My First Published Maths Article - Maths V Science

The secondary school I attended was a specialist science college, and so we had an in-house science magazine that came out every term. When I was in year 13, I submitted an article for it. This was a little bit weird because I wasn't taking any science A-levels, so I wasn't part of the science "crew". What was weirder was that the article was anti-science. It is about how maths is so much better than science.

It was the first time I'd written about maths and I loved it. I've always loved writing. I've kept a journal since I was 12 (when I first read the Princess Diaries) and I always did well in creative writing in English. Aged 17, my passion for maths was really starting to show itself. I'd been reading books by Simon Singh and Marcus du Sautoy and Ian Stewart on the bus to school and I was spending roughly 30 hours a week studying maths. Writing this piece made me realise that this was what I wanted to do: try to get people excited about maths through words. Which is what teaching is about, and also what this blog is about.

Anyway, here is the article.

MATHS V SCIENCE

A not-at-all-biased look at the strengths and weaknesses of two essential subjects*

Round One: Proof

It is an indisputable fact that mathematical proof is far more powerful and more rigorous than scientific “proof” (please note the use of ironic quote marks). This is because when a mathematician proves something to be true, we know that it will always be true and that there is not even a small, slight, snowball-in-hell chance that it will ever be false. Mathematical proof’s spotty younger brother, scientific proof, relies on evidence backing up a hypothesis. So if an enormous amount of evidence supports their claim, a scientist will say they have proved that it is right. However, consider this: we know that if you flip a coin 100 times you would expect 50 heads and 50 tails, but we know that it is possible to get all heads and no tails, or vice versa. Now if a scientist flipped a coin fifty million times and got all heads, they might well conclude that this coin will always produce heads. But how do they know that on the 50 000 001^st time they flip it it won’t be a tails?

Maths: 1 Science: 0

Round Two: Problem Solving

Amongst other things, mathematicians and scientists have to solve many problems. The main difference between the problems that mathematicians solve and those that scientists solve is that, most of the time, scientific problems are of great importance, whereas maths problems are usually as useless and unnecessary as sleeping pills in a geography lesson. One of the greatest mathematicians ever, GH Hardy once said “no discovery of mine has ever made or is likely to make…the least difference to the amenity of the world”. (in actual fact, Hardy’s discoveries have been of great use, but lots of the maths that number theorists do is really comparable to thermal-underpants-in-hell in terms of usefulness). Despite this, it is clear that mathematicians are better at problem solving than scientists are. Consider the following example: you have a normal 8x8 chessboard, and you cut the top left and bottom right corner squares off. What you are left with looks like this:

So there are now 62 squares remaining. You have been given 31 dominoes, the same size as a pair of squares. Can you put all 31 dominoes on the board so that every square is covered? A scientist would solve this problem by experimenting, by trying out different arrangements. (Have a go yourself- you can make a board and dominoes out of paper. It’s a fun alternative to listening to Mr Clargo in your maths lesson). After trying out a few different arrangements with no success, a scientist would conclude that it can’t be done. However, the scientist would never be sure whether they are right or not (unless they tried every one of the millions of arrangements-which would be a very pitiful existance). Now, a mathematician would tackle this problem logically : the corners that were cut off were both black. So there are now 30 black squares and 32 white. Each domino covers two adjacent (neighbouring) squares. Adjacent squares are always the opposite colour. That means that for every white square that is covered, a black square is also covered. So to cover all 32 white squares, 32 dominoes would be needed, and we only have 31. therefore, it can’t be done. Pure genius! The mathematical way of solving problems is often quicker than the scientific way, and you can always be sure that you are right.

Maths : 2 Science: 0

Round Three: Fame and Fortune

Most of you will probably find it hard to name a famous mathematician or scientist who is alive today (and no, the guys from Braniac do not count). However, if you look way back in time, the only people from that period who are remembered are the mathematicians. So whereas most of you don’t know who Andrew Wiles** is, in a thousand years time, he will be remembered and Jade Goody will not. Again I am going to refer to the great GH Hardy: “Immortality may be a silly word, but probably a mathematician has the best chance of achieving whatever it may mean”. To be fair, a few scientists will also be remembered. I suppose. Now as for fortune, you should know that there is a lot of money up for grabs for talented mathematicians (or someone with a calculator and way too much time on their hands). If you find a prime number with loads and loads of digits, the FBI will give you $1 million. However this is easier said than done (I have wasted many a Sunday afternoon number crunching to no avail).

Maths: 3 Science: 0

There we have it; maths wins hands down. So if you think maths is boring, hard, or pointless, think again!

*please note: maths is more essential than science.

 ** English mathematician famous for solving a 358 year old problem: Fermat’s Last Theorem

By the Way...

Several things in this article were edited without my permission for the magazine. The above is the unedited version. The bit about Mr Clargo was edited out. Also, the asterisked bit "*please note: maths is more essential than science" was changeed to: "*please note: maths is more essential than science, in my opinion" which completely ruins the joke. It was obviously written in a tongue-in-cheek way, I really don't know why they had to change it and make me look like a dork.

What do you think, not bad for a 17 year old?

Emma x x x 

"I'm rubbish at maths!"

Why is it socially acceptable to admit you're bad at maths?

In fact, it's not just socially acceptable, it's encouraged. You would feel more uncomfortable announcing that you're quite good at maths than saying you're bad at it. I'm right, aren't I?

We had an NQT/PGCE meeting the other day about numeracy. As soon as it started, there was a chorus of "Oh, I'm rubbish at maths"s from the non-maths or science teachers. One English specialist said, "actually, I like maths, I'm quite good at it", then hastily added, "I know, I'm weird!". Why do we feel the need for such an addendum?

The week before, we had a meeting about literacy. Now it will not surprise you at all to learn that the meeting did not start with a load of teachers saying "Oh, I'm rubbish at reading". I have never once heard an adult admit this. Admitting you are illiterate is very tabboo. So why are people falling over themselves to declare that they are innumerate?

It really annoys me when I meet parents of my pupils and they say to me, "I'm terrible at maths, I hated it at school". Children naturally copy their parents, so they pick up on things like this, and they think it's OK to say things like this. And as soon as a child gets it into their head that they're bad at maths, they become worse at maths. It's a self-fulfilling prophecy.

I hate it when people claim to be bad at maths. In my opinion there should be just as much stigma on being innumerate as there is on being illiterate. Maths is as much a life skill as reading and writing. Everybody in the UK should be able to do basic maths, just like everybody should be able to read and write. And if you can't, FOR SHAME.

Maybe I'm being a bit harsh. My worst subject is probably Geography. I'm sure I've said on more than one occasion, "I'm rubbish at geography". I think I've already mentioned on this blog the time my geography teacher wrote, "Is this some kind of joke?" next to an essay I'd written. I tell that story proudly. So I'm a complete hypocrite really. My head of department asked me the other day if I knew where Hull was. I admitted, unabashed, that I had no idea. I retell at any opportunity the time some pupils asked me if I knew where Bosnia was, and, looking my Bosnian student straight in the face, I said, "Of course, it's in Africa". Did I mention this student is blonde? I'm not ashamed of this. It was funny. So, yeah. Huge hypocrite.

OK, ignore everything I've said. Maybe it's obvious but I'll tell you anyway: I wrote this blog post on two separate days. The paragraph starting "Maybe I'm being a bit harsh" I wrote today, four days after the beginning paragraphs. I'm obviously in a better mood now (at the time of writing it's Good Friday, and hence the first day of my Easter break). I'll probably still publish this, in a few days, but I'm a bit disappointed that my argument fell flat on its face.

I hope you're enjoying your Easter holiday!

Emma x x x

Is Zero a Square Number?

In my faculty we have an unofficial lunchtime meeting every Thursday. What do you think happens when you put several cleverer-than-thou maths enthusiasts together in one room for forty minutes, along with some "normal" people who sit back and observe detatchedly, for fear that their views will be immediately shot down as they have not used phrases like "corollarlarily" or "without loss of generality" in expressing their argument?

You get some pretty heavy meta-mathematical debates which question the very core of mathematics.

You also get some pretty pointless debates, like: is zero a square number?

Now obviously the answer to this depends on how you define a square number in the first place. Here are some of the definitions my department discussed:

*A number, n, that can be written in the form n = a x a, where a is a natural number (slight variation, where a is an integer).

*A number n for which there exists a square with area n square units, and whose length is a whole/natural number.

*A number whose square root is a whole number.

None of these definitions is particularly satisfying. When I'm teaching my students about square numbers, I tell them that if you times a number by itself the result is a square number. The implication is that by "number" I mean natural number. When I list the square numbers, I never include zero. However, I feel pretty strongly that zero is a square number.

Take definition number one. Does zero satisfy 0 = a x a where a is natural? I say yes. Other members of my faculty say no. I believe that zero is a natural number. They do not. Why do I believe this? Well my professors at uni, for the most part, included zero. I know this because when they didn't want to include zero they'd put a little + next to the blackboard-bold N. If you define natural numbers as a "counting number", ie a number that could be the cardinality of a set, then zero is without a doubt a natural number. How many cows are there in this classroom? Zero. I have counted the cows in the room and the answer is zero.

On to the next definition: does there exist a square with area zero and natural-numbered sides? Leaving aside the "is zero natural?" debate, we first have to deal with: can there exist a square with area zero? My immediate response was, "Yes! There's one right here! In fact, I've got twelve of them!" (I was a bit hysterical at this point. You must bare in mind this was the last day of term). But it's an interesting question. It's almost like, "What is the sound of one hand clapping?", a Zen koan that for some reason doesn't interest me at all (hey monks, listen up: the answer is NO SOUND. Done.)

And the third definition. This one was offered by probably the most down-to-earth of the faculty. despite having a maths degree, she's somehow managed to maintain a grip on reality other maths grads signed away during Freshers' Week. If you square root zero, do you get a whole number? Yes. You do. Zero is a whole number. But using "whole number" didn't feel precise enough. Which then led to another debate: integer versus natural versus positive natural. Is i a whole number? If it is, then -1 would also be a square number. Can you have a square with an area of -1? Why not?

What do you think? Do you have another definition of square number that would settle this argument? Do you think zero is a natural number?

Extension:
Why is 1 a triangle number? You can't draw a triangle with one dot. Are the triangle numbers defined by the picture or by the formula (1/2)n(n+1)? And hence would zero be a triangle number?

Enjoy the Easter break!

Emma x x x

A Counting Game

Every so often I'll post about an activity I did or a technique I used that was good. I find I often forget these good things when I should be reusing them. Hopefully by recording them on the blog I'll remember them.

So, what I'm sharing with you this week is an activity I did with my year 7 class. I told them they were going to have a little competition (which was met with a few "YES!"s), and that it would be a competition against me. This excited them because they know I'm both extremely competitive and extremely clever. They would love the opportunity to take me down a peg or two (as would most people, probably).

The competition was this: I will play against one person at a time. We will start from zero, and we will take it inturns to add either 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10. The winner is the person who reaches 100. It's very similar to 21 dares, or those games where you have to avoid being the one who takes the last matchstick.

Now obviously, there is a trick that means you always win. Work this out for yourself, I don't want to spoil your fun! (Hint: whoever gets up to 89 has won, because whatever the other person says, you will be able to reach 100 on your next go. By extension, whoever gets to 78 has won, as they can get to 89, and so on). I took on about 7 students in total and beat all of them. As I did it, pupils started to notice things. This is what the pupils discovered, in order:

1) If Miss says 89, then she's won
2) Whoever goes first always wins (false!)
3) You need to stop her getting to 89
4) Get to 78, because then she can't go to 89!

I then extended them to realise that there were certain numbers that you should always try to aim for to make sure you win, and they spotted the pattern of these numbers. On the next go, a student beat me. And as much as it pains me to admit it, he beat me fair and square (I got my numbers mixed up). You can imagine how happy that made the class: "We beat Miss, and she got bare A*s!"

The class was then desperate to show off this newfound skill by challenging the head of maths to a competition. This hasn't been arranged yet but I think it's a really nice idea. I hope they beat him!

You can obviously extend the activity by considering a different target instead of 100, and different numbers that you're allowed to add.

I'm going to do the same activity with my other classes to see how it turns out. If you try out this activity, please let me know how it goes!

Emma x x x

Practise, Practise, Practise!

You have no idea how long I just spent deciding whether to spell practise the verb way (with an s) or the noun way (with a c) in the title. I think the expression is giving a command, so I decided to go for verb. And that's your literacy lesson for today.

Back to maths...

Imagine this situation: you're teaching, say, pie charts to, say, year 7s. You take them through an example and leave them to finish the question off individually. You then go through the answer. Then you get the pupils to hold up their traffic light cards to show whether they understand and got the right answer. Almost the whole class hold up their green card. You then say, "Good, in that case, do this one", indicating an identical question which has different numbers.

What is wrong with this scene?

In my opinion, nothing. However, some people, not mentioning any organisations (*cough* OfSTED *cough*) would say (did say, in fact) that there is a fatal flaw in the above.

All of the pupils hold up green, indicating they can do it. So why give them another question to do? Surely that's just a waste of time?

This might just sound like I'm being bitter about getting a worse observation grade than I would have liked (and let's face it, I am), but I just can't seem to come around to this way of thinking. In maths, you can't do something correctly once and assume you have secured that skill enough to remember it for the rest of the year, or even the rest of the day. My year 7s told me they could do that question. But transferring that to answering another question with different numbers, even if the method is exactly the same, is not trivial for most children. By rehearsing the skill again and again, it embeds it in their memories.

In addition, we all know what year 7s are like: they're either super-confident and claim that everything is easy and they can do it all, or they're the opposite and declare everything impossibly difficult. My class seems to be entirely made up of the former, perhaps because that's usually my own attitude towards learning anything new (which is why I injure myself so often in Pilates). I knew that my class holding up green traffic lights had to be taken with a pinch of salt. Green doesn't really indicate that they get something, it more indicates that they are willing to have a go and are in a positive learning mood. I knew this, I guess my "theoretical" observer didn't.

So I made my year 7s practise the skill for the rest of the lesson, apparently redundantly. I then set them the same type of question as homework. Do you want to guess how that piece of homework went? That's right, half of the class couldn't remember how to do it.

Now I can already sense a lot of you out there wanting to play devil's avocado on this one: repetition isn't necessarily the best way to embed something in memory. Doing just one question, but focusing heavily on developing understanding and making the experience memorable by including exaggeration, rhythm and movement, colour, order and patterns, laughter etc is more effective. Well yeah, you're right. I have absolutely no comeback for that one. Except: IF YOU'RE SO ****ING GOOD YOU COME AND TEACH THEM.

This week I've been thinking about pointless versus appropriate repetition. It's quite difficult to identify which it is. I'm going to make a special effort to avoid repetition and focus on deriving maximum understanding and memorability from one question.

What do you think? Should kids be going through a page of 10Ticks every lesson or is once enough for understanding to embed?

I'll leave you with the words of one of my professors from uni: "Practise, practise, practise: maths is not a spectator sport!"

Emma x x x

Ofsted: a Survival Guide for NQTs

Recently my Principal received "The Phonecall". We were told OFSTED would arrive in 48 hours. This is a scary thing for any teacher to hear, let alone an NQT.

I was anxious but also quite excited. It might not surprise some of you to know that I always enjoyed exams at school. I like getting the opportunity to prove how amazing I am. Of course, in school I was amazing at pretty much everything (except not being hated for being so arrogant), whereas now the only things I'm amazing at are accessorising to match my lesson plan, cube-rooting numbers in my head, and memorising my students' birthdays to freak them out*.

My academy was aiming for "Outstanding" so I felt like I was under a lot of pressure. I am *not* an Oustanding teacher. It pains me to admit that I'm not the best at something. I don't like it. But I have to accept that being a good teacher comes with experience, and very few teachers are Outstanding in their NQT. I don't like accepting this.

I am a very calm and laid back person , so I didn't worry about the inspection until the day before. Then I realised just how much preparation is needed for OFSTED. I had to produce:
-A lesson plan for all of my lessons that day.
-A compilation of data on each class.
-A seating plan for each class.
-A set of expertly marked exercise books for each class.
 I also had to tidy up my classroom and add to my displays.

This took bare a long time (do you find that students' language starts to invade your own?). I left school at 6:30pm and I can say with some certainty I was the earliest non-parent to leave. Some of my colleagues were there until 10pm. I got home and had dinner and then started working again. I worked until 9pm and then, being the procrastinator that I am, I thought, I'll do the rest tomorrow. I rarely stay awake past 9pm (even on weekends) so my eyes were starting to droop. I set my alarm for 3am and went to sleep. In the morning I worked for 3 hours, ran in and out of the shower like an instant carwash, and left for school at 6:30am. I was at school by 7am, and did another 1.5hours' work. It was then, 15 minutes before the first bell, that I was told I would be observed during period 1. Instatly relief washed over me: it felt so good to know. At that point, by the way, I still hadn't written my lesson plans for period 3 and 4 (I was free period 2). I didn't bother writing them in the end, because lightning doesn't strike twice, right?

So yeah, the observation. Well it was fine. My students behaved impeccably, even before the inspector arrived (they observed the second half). They were desperate to impress, more so than me I think!

I went and got feedback at the end of the day. The inspector told me lots of positive things:
-My strength is that I never tell kids how to do something or what the answer is, I question them until they can tell me.
-I have good relationships with students: they like me and there is an upbeat atmosphere in my classroom.
-I assessed regularly.

They also told me some things that could have been better, which were quite specific to the lesson but I'll try and tell you them without being too specific:
-After getting my pupils to traffic light and seeing lots of greens, I still got them to do another question similar to the one they'd just done. This was pointless because they already said they could do it. (I privately disagree with this to be honest. In maths you can't do something once and expect to be able to remember it forever. Plus my pupils are always over-confident in their self-assessments).
-When one student still didn't get it near the end, I should have gone back to the original long explanation from the start with the whole class, rather than explaining the method quickly again for him.

So there you go. The negatives weren't anything like, "there's no evidence of differentiation in your lesson plan" or "your learning objective should have included PLTS", it was very focused on moments in the lesson where I made a decision that was possibly not the right one. I think this is an extremely fair way of observing. It was very focused on the progress of the students, not on stupid gimmicks. I didn't do any group work, but who cares? There wasn't a moment in the lesson that would have benefitted from it. I think they focused on "hinge points" in the lesson, the bit where the lesson changes direction based on assessment. It's how you handle these hinges that's important.

Here are my top tips for surviving OFSTED:
- Make a seating plan (annotated with SEN, G&T, EAL, etc) and print off your class data now so that if OFSTED call you have a few less things to worry about.
-Plan normal lessons that you will feel comfortable delivering. Don't be flashy, that's not what they're looking for.
-Think about the hinge points in your lesson and how you're going to handle them. Think: what if they all get it/none of them gets it/half get it etc.
-Put a set of well-marked books near the place they're going to sit. I had some (about 5) full-up books from a good class that I deliberately put near an empty desk. I removed the books that were messy or marked badly. I noticed afterwards that the pile was in a different order so they must have been looked at. If your class is using their books the inspector can't really examine them much.
-I don't know if this is a good tip but when I got my class to traffic light I told them to turn around and show "our visitor" their colour too. I thought this would be helpful for their observation and I think they appreciated it but who knows.
-Brief your class beforehand. Mine were amazing, they really tried to look engaged and clever. One pupil called me over and whispered "Miss, he's standing behind me, ask me a really good question". Isn't that great?

Obviously the main thing is not to panic, but I think that's a useless bit of advice because you're either born a panicker or a non-panicker (like me). Remember there is no failure, only feedback. Even if you get rated "inadequate" (hate that word), you will get lots of useful feedback from it.

By the way, I was rated Satisfactory with Good features. That's an improvement on my last observation, so I'm happy. Also, it looks good because the inspectors will see that the school has rated me Satisfactory, so by beating that score, I validate the school's observation records. It's like moderating coursework: if the examiner thinks you've marked a piece of coursework too low, they raise all of your class's marks. So don't worry about getting Outstanding, just try to beat the score you have on record.

Good luck to anyone facing OFSTED this term. I found it a very positive experience overall, and I'm not just saying that.

Emma x x x


*I used to do this when I was at school. I still remember most of my form group's birthdays, or at least their star signs.

Satisfactory is Not Good Enough

I have just read a news article from the BBC which has made me rage. (link)

So they're planning to change the "Satisfactory" label to "Requires Improvement". Hmm.

I've had a few discussions with fellow teachers on why we shouldn't use the word "satisfactory". It sounds too insulting. Saying someone is a satsfactory teacher sounds a bit negative, even though the underlying implication is that the teacher is good enough, doing fine, performing as required, no need to worry. A better word might be "fine" or "ok".

But instead of changing satisfactory to a word with fewer negative connotations, they've made it worse! "Requires improvement". How dare they? If you are satisfactory, you do not require improvement. The definition of satisfactory is "Fulfilling expectations or needs; acceptable, though not outstanding or perfect." (thanks dictionary.com). If you require improvements, then surely you are not satisfactory? But Cameron has decided "just good enough is, frankly, not good enough". RAGE.

So basically you're moving the goalposts so that "Good" now means "Satisfactory" (and is hence no longer a compliment), and anything below that is bad. How incredibly demoralising. Is that what you want, Mr Gove?*

Obviously all teachers should be striving to improve, and observations help us to be reflective practitioners and develop our weaknesses. But telling us we're not good enough (when actually we are adequate) is demotivating and also just mean.

Teachers really don't get praised enough. We get better and better GCSE and A Level results every year, and you never hear anyone saying, "wow, teachers have done really well for us this year, the future of society is looking good thanks to our fine teachers". No, you hear "exams are getting easier" and "it's because teachers are teaching to the test, not teaching proper understanding". We never win.

You know what's going to happen next, don't you? "Good" become "requires improvement" and the only acceptable standard will be "Outstanding". More and more teachers will leave the profession for a job where you get the occasional pat on the back. Good graduates won't want to train as teachers when they could have cushy office jobs. We'll have a shortage, and then what?

On a more rational note, don't you think that "requires support" is a much nicer alternative to "requires improvement"? It puts the onus on the school leadership team to help the teacher improve, rather than the teacher (who is probably doing the best they can). One better would be to make it "would benefit from extra support" but that's a bit wordy. I just don't like that word "requires". It sort of implies an "...or else".

Anyway, this has been way too wordy, sorry about that. I'm just a little bit cheesed off.

I hope you all have a good week (because anything less than good would be unsatisfactory, obviously).

Emma x x x

*I realise this is not directly Mr Gove's initiative, but still.
PS I had to google to make sure Gove is the education guy, because I wasn't entirely sure. I fail politics forever.