The Punctuation of Maths

I do love a bit of cross-curricularism. Although maths will always be my first love, I must admit to being passionate about language too. One of my (many) skills is being able to identify the Latin/Greek/Whatever root of a word and explain the original meaning. Actually, that's just given me an idea for another post...

But anyway, this post isn't about words, it's about punctuation. I am still quite confused about when to use commas, semi-colons, colons, emdashes, endashes etc, although I do find them interesting. Language is very mathematical when you think about it.* But I'm not talking about English grammar here (which is probably a good thing, as I began this sentence with a conjunction), I'm talking about the punctuation of maths.

All of you reading this now, write down what a half is as a decimal. Have you all done it? Yes, even you. Good. Now, type what you just wrote into the search bar on your browser's toolbar (assuming you have one) without pressing enter. If you prefer, type it in a Notepad file. Done? OK.

Compare what you have written and typed. Are they the same? Look specifically at the decimal point. I'm guessing all of you used a full stop for your typed version. What about your handwritten version? Is your point at the bottom, resting on the line, or hovering in the middle? (Some of you may have used a comma instead. I'll get to you later).

In the US, it has always been the norm to use a full stop (or as they call it, a "period") as the decimal mark. So we can assume this is wrong. Kidding! (I have a few readers from the US who I must try not to insult). In Britain (and in British Empire nations) however, the mid dot, or "interpunct", was the standard symbol. However, the full stop was OK to use in typing and printing. The mid dot can be easier to read on lined paper because the point can't be hidden by the line.

There is a problem with the mid dot though. To me, such a mark indicates multiplication (although as far as I can tell this is never used in schools), and the top guys at the SI agree, because they rejected it as the standaard decimal mark.

In the end (well, in 2003), The 22nd General Conference on Weights and Measures declared that "the symbol for the decimal marker shall be either the point on the line or the comma on the line" (yes, comma users, I'll get to you), meaning that officially decimal points and full stops have the same appearance. Some older British people may angrily disagree with this decision. For once I'm not bothered.

OK, the people I was ignoring: some people use commas instead of dots for the decimal mark. Namely, people from non-British Europe, and also some random places like South Africa. This is because originally, a short, vertical mark dash was used as the decimal mark. This evolved into either the comma or the dot. France preferred the comma (presumably because it's more phallic) and the rest of Europe then took sides.

Now, some more writing for you to do. Write down, in figures, the number twenty seven million, five hundred and sixty four thousand, two hundred and fourteen. Done? Good.

Look between the seven and the five. What's there? A comma? A gap? Nothing? (For the Europeans: a dot?)

I was always taught at school that you should just leave a space between every three digits, and not put a comma. I had a few reasons given to me for doing this: because a comma could get confused with a decimal point (really?) or because Europeans might think it's a decimal (fair enough). Because I was taught this in school, I get really annoyed when I see people using commas. It's not what you're supposed to do!

I have researched this to check that I am correct, and I have found that the International Bureau of Weights and Measures states that "for numbers with many digits the digits may be divided into groups of three by a thin space, in order to facilitate reading. Neither dots nor commas are inserted in the spaces between groups of three". Officially this space is supposed to be a half space, although I don't know how to do that on MS Word.

I think most maths teachers in the UK would know better than to use commas to separate digits (they jolly well should!) but non-maths teachers may not know this rule or appreciate the importance of it and hence you might find commas in long numbers in Geography lessons. It should be part of a school's numeracy policy that every teacher conforms to this. Otherwise it would be as bad as a non-English teacher using an aberrant apostrophe (which I'm sure NEVER happens). I fully realise that by saying this I am practically begging people to criticise my grammar on this blog. Bring it on!!!

I will leave you with a nice little anagram puzzle:
Rearrange this expression to make another one: "I'm a dot in place".

Emma x x x


*One punctuation thing that annoys me because of its lack of mathematical sense is the " quotation mark. In a book, if someone is talking for a long time, there can be a paragraph break in the middle of their speech. You do not close the speech marks at the end of the paragraph, but you do have to put an open speech mark symbol at the start of the next paragraph. This annoys me because it's like having an open bracket that is never closed. In programming, such a thing is, to my knowledge, always disallowed.

A Warning to New NQTs

Many new NQTs will have already started their new jobs, just like I did last July. Because of this, I thought now might be an appropriate time to warn the newbies about the potential dangers ahead. Please note that, for once, I'm not joking.

I have been very lucky. My time as an NQT, whilst a million kilometres from easy, was completely straight forward and untroubled. This is because I was fortunate enough to choose (and be chosen by) an amazing academy with incredibly supportive professional and subject mentors. Some of my friends weren't so lucky.

There is a fairly scary rule that you might not be aware of: if you fail your induction year (your NQT year), you will never be allowed to be a teacher in a normal school. Ever. You are never allowed to retake the year, so you will remain unqualified forever (although you will still technically have QTS - Qualified Teacher Status).

So, yeah, the stakes are pretty high. You do not want to fail this year.

I remember thinking this time last year that there was absolutely no risk of me failing. I thought you would only fail your NQT if you were seriously bad. I mean, we've all seen really bad teachers, and they've managed to pass. But it turns out, failing is easier than I thought.

Even if you got an "outstanding" or "good" rating for your PGCE, you are still at risk of failing. I know this from experience. You won't fail due to poor teaching, but you may well fail due to poor support from your school.

Here are my tips for avoiding failing:

• You are entitled to meetings with your induction tutor. Make sure you ask for these! If possible, request these meetings via email, so that you have proof that you have asked for them. That way if they do not give you these meetings (because your tutor is too busy) then at least you can say you asked. The importance of this will become more obvious later.

• When it is time to fill in an assessment period form (at the end of each term), make sure the school fills it in, in consultation with you, and sends it off. Ask them outright whether you are at risk of failing. If they say yes, ask for the support plan that you are entitled to. If they say no, make sure that is clearly expressed on the paperwork.

• If the school does decide to fail you (because, perhaps, it's cheaper to get you fired and hire a new NQT, or because it's cheaper to fail you than make you redundant), then you have two options:

1. Accept it, and ask if you can resign early (before mid-June). That way, it won't count as you failing, it will count as your year being incomplete. You can then get a job elsewhere and finish your induction period.
2. Dispute it. This is where the documentation comes in: if you have evidence that the school has refused to give you regular induction tutor meetings, or if they refused to make a support plan for you after identifying that you were failing, or if they wrote on your previous paperwork that you were not at risk of failing, then you have a good case against your school. If you are a member of a union then they can help you.

Know what you're entitled to:
 (taken from the TES website)

Under induction NQTs should have the following:
1.       A job description that does not make unreasonable demands.
2.       An induction tutor.
3.       Meetings with the induction tutor.
4.       The Career Entry and Development Profile discussed by the NQT and induction tutor.
5.       Objectives, informed by the strengths and areas for development identified in the CEDP, to help NQTs improve so that they meet the standards for the induction period.
6.       A ten per cent reduction in timetable - this will be in addition to PPA time.
7.       A planned programme of how to spend that time, such as observations of other teachers.
8.       At least one observation each half term with oral and written feedback, meaning a minimum of at least six a year.
9.       An assessment meeting and report towards the end of each term.
10.    Procedures for NQTs to air grievances about their induction provision at school and a named person to contact at the Appropriate Body, which is the local authority or the independent schools’ council teacher induction panel (ISCTIP).

Your induction tutor will probably be really busy and may forget to do these things, or try to avoid doing these things. Ask for them! They are not allowed to say no!


I don't mean to be all pessimistic and scaremongery but I'm pretty sure my friends who have narrowly avoided losing their ability to teach forever never thought this time last year that this would happen to them. Unfortunately some schools are just not supportive of NQTs. If you do have a bad experience, make sure you let your ITT provider (the university you got your teaching qualification from) know so that they can dissuade future NQTs from starting there.

I wish all new NQTs the best of luck in their new jobs!

Emma x x x
(still an NQT for two more weeks!)

Is Zero an Imaginary Number?

If you're thinking that this title sounds familiar, it's because you're thinking of my famous post "Is Zero a Square number?". You can think of this post as the sequel. I can foresee a whole franchise of posts of this nature in my future.

To answer today's question, we first need to decide how we are going to define "imaginary number".
No prizes for guessing which website I'll be going to for that!

"An imaginary number is a number whose square is less than or equal to zero" (Wikipedia)

Oh. Well, I guess we're done here. Zero is an imaginary number, as zero squared is zero, which is obviously less than or equal to zero. Job done.

Oh come on, you can't possibly be satisfied with that!

What is the rationale behind including zero? If you don't include zero, what things are affected? Would it really make a difference? I need to know WHY! Darn you Wikipedia!!!

I need another definition!

So I will check out my usual second port of call,  Wolfram Math World:

"A (purely) imaginary number can be written as a real number multiplied by the "imaginary unit" i (equal to the square root ), i.e., in the form ."

OK, so that's a little bit better. Again, zero would fit this definition because you could write it as 0i (or 0j if you prefer) and zero is certainly a real number.

But I'm still not happy. It just doesn't feel right. 

Let's think graphically for a moment. On an Argand diagram, real numbers are numbers along the real axis (the horizontal axis) and imaginary numbers are along the imaginary axis (the vertical axis). The number 0 + 0i has coordinates (0,0) and is hence on the origin, both on the real and the imaginary axis. So this means it is either real and imaginary, or neither. We know zero is real, so the best option would be to take it as both.

The evidence is stacking up on one side: zero is an imaginary number. But I'm still not convinced.

In fact, the only thing that I even find slightly convincing is the argument put to me by one of my A level students. He said to me: zero can't be an imaginary number, because all imaginary numbers have an argument, but zero's argument would be undefined.

I really liked this. What is the angle between the positive real axis and the line connecting (0, 0) to (0, 0)? It could be anything, so it's undefined. Can an imaginary number have an undefined argument? A quick Google suggests actually yes, it can. Foiled again.

Fine, I give up. Zero is an imaginary number. It is also a real number. I can definitely see why including zero would be useful in terms of subspaces etc as it allows the imaginary numbers to have an additive identity.

So, as my students would say, "Allow it".

Emma x x x 

Are You a Maths Teacher or a Mathematician?

Me and some colleagues were discussing this today: do you consider yourself a maths teacher or a mathematician?

If you reply by saying maths teacher, then the follow up question would be this: should non-mathematicians really be teaching maths?

To me, being a mathematician does not mean being really really good at maths. Have you ever heard of an artist being described as "someone who is really good at art"? Of course not. Instead, it's about passion. In my mind, a mathematician is someone who loves maths, and is limitlessly curious about all things mathematical. It's someone who doesn't just enjoy doing a maths puzzle, but who simply cannot stop doing a maths puzzle until it is fully explored, solved, and extended.

If you're not like that, then you probably see yourself as a maths teacher and not a mathematician (or as neither, I know I have a lot of non-maths teacher readers).  Is it important for maths teachers to be mathematicians? Remember I'm not talking about subject knowledge here, I'm talking about passion. Would you want someone who says "I can think of a thousand more things I'd rather spend my free time doing than doing a maths puzzle" to be teaching your children maths?

I'm uncharacteristically on the fence with this one. If you'd asked me a year ago I would have been adamant that only mathematicians should teach maths. But after having worked with a wide variety of teachers in this past year, a few of whom would fall into the other category, I'd have to say, if they can teach it well, who cares? Why be all snobbish and superior about it? Does it matter that they don't know a hundred and two amazing things about Pascal's triangle? If they can make students feel positive about maths, that's good enough.

However, I think my strong point as a teacher is my passion for maths (which my students would refer to as my geekiness). I do think this makes me a better teacher, because I think enthusiasm is catching, and it helps pupils to see maths as a much wider field than just "sums". Grrr, how I hate that word!

I've tried my very hardest not to offend anyone in this post, and as a result it's not that interesting to read. Maybe I should consider adopting a more provocative writing style and taking more extreme views such as: If you didn't get an A* in your maths GCSE then you shouldn't be a maths teacher! (To be fair a small part of me secretly believes this).

I'll hand the debate over to you: do you have to be a mathematician to be a good maths teacher? Comment below.

Emma x x x

What is a Line?

On Monday I was tutoring a girl in year nine, and teaching her about linear graphs, and the whole y = mx + c thing. She asked what linear meant, and I told her that it comes from the Latin word linearis, which means like a line. She was momentarily puzzled: "Aren't all graphs lines?", she asked. I drew a vaguely-quadratic looking graph on the mini whiteboard and said, "This one's not". To which she replied, "Er, yeah it is".

This made me question my concept image of a line. Can a line be curved? If no, then why do we bother saying "straight line" when this would apparently be tautologous. If yes, then how can we use the word "linear" to describe relationships that correspond to strictly straight line graphs?

I decided to do some research. I asked a few members of my department how they would define a line. Here are some of the answers I got, along with an evaluation of each.

Something that joins together two points
The person who gave me this definition said that a curve would count as a line by this definition, as a curve can join two points. However, this would mean that the line joining two points would not be unique, and that makes me feel uneasy. Also, this definition seems to be more for a line segment, as opposed to a line: I have always been taught that lines are infinitely long in two directions.

The shortest path between two points
A little bit nicer, as it now has uniqueness (if by "shortest" we're referring to the usual metric). However again this is just a line segment, not a line.

The locus of points where...
This definition was never finished, because he couldn't think what the locus would be. I decided to still include this definition because I think he's on to something.

A sequence of points
Take any sequence of numbers and plot them on cartesian axes. Then join these up, and what you have is a line. I would refine this by saying "linear sequence", but that would be a bit of a circular definition, because I would define a linear sequence as one that would make a straight line when drawn on a graph. This definition isn't very satisfying because it only makes sense on a set of axes, whereas obviously lines occur elsewhere. In geometry, for example, lines can exist with absolutely no plane of reference.

I suppose the definition I would go with would be the second one, except I'd alter it to make it infinitely long:
"A line is the infinitely long path that goes between two points, such that the path from one point to the other is the shortest possible" Arggh this doesn't work! The line could be squiggly outside of the two points and this could still hold! How do I say "it's straight" when I'm trying to define straight in the first place?

I'm afraid I'll have to do what I always end up doing when it comes to maths debates:

Wikipedia to the rescue!

"Line (geometry), an infinitely-extending one-dimensional figure that has no curvature:

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects.

Thus, until seventeenth century, lines were defined like this: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...] will leave from its imaginary moving some vestige in length, exempt of any width. [...] The straight line is that which is equally extended between its points"

In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.

Is it just me or is that a really hand-wavey definition? And it's still not clear whether a line has to be straight: the pre-seventeenth century definition allows for curvature, but the analytic geometry definition does not. And what does "equally extended" mean anyway?

You might wonder why on earth I care about these things. I do too sometimes. But when I asked people in my department, we got a good discussion going, and that for me is an amazing thing: people arguing about maths. I love it! I remember year 11 RE lessons when I was at school: we used to argue all the time (and every comment would start with "Surely..." - Do any of my ex-classmates remember this?) but we never argued in maths. I love having a good debate (pro tip: never use the expression "maths debate" in a lesson. Trust me) and it is something we should be doing more of in maths. Maths is not about blindly accepting rules and definitions. We should question them, challenge them! Ask a child this and watch their mind explode: "How did the first ruler making factory make the ruler perfectly straight?"

As you can probably tell from the length of this post, I have a lot of work to be doing which I am trying to avoid. There is something quite stressful happening next week in my department, and my stress reduction technique is to immerse myself in mathematical pedantries.

Bye for now,

Emma x x x

I thought you might be interested to read the hilarious text message conversation that took place between me and my dad as I was researching this post:

Me: how would you define a line?
Dad: a set of points determined by a linear function.
Me: define "linear".
Dad: an equation that forms a line.
Me: that's a circular definition.
Dad: define "circular".
Me: a shape that contains no lines.
Dad: define "line".
Me: our conversation has come full circle!
Dad: define "circle".

My dad never should have been given a Blackberry.

Emma x x x

A Grouping Activity

In today's blog post, I'm going to be sharing with you one of the first classroom activities I was ever taught. As a would-be teacher I mean, not as a student.

It was the second day of my PGCE, the first day that we were separated into subject groups, although a lot of us maths lot managed to find each other on the first day anyway. I suppose geeks, like insects, give off pheromones that signal to others of the same species.

We were each given a number. For the sake of simplicity, let's say we were given the numbers 1 to 36. As memory serves, there were actually over forty of us. But let's not let accuracy get in the way of a good story.

On each of the six tables there was a piece of paper saying one of the following:
Prime numbers
Square numbers
Triangle numbers
Numbers greater than 20
Even Numbers
Numbers with more than four factors.

We had to join a group corresponding to our number. As I recall my number was 3. I could have joined either the prime group or the triangle group. Some numbers could have belonged to three different groups. The number 5, for example, had no choice in the matter. The catch was this: every group had to have exactly six members.

I could tell you the solution to this problem, but I'll leave that for you to work out. Feel free to email me (nqtpi@gmail.com) if you get stuck. The one hint I will give you is that a girl named Lydia worked out pretty quickly that because both of our numbers were triangular but not square, we would have to be in the triangle group. And that is the story of how how me and Lydia became friends* and how Team Hopper (previously known as Team Triangle Numbers) was formed.

*OK not really, but I repeat: let's not let accuracy get in the way of a good story.

I think this activity would be a cool way of getting a class of pupils into groups. It's pretty challenging, so maybe best saved for A level classes? Obviously it will need adapting, as if you have 36 students in your A level maths class, you are clearly working at the wrong school.


Emma x x x

PS My blogiversary is coming up, so expect some interesting posts in the near future.

Why Are You Just a Teacher?

One of my students asked me that the other day. With particular emphasis on the "just". It's quite insulting really, although I think they meant it as a compliment.

I had just told the class what I had got in my gcses and A-levels. I'd like to point out that they did ask me, I wasn't just showing off. I told them my results, and they were suitably impressed. As usual they found it hilarious that I managed to get nine A*s and one D. They found it even more funny when I told them the D was for art. Now whenever I draw a diagram on the board they make snide comments. Anyway, after hearing how well I had done at school, one of the pupils asked, "If you got such good results, why are you just a teacher?"

The thing that immediately struck me was that the pupils in front of me (set one, as it happened) did not see me as a successful person. To them, I was "just" a teacher. Not a dentist, a doctor, or the owner of a business, just a teacher. You don't have to be "bare clever" to go into teaching, unlike dentistry or medicine. Nor do you earn a lot of money (to them, a lot of money means considerably more than my £21k, which to me still feels like I'm winning the lottery every month). They don't see teaching as the sort of job a high achiever should aim for. This does not make me feel good.

I found myself justifying my career choice to a group of precocious sixteen year olds. I shouldn't have to! Teaching should be a well-respected profession. Teachers should be seen as the cream of society. I was talking to a colleague about this, and she told me that her grandfather was a teacher, and was the most looked-up-to member of the village. The whole village turned up to pay their respects at his funeral. This is how it should be. We deserve as much respect as dentists, at least. The job title should carry as much clout as "solicitor" or "executive". Students should assume their teachers have amazing exam results, and look up to them as beings of superior intellect and moral fibre. Students should aspire to be teachers. I'm fed up of hearing "I want to be a lawyer/dentist/doctor, but if that fails I'll just be a teacher". It should be:  "I want to be a teacher, but if that fails I'll just be a PE teacher". Ha ha. That was a joke. Please don't beat me up.

Maybe this attitude just applies to students from my school, where the student demographic is mostly Asian. If you'll allow me to stereotype wildly: Asian parents often push their children into medicine, dentistry, pharmacy, etc perhaps to the exclusion of all other careers? Maybe this accounts for this attitude.

Do you know what your students feel about teaching as a profession? Ask them, but be prepared to be offended!

Emma x x x