A View From Abroad
Although coming from a different background and a different country I have realized that the teaching methods aren’t so different at all. I learned Maths through the formulae or instrumental methods. In high school they teach you maths but really they are only preparing you for the big exam will get you to university. Also, in your last year, when you are focused on the A levels, there are additional private institutes that offer year courses on pure exam preparation. These institutes, are very good at showing how to use the formulas and how to use the short cuts to get to the right results quickly – the more questions answered correctly in the exam, the better chance of getting into a prestigious university (the exam is 3.5 hours long).
So you find yourself in a rat-race, competing with millions of other applicants, with the additional pressure and demand from your family, in a way you are assessed and judged on how intelligent you are only on the result of this ‘instrumental’ biased exam, but nothing else. However, to me compacting everything you have learnt over the 13-14 years at school, and let them decide your future in this short period of time is cruel to young beautiful minds that have so much more potential.
This is what I know of Maths teaching and I have to say, just like many other students I don’t like the word problems either, when there is too many words there was always more confusion. However, I loved using formulae and getting to the answer from the short cuts, because time is precious, as it always have been! However, now that I am able to see the other side of the coin, now that I have been shown the difference between instrumental and relational, I wish I had learnt relational first. Then with the greater understanding of maths perhaps I would come up with my own short-cuts and formulae which would of make more sense.
I believe there is a big revolution beginning to happen and it will eventually be understood and applied in schools that the relational way of teaching maths must be the best way or at least I’d like to think so. When I have been shown the logic behind the formulae I would the ‘Eureka’ moment; everything became perfectly clear and whole lot simpler. From my personal experience I strongly believe that without creating a good foundation you cannot build a solid body of knowledge. Now the problem is, how can you build a good foundation if your students are not perceptive to the methods you apply?
Meyer had cleverly rebuilt an experiment from a textbook on the connection (link) between the instrumental and relational understanding of maths. Using this experiment (how long will it take to fill the container?) he was able to make the students understand the problem first. Once you clearly understand the problem you ask the right questions to its solution and you are half-way through to solving that problem. And his experiment shows, despite that everything is clearly stated and written down in the textbook we often miss the opportunity to understand the problem. Then it leaves us only with an instrumental understanding and a reliance on that knowledge to get solutions without any true understanding, in reality we won’t have a text book or teacher telling us the correct answer.