Thursday, August 16, 2007

Mathematical Mentality

As I sat, nursing my cup of tea, bitterly watching MOTD (Match of the Day), mathematical thoughts intruded the slow frustration building up inside me. (Part 2 of the post before. :p)

What mathematical thoughts could they be? Well that of a mathematicians mentality. This is not about the game football, but a couple of years ago I was asked to help train a bunch of kids. (Call it 'community work'!) The group was rather big and I couldn't filter out those students who didn't know how to play football. We were indirectly told to prepare the kids for a tournament, and that was it. Like I said above, I don't consider myself to be a great footballer- I just love playing the game, and you don't have to be great to play it. In the end, the other person who was meant to take the other team didn't turn up. This was most annoying since I ended up having to work with both teams.

You see there also exists a thing called football mentality. Football awareness which you use on the field. Some kids had it, but a majority had no idea. The first thing I had done was to organise a quick friendly to assess the ability of the kids. I don't think some of them had wanted to be there, but had not been given much of a choice.* (You can spot them a mile away!) Through this friendly I learned a lot. I saw that only about five or six really had that footballers mentality. Some of them didn't know how to pass, and whenever the ball came to them they had kicked it anywhere. This was most frustrating for we had a tournament to prepare for.

What could be done, if not to go back to square one. So I went to the basics. Even for those who could play, I showed them how to pass. I put them into pairs and made them first control the ball and then pass -using the inside and the outside of the foot. The main point of this activity had been to first control the ball. If they passed without controlling, they were in trouble (i.e. a few laps). I was told that many of the kids had 'two left feet', but that was unimportant. What I wasn't told was that a majority of them had sneered when I'd mentioned how great football was. My job automatically became harder. Not only did I have to prepare them for the tournament, but to do that I had to initially rid this negative attitude they had of football. I had to make them enjoy the game.

I tried to do both of these tasks, but it seemed that only the footballers amongst the students really cared for my job. They tried their best to make a go of things but then began doing everything themselves, and not playing in a team. I didn't like that. Every session started with controlled passing, shooting and moving with the ball. They hated me for it - it made their hands sting - but this pain was necessary. The pain of having a ball kicked in your face was also necessary. Not as much, but it all contributed towards their footballer mentality. (You see there comes a point when a ball in your face no longer hurts!)

Anyway, everyday we spent the first half hour or so doing these routine things. However, I always tended to add another extra routine into this work out. As they learnt to control the ball, I told them to no longer control it before passing, but to try and pass with one touch. Quick passing. This was a big ask and many of them stumbled. They could pass, but the problem was passing to a team mate. The pass had to be weighted you see, just right for your team mate. They used to 'blast' it towards their team mates, and most often than not the team mates got a ball in the face. This was when asked not to control it. (However some progressed nicely).

After this half an hour of basic routine building block exercise, a friendly game occurred. I had mentally created the two balanced teams that might work, and made any changes that might be needed. One kid always stood and played near the outline of the pitch. The kid was afraid of the ball and if, during a game, the ball came in the students path, all training was forgotten and the ball was kicked without any direction. That player became a sub. Some resilient players impressed me so I then created positions to suit their playing 'mentality'.

Now you might be thinking, this is all football so what the heck was the point of this post - but I must pause you there. I have on occasions compared a 'footballers mentality' to a mathematicians mentality, as other people compare it with a musicians. Now, if I was to read through my post again using the 'code' mathematics= football, then most of it would still make sense.

In schools we try to help children to build their own mentality of maths. We teach them the basics - to add, subtract, multiply etc. We repeatedly teach them these basics, but when they begin to understand the basics we push on. They may hate every minute of it, and not like what they're doing, but part of our job is not just to make them do the 'routine' tasks, but to make them enjoy doing so. I mean we have to impart on them this mathematical mentality which will allow them to enjoy and learn maths. It is a hard job indeed, but I will say again- the hardest part is in making the students attitudes towards maths positive, as it was with football. After you've overcome the attitude hurdle, things become nicer.

I tell my friends about the joys of football, but they'll never understand until they play it. They may watch it, as kids watch their teachers doing maths, but until you yourself haven't gone through it, you'll be ignorant of it. Watching the game is as important as playing it. Watching mathematicians doing maths is as important as well. But playing it and doing maths is an altogether different story.

You see I've experienced both scenarios. I've tried to make 11, 12 year olds enthusiastic about maths and I seem to have helped a few people. This doesn't inspire me, rather it makes me wonder what can be done for those who 'hate' maths and find it boring. We need to make sure that most students have a sound mathematical mentality. So that on the field, they can sense the pace of the game. They can sense what a question asks them and then do their business. I don't think much explanation is required about this mentality, since it is very much like the football one. If you don't play football for a while you become rusty. In games your first touch becomes poor. Your perception of things is found wanting.

In these holidays my mathematical mentality has become rusty. Questions which I could do without a problem before, are now causing me problems. But once I get into the swing of things and the year starts, the rust will hopefully decrease. What I've ignored in this post are those students who have this mathematical mentality. Those who appreciate maths and are positive about it. The main thing with these students is that they need to be taken down the right path. If that is done, then 'we are done'. There are cases when students have this mathematical mentality but the rust which gathers, never goes away. They may choose a different path in life, and so be it.

As I return to this post, I look at the football academies that exist for teenagers to nurture their footballing mentality. Kids as young as eleven can be seen enrolled in these academies, each promised a contract with a different club if they evolve into a great player. These young players have that motivation. 'I will be playing at Old Trafford one day,' they say. This may be unrealistic, but they don't have to be playing for United to be playing at Old Trafford. These kids have ambitions and they are vigorously trained to help them achieve their ambitions.

Do such 'academies' or schools exist for teenagers to nurture their mathematical mentality? No - well none that I know of. A lot can be argued about mathematical ability, but in my opinion if you're not 'naturally' talented then you can fall into two (or three) categories. You have the lazy teenage mathematicians, the hard working ones, and well the third category I was going to write was the ones who love maths but they belong to the hard working category (at times). I've always thought that the problems started at secondary school - they don't. They start at primary school. Nuno's friend is seven or eight years old and hates 'numeracy'. Another of Nuno's friends (six, seven) wants to be an 'actor'. 'Do you like maths?' I ask, only to be met with a shake of the head and a slowly said paragraph about why maths isn't liked, but history and English are. Thankfully these kids were to young to 'mock me' for even suggesting that they should enjoy maths!

Now I realise that my 'analysis' is empty, since there do exist students who love and enjoy maths at that age. If they're unlucky, they end up having problems in secondary school (Noddy), and never like maths in the same way again.

Does the difficulty lie in nurturing mathematical mentality, or trying to create a way in which this is possible? Will all the students who do end up studying maths always say, 'I got lucky and had great maths teachers.' upon being asked why? That is true, but is it me exaggerating things again, or have I yet to meet a teenager who loves maths. Since I left college I've not met anyone who wants to study maths at University - no one. I know I don't exactly travel the world, but from where I live and my school, no one. I know that one person from my college applied for maths (hope they got in), but is it the lack of 'mathematical academies' that is off putting to students? Should universities - the big clubs, do something about it rather than the Government and other organisations? Actually what can be done?

I have gone off another tangent (since I didn't complete this last night), but what Polya says in his book should be instilled in children at a young age. That book is a great for learning about mathematical mentality. I think I fall into the category of the students who love maths, but this has always been connected to the teachers I had. I have been lucky, and as I always add to the unfortunate questioner, 'I had the right mathematical accidents in life.'

Is this post ever going to end? (rhetorical question!) So to conclude(!), I think the most important thing of a mathematician is his attitude, which is closely related to their mathematical mentality. A healthy attitude towards maths, leaves the door open for everything else. I like the idea of comparing this mentality to a footballers, since when a player has the ball they are also faced with questions. Whose free? Can I find him? What are the unknowns? What do we know? For the person with a superior mathematical mentality these questions will be second nature to them, as will be the avenues they explore and the tackles they make. For the young aspiring mathematician, there will be the hesitancy before launching into a tackle. A naiveness in the next step of their calculations, which hopefully evolves. Mathematics and football is all heart. (If you watched the second video linked above, you'll know what I mean. :D)


beans said...

Actually, primary school might be pushing it. Noddy just said that what I've said is 'bull.....'! According to Noddy, kids enjoy all subjects at primary school and don't really care.

I disagree - we kids tend to always remember what we dislike, and I do recall disliking English at primary school (and Geography).

But yeah - things probably start going wrong at secondary school!

kh said...

I'm 16 and from the UK. I love maths (why else would I be reading maths blogs?) - partly because of my teachers but mainly because I find maths naturally interesting. It has to be said that secondary school does a pretty good job of making maths look boring and completely dry as a subject. A lot of it is 'cut-and-paste', i.e. people apply the methods without understanding what at all they're doing.

So when a question is given in a slightly different format, they get frustrated and give up. They might learn how to do that kind of question too, even though it's the same thing in a slightly different way, but never seem to connect the two ideas, and never build properly on what they already know.

I do know a lot of friends who don't see how, say, learning trigonmetry has any practical applications. Of course there's loads (and with a little imagination I'm sure they could think of some) and mathematics doesn't exactly need a real-life application to be important. It's more an excuse.

Some people just don't want to learn.

'Course, as much as I love maths, it doesn't necessarially mean I'm good at it. Ahem.

beans said...

Hey kh,

I agree with your last line. I think loving it makes you more determined to work harder at it. :)

I'm glad that you find maths interesting, and once again, having read a paper from Maths Under the Microscope (I'll try to find the link again), in Hungary less of the applications of maths is taught but more of 'real maths'. It is a shame that in England, or within my social circle, the first question one is asked about studying maths is, 'What's the point?'.

Maybe a certain 'cut-and-pasting' is required (initially), but it never stops. My school prepared everyone for the intermediate paper, and in year 11 when it was our GCSE year, those who were entered for the higher paper had extra lessons for that material. I really enjoyed them extra lessons, but rather than preparing us all for the intermediate paper, could they not prepare everyone for the higher paper? This way, the students will be stretched mathematically and can still opt to do the intermediate paper.

BTW, if you are hoping to study maths at university and do A Levls, then definitely take further maths. It's hard but great fun. :)

Best of luck with your GCSE results as well.