Wednesday, 2 June 2010

The Maths Teacher's Paradox (or why some of our best teachers might be failing their students)

I've been sat in front of this blank blog page for a while now wondering how I'm going to start this piece, and I've decided to start with a quote, given to me over a decade ago which I will always remember.  It was told to me by a very wise man, who happened to be the headmaster at a school I was being interviewed for.  He said to me:

In life, every person you meet you will be better then them at 10 things at the very least, but never forget that same person will be better then you at 10 different things at least.

It's something I take remember during my life, and will probably take to my grave and it's nice to share it with the world.

Having said that though, in my opinion there are two people in life, people who naturally understand algebraic concepts, and those who do not.

One of my closest cousins is currently undertaking a teaching degree, and she unfortunately is one of the people who struggle with maths, especially algebraic concepts (but boy you should see her ride a horse she is awesome!).  Fortunately for her I am pretty good at maths (but can't ride horses!!), and yesterday I got a message from her asking for help.  She was doing science coursework relating to the Simpson's Diversity Index, and was asking if I could simplify the maths for her, so she could understand it.  Instantly I panicked, having never even heard of the Simpson's Diversity Index I though I was going to be of no help.  I instantly got onto Google, and found the formula:

Now to the untrained eye, that would look pretty terrifying.  There are a lot of letters in there and not many numbers.  When I tell you the D is the diversity index (which ranges between 0 [meaning no diversity], and 1 [meaning infinite diversity]), n is the total number of organisms within a particular species, and N is the total number of organisms of all species; we should realise that this is mostly a simple matter of substituting numbers for letters and then solving for D (Sigma - the greek letter just means sum of (so we need the sum of all the n(n-1)s)

So, us people who naturally understand maths, or those that have just had a lot of practice can look at that formula and instantly realise that the Simpson's Diversity Index is actually a fairly simple formula.

After going through this formula, and writing out 3 pages of notes on it to try and simplify it for my cousin, I decided to offer my services to her, if she needs it.  She will be going into her final year soon, and if she needs to understand maths better I have offered to give her some tutoring (assuming I have time) over the summer in basic/intermediate algebra.  Why did I offer this? Because I truely believe that anyone can learn algebra.  All you need for algebra is to learn a bunch of rules, and then have plenty of practice.

Which got me thinking, if that's all you really need to do, why does my cousin struggle with it?  She's bright, and intelligent, so there must be something underlying that confuses her; so I racked my brain and  think I came up with the answer.  It's the letters, that's what people don't understand...the letters!! Maths is about numbers; where are all these letters coming from.  That's when I came up with what I have called the "Maths Teacher's Paradox", I don't know why I called it that, it just kinda sounds cool I suppose.

The Paradox

When I was thinking back through my past, especially my maths tuition, I was never told the why or the what of the letters, I just understood we had to work them out...I naturally understood algebra.  Some people do not naturally understand algebra remember, and last night I found myself for the first time actively thinking about what the letters mean.

I don't know if this was just my teaching experience, but I would guess that most people are never told (especially early on in their maths life) what the point of the letters are for.

We are never just sat down and told all in one swoop:

Where there is a letter in a maths problem, it is still representative of a number, and it can mean a few different things:

1.  That we don't know what the number is, and we need to find out what that number is - as in; 3 + b = 7; what is b? We need to solve for b, and we obviously get 4.

2.  That the numbers are variable, and therefore one single formula is needed, which we can change the numbers for depending on our dataset, and then solve for one letter, as in the formula bellow, or the most famous being: y = mx + C our famous graphical equation, where m and C are constants (i,e, being any number you wish, to make the correct graph), then we substitute 1 then 2 then 3 then 4 then 5 etc as the x (for our x axis), and that gives us the points on the y axis to make the graph

3.  The numbers are either very, very big, or very, very small.  That is to say that in a formula it is far easier to put in a c, instead of 12349533172331 as a just makes it easier to read...

4.  The letter represents a known constant, in astrophysics equation you might find yourself with a c, this could in actuality be the speed of light, a constant and something you do not need to solve for

5.  Where the letters are representative of a number, but also have a meaning behind them; d = s x t - this is a famous formula which means Distance = Speed x Time

......we are just expected to evolve our learning to understand these concepts

So why are we never told this?  Probably for the same reason I've never thought about it before.  Presumably (and hopefully - although saying that someone I know did a geography degree and during her interview for her PGCE she couldn't point out where the North Sea was...she's now a geography teacher...), maths teachers become maths teachers because they're good at maths!  Which means, like me, they've probably never even thought about the letters, the why and the what, they've always just understood the how.  How to get the maths to work, without needing to have it explained to them, what the point of all the letters actually are.

Unfortunately not all people are like that like I've said before, and I think if somebody turned round and really explained to the people who don't fully understand the concepts of algebra, the why, and the what of all these letters it would be easier for them to understand.

Hopefully this makes sense, and is merely speculative from my own experiences of 1-to-1 tutoring, and my own maths lessons when I was first learning algebra.  Very good maths teachers, are too good to see the struggles faced by some of the kids, and they need to take it back to the very basics at the start of algebra tuition so that the people with lower ability to just "pick it up" can fully understand not just the rules around algebra, but the actual reasons behind it.

Much love to you all,