The art of mathematics comes in the necessity of practice. Not only is following the rules necessary, but it is knowing why the rules work in the first place. Look at this sequence of fractions:
3/3 = 1/1 = 1
4/8 = 1/2
5/15 = 1/3
6/24 = 1/4
7/35 = 1/5
Is this just pure coincidence, or is there something about fractions of the form (x+1)/(x2-1) that mean that they always cancel down to 1/(x-1)? For the mathematically experienced, this is an obvious result that comes from looking at algebra rather than arithmetic. It's also not a coincidence that the sum of the first n odd numbers is n2. Many of my pupils will forget this fact quite quickly. What I hope they don't forget is the ability to at least formulate the question properly.
The child may ask "if God made the world then who made God?" which is a perfectly reasonable question on the face of it. However, but drawing out from that child the ideas that underpin that question, the child will see the answer for herself. This method of "drawing out" is the literal meaning of education.
This does seem to be the reverse of the exam system. Rather than drawing out ideas from the child, efforts seem to be made in cramming information in. The average human being can get by in life without ever needing the knowledge that the Battle of Borodino took place in 1812 or that the giant squid is architeuthis dux, or that E flat major is the relative major to the key of C minor. Yet, these facts may indeed be examined upon and one's place at university may rest upon knowing these facts.
The trouble with cramming in facts is that it creates an imposition on a pupil to which the pupil may not necessarily have subscribed. I am sympathetic to my fifth form student who takes one look at the differential calculus and says, "when will I ever need to know that!?" I try to explain about the understanding of rates of change, of velocities and speeds, and even the ideas of maximising commodities subject to constraints, yet I do question whether the syllabus allows for true education or relies on just simple cramming.
Ah, but education should be about preparing children for dealing with real life problems. I find it interesting that, despite lots of education, we still suffer from the same real-life problems without a satisfactory solution. Perhaps our education system is to blame, or perhaps we keep looking in the wrong place for the solutions and never allow ourselves to be educated in the right way. We bewail the competitive nature of employment and the rat-race, seeking a quieter and calmer lifestyle, and yet the levels of competition for jobs and university places is growing. Is this competition right? Does a good education give one a competitive edge? If it does than why do people continue to compete for a resource rather than find ways to distribute that resource fairly?
Albert Einstein said, "To obtain an assured favourable response from people, it is better to offer them something for their stomachs instead of their brains." Perhaps, then, the Education System should be more a training in utilising resources to attain favourable responses from people. However, then we would find ourselves in the position in which people become means to ends rather than ends in themselves. Already we seem to have done away with any notion of learning for the sake of learning.
There is something inherently wonderful about education in itself. To take pleasure in knowing why something works is wonderful, but so is the pleasure in not knowing why something works and engaging in the task of finding out. This really does allow each one of us to engage with the world around us as we are rather than what we are prescribed to be. We only find out who we really are when we learn to interact with the world around us. We only ever find our passions in life by trial and error, by painting pictures and realising that we've not got the colours quite right. The child that always asks, "please sir, have I done it right?" needs to develop the confidence to know precisely for himself whether he is right or wrong and how it matters to him. We know that 2+2=5 is wrong mathematically, but the act of appreciating why it's wrong possesses a beauty in itself and that beauty is not smugness..
With a world that is constantly geared to the dread dichotomies of success/failure, right/wrong, true/false, we seldom see failure as an education experience. "This organisation does not tolerate failure", said Blofeld, and his thinking ensures that any error is met with the fires of Hell itself. This is why Christianity is quite the reverse. It is only in the task of recognising our errors and turning to God that we realise who we really are. We only find ourselves when we allow ourselves to be drawn out into the arena of Creation and, trusting in God's mercy when we fail, realise who we are through understanding our capabilities and limitations. We need to educate ourselves out of the pass/fail dichotomy if we want to see the true colours of living.
Ulitmately, all competition is meaningless in itself as, at the end of the game, all the pieces go back into the box. It is what remains after the game is played and how we have been drawn out of ourselves by the process. True education will seek to give a pupil that which extends beyond the narrow confines of the work place, and the best education will lead that pupil to become a disciple of God Himself.