Tuesday 24 July 2012

Why Is It Called a Quadratic?

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


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

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

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

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

Emma x x x

Friday 20 July 2012

True or False PART II

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

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

True or False:

If xa = xb  then a = b. 

Let me know your answer in the comments below!
I'll do a full post on this once I have some answers.

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

The "logic" of the question

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

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

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

The types of number allowed in the question

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

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

 Complex numbers and e

The question had first occurred to me when I was teaching a Further Pure 2 lesson, aand we had to solve the equation e3 = ez which prompted one of my students to say, surely the answer's just z = 3? I had already gone through the solution on the board and we had come to the answer which by the way is z = 3 + 2kπj (for integer k). My student didn't believe this could be true, and I even started doubting it myself. I couldn't see how z could be anything other than 3. e is just a number, after all. It's kind of amazing when you think about it. That is what is so awesome about complex numbers, they always manage to surprise you. The reason why it works is because you can write
e3+2kπj  as   ex e2kπj   and it can be shown that e2kπj = 1.

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

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

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

Fellow teachers, enjoy your freedom!

Emma x x x

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

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

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

Wednesday 18 July 2012

True or False...?

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

True or False:

If xa = xb  then a = b. 

Let me know your answer in the comments below!
I'll do a full post on this once I have some answers.

Emma x x x 

Wednesday 11 July 2012

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.

Monday 9 July 2012

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!)

Thursday 5 July 2012

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 sqrt(-1)), i.e., in the form z=iy."

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 

Tuesday 3 July 2012

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