post. I won't bother repeating myself.

"Never touch a student"

Way too many teachers are terrified of being called the P word. A gentle hand on the arm isn't going to cause any trouble! A pat on the back isn't going to put you in jail! (Obviously if the student in question has ASD or is a CP case then exercise caution).

On January results day just gone, one of my year 12s asked me for a hug. I am ashamed to say I almost said no. But then I thought, screw it, this girl was so nervous this morning and now she's overcome with relief, of course I should give her a hug! She shouldn't have had to have asked, I should have offered her a hug! It's a one-off ting, it's not a big deal. Stop being scared.

"Show students the insides of your wrists to show them you trust them"

This wee gem was imparted on me by someone at university when I was doing my PGCE. Not only is this advice bizarre, it is also incredibly hard to do. I challenge you all to attempt this in one of your lessons this week. Unless you have a curiously-shaped mole on your wrist you can somehow incorporate into your starter activity, I don't think you'll manage it.

"Keep your hands above waist height to demonstrate power"

Another PGCE one. I should point out these were external speakers, not my actual teachers, who were all brilliant.

My question is this: where can you put your hands above your waist? On your shoulders? Under your armpits, in the manner of a gorilla? Stroking your chin with your left and scratching your head with your right?  Or arms spread wide, as if welcoming in your students' ideas and basking in their insights. Hmm.

Oh dear, a quick google has revealed to me that this advice is not only given to teachers but to business people who have to make important presentations and speeches. Listen up guys, here's my public-speaking advice: if you need to resort to body-language tricks to get people to agree with you, the content of your speech is probably rubbish.

"Do vocal exercises in the shower every morning"

We had quite a few sessions on looking after your voice during my PGCE. Teachers talk a lot compared to people in other jobs, and voice strain is definitely something that could be an issue. I know that the first day back after a holiday my voice always hurts by the end of the day.

But do any teachers ever actually do vocal exercises? Every morning? In the shower? Don't get me wrong, I enjoy my sessions with the loofah-microphone as much as the next person, but certainly not at 5:45am, and anyway, my performances are limited to actual songs, not odd bird-like squawks and exaggerated vowel sounds. I can't believe I attended those lectures.

"Get a gun"

No, I've not gone crazy. Read this article. I'm not even going to pass comment. Except perhaps to say: oh dear, America.

Got any stupid teaching tips for me? Post them in the comments!

Emma x x x

Saturday, 20 April 2013

Factorising: a Divisive Topic

See what I did there?

We had a faculty meeting last week, which can only mean one thing: a raging argument about the best way to teach something. We're an opinionated bunch.

The topic in question was quadratic equations. More specifically, factorising, although we touched on solving them in general.

What method would you use to factorise this?
6x2 + 26x + 20 = 0

The mathematician answer, of course, is I wouldn't, I would just use the quadratic formula.
The more observant amongst you will have divided by two first and then found the answer quite easily.
For the purpose of this exercise, dividing by two is not allowed, and neither is taking two out as a factor in the beginning.

This is the method that was put up on the board for us to discuss:

6x2 + 26x + 20 = 0
20 * 6 = 120
Two factors of 120 which add to make 26 are 20 and 6.
6x2 + 20x + 6x + 20 = 0
2x (3x + 10) + 2(3x + 10) = 0
(2x + 2)(3x + 10) = 0

I would estimate that about a third of our faculty looked at this and immediately said, yes, that's how I teach it. Another third said, I know that method but I don't really teach it, and the last third said, I've never understood that method, I think it's stupid.

OK, that last group was probably smaller than the other two, but it felt bigger because I was in it.

I hate this method! It took me ages to get my head round it. I finally got round to proving it to myself so I feel happier, but there is no way I would teach this to a class, because first I'd have to prove it to them to show them where it comes from. I don't think they'd be able to use this method without understanding how it works.

But apparently, I am wrong. There are a lot of students who are successful using this method, and many teachers swear by it. The alternative, trial and error kind of method that I use is obviously quite annoying (hence why in those cases where the coefficient of the x squared term is not 1 or a prime, I would never bother factorising) and it probably puts some students off.

So here's the big question: should we teach students a method they do not understand if it makes getting the answer faster?

Primary schools have moved away from teaching un-understandable methods like column subtraction, bus-stop division, long multiplication in columns, etc towards using methods that students can understand how they work, like chunking, partitioning, open numberlines, the grid method, etc. Surely we should be doing the same higher up the key stages?

I would love to know your opinions on this. If you're a maths teacher, which method do you teach? If you're not, what method were you taught when you were in school?

Emma x x x

Appendix: Explanation of above method

ax2 + bx + c
= (kx + l)(mx + n)
=kmx2 + knx + lmx + ln
=kmx2 + (kn + lm)x + ln

So we are looking for k, l, m and n such that:
km = a
kn + lm = b
ln = c

Aim to split b up into kn + lm
ac = kmln = kn * lm
So we can factorise ac such that the two factors sum to make b.
Working forwards:
ax2 + bx + c
= kmx2 + knx + lmx + ln
=kx(mx + n) + l(mx + n)
= (kx + l)(mx + n)

Tuesday, 2 April 2013

Minus versus Negative: Some Mathematical Grammar

Ooh, today you're getting a discussion of maths and grammar, aren't you lucky?

We had a "moderation day" last week. It's an INSET day where teachers moderate their coursework. As you can probably imagine, the maths department was incredibly swamped that day. NOT!

Maths doesn't have coursework, so theoretically, we didn't have anything to do. In practice, however, we had absolutely loads to do, because we are still teachers, and a teacher's work is never done. That sentence has far too many commas. Should I really be writing a post about grammar? You can always put bad grammar down to style, can't you? My style is to use too many commas. Like, this.

Anyway, the maths department decided to take a long lunch on this moderation day, and went to a popular pizza restaurant armed with multiple two-for-one codes. The joke "how many maths teachers does it take to split a restaurant bill?" comes to mind.

We spent most of the meal maths debating. This often happens to us. Luckily the restaurant was almost empty, or it could have been quite embarrassing. We were scrawling equations on napkins using board markers (the only pen we ever have on us) by the end.

The subject of the debate? Should the word "minus" only ever be used as a verb?

Read the following sentence out loud: The weather today is -2 degrees. How did you say it? Did you say "minus 2 degrees"? Or did you say "negative 2"? I would guess that if you are from the UK you probably said minus. I know that's what I say. It's definitely what the weather people say on TV.

Now read this out loud: x - 5 = 7. Did you say "minus" again? You might have said "take-away", possibly "subtract", or even "less".  I would say minus, probably because this is what all of my maths teachers used to say to me.

Third test: what rule were you told about why -5 x -2 = 10 not -10? Say this rule out loud. Did you just say something like "a minus times a minus makes a plus" or "two minuses makes a plus"?

Can you see a slight issue? We're using "minus" as an adjective, meaning negative, and we're also using it as a verb* meaning subtract. And thirdly, we're using it as a noun when we say "a minus" meaning a number less than zero.

We're all quite comfortable with this word having several meanings. But what about students? When they first learn about "directed numbers" (as they're known), does this odd quirk of English confuse them?

I can see why some might think this. I understand that the language of mathematics should be used very carefully. I've always been very interested in grammar, which was why I did A level French (which, incidentally, everyone thought was weird: most of my teachers assumed I would be studying maths and the three sciences). At uni I took some modules that were about logic, which is pretty much just another word for language. I've taught enough EAL students to know that you need to choose your words carefully. But to be honest... I'm not completely convinced.

In a "number sentence" (a wonderful expression, thank you primary school teachers), I will always pronounce a dash (or hyphen, or em dash, or en dash) as "minus". Let me tell you why: that little symbol represents two things at once; it's the operation of taking away, and it's also to indicate something that is being negated (notice that both of these things are actions: I'm not saying it represents a negative number, I'm saying it represents something that is being negated). You absolutely need this symbol to represent both at once, because you want to be able to swap between the two meanings depending on how you feel.

Take this example:

3 - 4 (x - 7) = 10

If you wanted to solve the above equation, there are a few ways you could do it. Before reading ahead, please solve it.

How did you treat the minus before the 4? Did you see it as indicating something you're taking away from 3? Or did you see it as attached to the 4, making it a negative 4?

Did you do this:
3 - [ 4x - 28] = 10 (expanding the bracket with 4 as the multiplier)
31 - 4x = 10

or this:
3 [- 4x -- 28] = 10 (expanding the bracket with -4 as the multiplier)
3 - 4 x + 28 = 10

Or something different?

Can you see that if I, as a teacher, had indicated in some way that the minus before the 4 was a negative symbol, the first method wouldn't really make sense? And if I had indicated it was a take away, the second method doesn't really make sense, because students aren't taught that subtraction follows the distributive law.

The duplicity of the minus is one of those mathematical things that makes sense when you are mathematically fluent. Just like in English, how we have words that look the same and sound the same but mean two different things. As a fluent speaker of English, I don't even notice these. Look, I just used one! Notice! I didn't have to think: wait, is this the verb to notice, or a kind of sign stuck on a wall? I just used the word. And guess what, when I learnt French, my professeurs didn't just remove all homophones from the syllabus so that, as a learner, I wouldn't get confused, they left them in, so that I could aim to become fluent. Why should maths teachers do this? Don't we want our students to become fluent in maths?

Yes, we should be careful with our language in maths lessons. We should make sure when we say "line" we don't mean "line segment". But we cannot protect our students from the difficult to understand bits. We need to expose them to these things.

What do you think?

Emma x x x

*Technically it is a preposition rather than a verb. But these days we use it as a verb, saying things like "minusing" and "you minus the five from both sides". I know technically these uses are wrong, but it's what we say. Just like how we say "timesing" and "timesed" because we use the word "times" as a synonym for multiply now.

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

Tuesday, 19 February 2013

You Are Not a Geek.

I love shopping for new clothes almost as much as I love doing integration by parts, and in the past I have particularly delighted in finding geeky clothes, especially in mainstream shops. However, whilst shopping yesterday, I noticed a new trend that seems to be in all the shops: Topshop, New Look, Primark, even those cheap-looking factory shops that I would have loved as a thirteen year old. The trend is simply this: the word Geek.

So many t-shirts saying "GEEK" in bold capitals. Sweatshirts, hoodies, everything really. And all around me were trendy girls with their hair in gravity-defying topknots, wearing Vans, ankle-grazers, biker jackets, and these GEEK tees.

I had this barely repressable  urge to go up to them and say:
"Do you think p = np will ever be solved?"
"Who is your favourite A Song of Ice and Fire character?"
"Did you get the reference to the Contra code in Wreck-it-Ralph?"
"Have you tried running Steam on your Raspberry Pi yet?"
or even "Do you know Pythagoras' theorem?"

And if they weren't able to answer these questions I would have shouted in their faces: "YOU ARE NOT A GEEK!"

I take pride in being a geek. It is a label I wear proudly (although I don't literally wear it). It makes me angry when other people claim to be geeks. Being a geek can be hard, especially as a teenager. The title of Geek should be your reward.

So please, don't dilute the meaning of geekism by applying it to unworthy things.

Annoying Facebook Girl - Spend 3 hours on facebook LOL i'm such a geek

Emma x x x

Monday, 21 January 2013

The Two Children Problem

Sorry I haven't posted in so long. For a while I became a bit demotivated and sort of fell out of love with teaching. I also moved house, leaving me without internets for over a month. I'm not gonna lie, that was tough.

So here goes my first post of 2013 (and I'll spare you the mathematical factoids about the number 2013: my students seemed to know them all from Facebook before I told them, anyway).

A few days ago a friend with whom I did my PGCE posted a really rather excellent maths problem on Facebook, claiming it to be better than Monty Hall. It went like this:

I have two children. One of them is a boy born on Tuesday. What is the probability I have two sons?
I managed to work out the answer within twenty minutes (not bragging, just sayin') but I kept thinking about it for several hours afterwards, and then spent all week talking about it to anyone who would listen, and even those who wouldn't (sorry colleagues, sorry boyfriend, sorry dad...).

But that's not actually the problem I'm going to discuss. As interesting as I found it, I found that the people I discussed it with got themselves hung up on a less interesting part of the puzzle. I couldn't believe that something I thought was so clear cut was causing so many people such cognitive conflict.

So I am going to discuss the Two Children Problem (the problem above but without the Tuesday bit). 

I thought it would be fun to lay this out in a Sophie's World kind of format. I am going to discuss the problem via the emails sent between my dad and myself for the past few days. These are almost entirely unedited. Enjoy!

Me: I have a good probability puzzle for you. I have two children. One of them is a boy. What is the probability I have two boys?

Dad: Off the top of my head, I would guess 50 – 50.

Me: It's not.

Dad: I don’t understand.  The other child is either a boy or a girl.  That means there’s a 50% chance of it being a boy. What’s the snow like in Coventry is it bad?

Me: There are four options: BB, BG, GB, GG.
I have eliminated GG by telling you one is a boy.
So the probability would be 1/3.
It is a bit like the Monty Hall problem (two goats, one car).
The snow is mental!! Got sent home from school at 12:30. Walked home. In un-walked areas it was about 15cm deep.

Dad: No there are only two options because:
GG is not possible and GB and BG are the same option (nothing is mentioned here about whether one is older or younger).  Hence the only options are BG or BB.

Me: But they are not the same. I didn't point at a child and say this one is a boy. I said one of them is a boy.

Dad: Yes and so if one of them is a boy, the other is either a boy or a girl. 

Me: No!
Flip two coins. If both are tails, re-flip them both. If not, then one of them is heads. What is the chance the other one is also heads?

Dad: I have to go home now.  Send a more explicit explanation to my home e-mail.  I still don’t believe you.

Me: f you flip two coins, and reflip them if they're both tails, then at least one of your coins is heads. What is more likely, they're the same or they're different? 

Dad: I don't understand this analogy with coins.  Surely the analogy is that:
You flip one coin and it turms out tails (a boy).  That is now a known.  If you then flip another coin it may be heads ( a girl) or tails (a boy).  So the question is if you flip one coin and it turns out to be tails, and then you flip another coin what are the chances of it being the same or different as the first coin?  With the second coin the probability is 50 - 50 of it being either heads or tails.
Is this not the case?

Me: This is not the same. You are right that if you flip one coin and it is heads, then the other coin is 50-50. But what we have done is flipped both coins, looked at both of them, and noticed that at least one of them is a head. So there are three possibilities: HH, HT, TH.

Dad: All right, I think I get it now. 
If you flip 2 coins at the same time, what are the chances of them both being heads, if one of them is a head.  Then the answer is 1 in 3.  

I think that what troubled me here was a notion of sequence.  I am still thinking about this.

What about this? There are 2 goals in a football match. Palace have scored 1 of the goals. What is the probability that Palace have scored both goals?

What is the probability of Palace scoring both goals? Surely the score is either

Palace1 : 1 Bolton or
Palace 2 : 0 Bolton

Therefore there is a 50 - 50 probability that Palace have scored both goals, (and in the real world Glenn Murray probably scored both of them).

What is the fallacy if,

Goals equals children
Palace goals equals boys

Therefore the score is either

1 boy : 1 girl
2 boys : 0 girls

Me: Hmmm this is a really interesting question. I am thinking about it.

Dad: What about this.  If you put two blue counters (boys) and two red counters (girls) in a bag.  If you take out a blue counter, it immediately negates the existence of one of the red counters, because it is no longer a possibility.  Therefore for next pick there is one red and one blue counter in the bag and therefore a 50% chance of it being blue.
With the coins, if for example, the first coin is a H then it negates the possibility of
therefore there are only two possibilities;
H  T or H  H.
If the second coin is a H, it similarly negates the possibility of  H T and so there are only two possibilities T H or H H.  Therefore in both cases if you know that at least one of them is a H, there is a 50% chance that the other is also H.

Me: You are correct in what you have said above, but that is a completely different situation.
The original problem can be thought of like this:
Pick a million families that have two children. Discard all those that have two girls.
Are you really saying that half of these families will have two boys, and the other half will have one of each?
Surely you can see that that is not true in real life. 

As for the football question, let's think of it in the same way:
Pick a hundred Palace matches where two goals have been scored, eliminating all those where Palace did not score.
Of these matches, do you think Palace will have won half and drawn half, or do you think they are more likely to draw? 

Dad: Let's say 990,000 families instead of 1,000,000.  Discard families with two girls. Is that 330,000?  That leaves 660,000.  Half of those are families with two boys.  That is 330,000.  Wouldn't that fit with real life?  You end end up with 50% boys and 50% girls.  (Actually in the real world I think there are slightly more female births than male births).

Me: You are wrong. I think it's more like this:

1 000 000 families.
250 000 two girls
500 000 mixture
250 000 two boys.

If you don't agree, consider a family with 4 children.
Combos are:
0 boys 4 girls
1 boy 3 girls
2 boys 2 girls
3 boys 1 girl
4 boys 0 girls.

You might think these 5 are equally likely. But actually they're not.

The 2 of each option is actually 3 times more likely than having 4 of one. 

This is based on the Binomial theorem. Are you familiar with that? Do you know what Pascal's triangle is?

Dad: If you flip a coin and it turns out H does it make it more likely that when you flip another coin, it turns T? If so, why?

Yes I can see that the combinations are

And therefore a 25 percent distribution. But again if each birth invoves a 50 - 50 chance, how does one birth apparently influence another? 

Is pascal's triangle a kind of musical instrument?

Me: One birth does not influence the other. But knowing that one of them is boy only tells you that GG is not an option. There are still three equally likely options left. Note that by "one of them" what we really mean is "at least one of them". 

Dad: Interestingly in 40 football matches where there were at least two goals, 23 of which were 1- 1 after 2 goals and 17 were 2-0 or 0-2. Therefore approx 58 perc to 42.

Me: Well that supports my argument! Convinced yet?

The conversation ends there (for now). Do you think the answer is a third or a half? I find it weird that my dad still doesn't quite believe me. I suppose that is the difference between a mathematician (me) and a linguist (him). I look for meaning by extracting the bare minimum of information to simplify the problem, whereas he looks deeper in between the lines to find meaning. 

It goes without saying that my way of thinking is better. :-)

Emma x x x 

PS Look out for part two, where I'll discuss the Tuesday boy problem.