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

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