Friday, February 12, 2010

Math part III: Algebra and Calculus

So after numbers, arithmetic and geometry I get to the last part of mathematic that is taught in high school science: algebra. It is also the point where most people that already dislike math move up to hate and what they hell will that ever be usefull for. So what happen at this point, well most people will tell you that it’s the addition of letter to the number that just sound illogical and learning all those new “rules” to work with these letter. Now, you can probably guess I don’t agree, first off because I always found (elementary) algebra easy and second because I know that it is found everywhere and that almost everyone (even those that have not been taught algebra) use it everyday.

First off because, I think, the problem is how algebra is taught in school. In my opinion algebra is taugh way too late in the curiculum, it should be taught right at the beginning when we are learning the rules of arithmetics and numbers. You see, algebra introduce in math the concept of variables, i.e. unknown quantities. The problem is when you are first taught about this concept you have already been using math with unknown quantities when solving problem. So you have to relearn how to incorporate unknown in your math and the “special” rules you need to use to apply arithmetic on unknown quantities. Lots of duplicated work and lots of confusion from student as you have to basically rewrire the brain. Also, what we are taught is just the surface of algebra. So in the spirit of these post about math what can we do with algebra that we could not do before?

Well, a lot. You see algebra is not just the study of what happen when you replace number with variable, it is also the study of what happen when you change the operation and rules relatated to them and (more importantly, I think) how to apply mathematical operation on thing other then numbers. This last thing gives us chemical equations that give us the opportunity to understand what happen in chemical reaction, such as combustion. In fact, most of not all of science need algebra to be represented in comprehensive mathematical form. So basically, algebra add to the ability to representing physical and imaginary, manipulating these quantities and representing their shape, position and size, the ability to work with unknown or variable quantities, the ability to see how other type and set of manipulation would work and most importantly how to apply mathematical logic beyond numbers.

So with all that power, math must surely being done for right? Wrong, we now get to calculus, a part of math some of you might only have heard through horrified mention of college student having their first taste. So what is this dreaded discipline? Well calculus is slightly akind to geometry, in the way that while geometry deals with the shape and size of thing calculus deal with the movement and change of things. It involve such concept as the limit (seeing how as you approch a quantity another varies), fonction (describing a quantity in relation to others), derivative (describing the change in a fonction as it’s input change), integral (the reverse of the derivative and also the area under a fonction) and infinite series (numbers with recursive proprieties between themselve).

Lost of component that have an unparallel power, just about everything in the univers can be represented by a fonction, a derivative of a fonction or it’s integral (depending on what the phenomum involve). Most of advanced physics, engeenering, economic, all science in general involve calculus to a great degree. To continu with the language analogy, if arithmetics, numbers, and algebra form the “letters” of the language in our models of relality, geometry and calculus (and some would argue calculus alone) is the gramar, words and text structures. Throught this we can for exemple show how blue and yellow light form green light (it involve the transmission fonction of our eyes photoreceptor coupled with the wavelenght of the light we are seeing and how our brain intepret the resulting electro-chemical signals). Maths can be used to model much more complex phenomenum, such as the nuclear fusion in stars, chemical process involving complexe molecules (such sugars) and would appear at this point to be limited to predictable, deterministic process. In the next post I will talk about how maths deal with undeterministic process when I talk about statistic, chaos theory and probably the more advanced maths that I am only slightly famillar with (such as group theory).


Algebra tutoring said...

The problem is not that Algebra is taught too late. The problem is that it is taught improperly. Students are not taught to think. Instead, they are taught to memorize.

yofed said...

I loved algebra, to me it was like a nice puzzle... what I hated was to memorize those %#@? times tables! I'm so glad my boy is learning them without being to be coerced! Hopefully, that will help him retain his love for the numbers!