Saturday, 20 August 2011

Hidden Genius

It would appear that many mathematics teachers have forgotten (or worse still, never knew) that teaching mathematics does not have to mean transmitting lots of facts at a class full of silent children.
There has been a dumbing down in mathematics education. I see it in the UK, US, Australia, France, New Zealand, Germany and many other North-West cultures, but also in the Gulf Nations where they have fallen victim of allowing themselves to believe that companies from these North-West nations can arrive, unpack their education systems and impose them in a different society (many of the Gulf Nations have been fooled in to thinking the West knows best!)
The dumbing down is a result of a fundamental misunderstanding by policymakers of what mathematics actually is. So, more and more, mathematics is being reduced to calculations and using numbers ("numeracy" as it has been dubbed in several countries).
The main symptom of this dumbing down is that students expect to be spoon fed information and that solutions to mathematical problems should be immediate, obvious and algorithmic. But mathematics is not like that; never has been, and shouldn't be allowed to become. Mathematical problems can take a very long time to solve, or indeed remain unsolved, but along the way, through the head scratching, experimenting, errors, iterations and creative process a heck of a lot of mathematics can be learned. This learning through attempting a problem is age old in mathematics and leads to a much greater understanding of what is learned. There is no need to always tell students how to do something, instead give them a difficult problem and let them play with it.
I believe that there are many hidden geniuses out there (in fact, I believe everyone is a genius – just that we all have different fields of expertise). They are hidden because the education system that they are a part of does not provide opportunities for unlocking the genius.
Here are a few examples of moments that have warmed my heart over the years:

12 year old girl, England, 2008
I asked the child that very same question that the mathematician Guass is famed to have tackled (though whether he actually did or not is debatable to say the least!)
Add together all of the numbers from 1 to 100.
She looked at me with consternation, not best pleased about being asked to do something so apparently pointless. "I'll do 1 to 10" she told me matter-of-factly.
She wrote down the ten numbers in a column addition and then huffed and puffed as she tried to go through adding them mentally. After three attempts of losing count, she suddenly stopped and her eyes sparkled. "It's 55," she said with confidence and pride in her voice.
I beamed and asked her to tell me how she knew that.
"They're elevens, aren't they? Look."
The girl wrote down the numbers 1 to 10 again but this time in two columns:
1 10
2 9
3 8
4 7
5 6
"They always pair up to make 11. 1 and 10 is 11. 2 and 9 is 11." Her eyes were wide with delight. "And there are 5 pairs. So it's five elevens."
And this is what I mean by genius. Genius is about seeing the world differently, it is looking beyond what we have been taught, it is about breaking convention.
Adding the numbers from 1 to 10 together does not take genius. But ignoring all that you have been taught about methodology and seeing a problem in an entirely new light does.
I asked the little girl the original problem again; add together the numbers 1 to 100. She jotted down only three lines of working:
1+100= 101
"I think it's 5050," she said, a little unsure at first of her multiplication of 101 by 50. "Yep. Yep. It's 5050."
Most kids can solve this problem if given enough time and space to think. Most will need prompting to look for a different method, but occasionally you will come across a hidden genius in your class.
But to find them, you need to provide opportunities. How often do maths teachers give this space and time to just think? To create problems? Why has mathematics become so entrenched with a didactic approach? Space to see beyond the convential, to re-write a problem in a new way is incredibly important to truly learning mathematics. As Einstein said:
"The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill"

11 year old boy, England, 2007
The class teacher had posed the problem "How far can I get for £50?" The students had access to local bus timetables, maps and the infinite world of the internet.
This is an age old question in maths teaching. The kids enjoy it. The open-ended nature of the task allows them to experiment and play.
At the end of the lesson, the teacher asked for final destinations from the children. Eager hands shot up and a variety of exciting and interesting locations were happily announced.
At the back of the room, near to where I was sitting (I was observing), a small, pale, ginger haired boy sat with his faced pressed to his book still working. He had opted to work alone in the lesson (students were allowed to form teams) and had remained entirely silent throughout.
The teacher gently asked the boy to sit up and take part. "How far did you get?" he asked the shy lad. The rest of the class turned and stared. The boy gulped.
At the start of the lesson the teacher had promised a prize bounty of chocolate for the student or team that managed to get the furthest away. The routes given so far had been diverse and included bus and train fares, walking for days with the £50 used to buy food and drink, and a whole host of suggestions about hitch-hiking, which led to a great discussion about safety.
The red haired boy put down his pencil, gulped again, and then announced "Sydney. Austrailia."
The class erupted with laughter. The boy sat resolute. The teacher calmed the class and asked the boy to explain.
"I found a really nice bike in the free-ads," he started and the class burst in to laughter again. One child barked "you can't get to Australia on a bike!" Red head ignored the comment and ploughed on regardless.
"Then there was this advert in the paper," he held it up for effect, "looking for a paper boy in town. But you have to be 12. So I kept the bike until December – that's my birthday. Then I got the paper round job."
The class now were silent, the teacher grinning from ear to ear knowing what was coming.
"On the internet, Sir, you can book a flight all the way in to next year. And it's well cheap if you do. You can get an Australia flight with... er..." the boy consulted his notes, "Emirites," he said a little uncertain of the pronunciation, "and it's only £425."
The other children hung on every word.
"So. I did this paper round until June and saved up all my money. I got paid £18 per week. So I needed to work for 24 weeks and that gave me £432. I used £3.70 to get the bus to the Airport and then I even had £3.20 left to get something to eat."
The little red head won the chocolate as well as one heck of a round of applause from his fellow students.
The mathematics that the boy did was not complicated. It did not require any great mathematical skills. But I see this as a moment of genius because the boy ignored the conventional. He dared to think differently.
This is the sort of genius that we are all born with. Watch a young infant at play with some inanimate object and you will soon realise that they see more than you do. They will think of hundreds of different uses for a cardboard box, but to you it is just a cardboard box.
Somewhere along the line, this inventiveness, this genius leaves most of us. Perhaps part of the reason that we lose the ability to see a hundred possibilities, a unique approach, a different perspective in every day problems is that we all experience an education system that does not value genius. An education system that spoon feeds and gives no time to breath and create.

15 year old boy, Qatar, 2011
I sat with a boy as his class were being shown (for the nth time) how to find the area of fairly simple compound shapes, made up only of rectangles. He doodled in his book and made a show of being mindnumbingly bored by the whole experience.
"Do you have some graph paper?" I asked him, seemingly heading off on a tangent.
"Sure." He flipped open his level-arch file and unclipped a leaf of pristine paper.
"Draw a set of axes. First quadrant only." I told him with no further explanation.
He stared at the paper for a moment in thought, then produced a pencil and ruler from his case and neatly drew out the two lines required to form the x- and y-axis.
I took the paper from him and randomly drew a squiggle across the page going from left to right. He looked at me bemused.
"I want you to tell me exactly what the area under that curvey line is." I said.
Now this is a fairly standard investigation that I have done with hundreds of kids. They will start by counting the squares and realise that that is frustrating, then they will split the space in to blocks that they can calculate – imagine rectangles all over the place. Some will just estimate, some will be painstakingly accurate. Some will use square centimeters, some will invent their own units of measure.
The boy was no different. He faffed about with the problem for a while and became annoyed. "There has got to be a better way of doing this!" he laughed. After a few moments of experimenting further, he struck upon the notion of dividing the whole thing up in to vertical strips. "You can pretend each one is a trapezium," he said, justifying his approach, "and then find their areas"
He had made each vertical strip the same width. I asked him why he had chosen to do that and he explained that it was just because he was lazy, "that way they all have the same dimension, so you can do all the calculations at once."
After a few more minutes, he said to me in a conversational tone "It gets more accurate the smaller you make the strips, doesn't it?" I raised an eyebrow inquisitively. "Yes. Yes. I'm sure that's what it is. So if you could make them really small,"
"Really?" I interrupted.
"Yes. Infinitesimally?" I nodded. "Yes. If they were infinitesimally small then it would be very accurate."
"Just very?" I asked.
"No. No. It would be exact."
I have used this task countless times, and groups of children time and again will come to a similar conclusion. But it was the boy's ability to see through the structure, to imagine the abstract that makes him a genius in my eyes. Newton and Leibniz both came to the same conclusion. And this conclusion leads to the formation of The Calculus. Not bad for a 15 minute discussion on a hot May afternoon.

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