I’m hoping somebody will comment on this post to help me understand why math is taught the way it is and start a constructive discussion on the matter.
So, here’s how I have been taught math from 1st grade all the way up to college:
- You learn the basics of whatever part of math you are studying (theorems, formulas, basic algebra, etc.)
- You solve hundreds of exercises to strengthen your skills on the part you’ve learned the basics of. This includes tests, quizzes and homework.
The time allocation of these two activities was always in favor of the second one (solving exercises), rather than learning the basics. It usually took little time to cover in class a theorem, but it would usually take several hours to finish the homework. Nonetheless, I honestly do not remember how to solve by hand most integrals from my Calculus I and II classes and I do not remember most formulas from my geometry classes. I look formulas on Wikipedia when I need them and I use Mathematica or Wolframalpha so solve my integrals. I had an A in my Calculus classes, I was good at solving integrals. If right now you put in front of a Calculus exam I would probably fail it.
I’m a computer developer and as such I recognize the importance of not “reinventing the wheel”, which is a term referred to the fact that many problems in computer science have already been solved and that you should not spend your time resolving it (unless you want to learn how it works). Now it really upsets me that I spent so much time (and money) reinventing the wheel in my math classes. I think that if I spent most of my time setting up problems, formulating hypothesis and calculating my results with a computer instead of focusing on the numerical resolution of it I would have learned much, much more. Now don’t get me wrong, I think that being able to do basic algebra in your head can be important (and convenient) for many tasks in life, but to what point?
In an educational environment we put so much time in the actual calculations to find a solution to a problem, when that time could be spent elsewhere understanding more important things, like setting up a problem, challenging students to solve a problem in new or multiple ways or simply covering more topics. I heard countless times people saying that “oh, using a calculator is too easy”. During my elementary, middle and high school education in Italy, I was strictly forbidden to use a calculator (square roots, trig and log functions were always setup to be trivial cases like Sqrt(9) = 3). This is completely absurd. Calculators were invented for the purpose of making calculations quicker and allow us to concentrate on the more important things, setting up our problems, understanding what our objective is, finding new ways to solve a problem. Before we had calculators, to find the value of a square root you looked it up in a table or you would calculate it manually. We don’t use tables anymore to calculate square roots, because a cheap $5 calculator does it for you.
Now in 2011 we have computers that can do much more than just add, subtract or calculate square roots. Mathlab, Mathematica, Wolframalpha and many more software out there allow you to setup and solve very complex calculations, including integrals, derivatives, second order linear equations and the list can go on and on. Why is an integral any different than a square root? Just plug it in the computer, find the values, make sure they are correct (spend time verifying the correctness of your solution) and move on! Why not spend our time more productively?0