I was sent this article from Café Casey. It discusses a common difficulty faced by those who have to teach mathematics. What started off as a comment on the blog, turned into the ramble below. Enjoy and do make sure to read the linked article first to understand the context!
When Will I Ever Use This?
What’s tough about answering the “when will I ever use this” question is a result of the fact that students take math classes one at a time by “subject” area and that those subject areas are not often tied together.
People, in general, aren’t going to find a use for, say, calculus as a stand alone topic, nor linear algebra, nor geometry, nor trigonometry, nor differential equations, etc. However, if a person spends a little bit of time learning all of these things, then there is a tremendous amount of application. And mathematics combined with another discipline is extremely powerful. Unfortunately, school is mostly useless at doing this.
Some Examples
I am writing some software right now to allow people to draw on their computer (and yes, this has already been done by many, many other people, but it’s a “fun” learning exercise for me). But let’s say I want them to draw a square and give them the ability to rotate it. Unlike pencil and paper where we can simply draw the rotated figure at our desire, drawing on a computer is not so simple. Rotation requires geometry and trigonometry and with a little bit of sophistication it’s linear algebra with trigonometry. But then we have to consider boundary conditions … there’s a canvas. I don’t want the person to draw a square and then rotate it so that part of the square is off the viewing screen. Handling those boundary conditions is algebra and geometry and trigonometry. And now suppose I don’t want to do this for a square, but for any polygon (convex or concave) of any size. Oh yeah, and it’s helpful if you can program!
To further the programming example, suppose I want to writing a simple “shooting” game. Like the old school Atari games where spaceships come from the right towards you on the left and you have to shoot them down. Conceptually it’s a simple problem, “when the bullet hits the enemy spaceship, the enemy spaceship is destroyed”. But this is far more complicated to implement than it is to say. How do you know when your bullet has entered into the same region as the enemy spaceship? Again, visually it’s so simple and doesn’t even sound like a problem worth considering. But try programming this. Without mathematics, it’s practically a lost cause. This is called “collision detection” (and in truth, there are ways to get around this as well, but guess what … that’s more knowledge). Unfortunately, we call all this programming, not mathematics.
A friend of mine wants to buy a house. How much will she have to pay? Can she afford it? There are numerous mortgage calculators on the internet. But guess what, those calculators only give a ball park figure and they do not consider her particular financial situation — things like her tax bracket, what she can take as a tax deduction, the influence of property taxes, possible reinvestment of the tax deductions, etc. That is all mathematics, but we call it finance. And now suppose she wants to consider renting out a room. What then? Of course, we can say, “but there are real estate professionals that we can consult”. Yes, but consider this: a) that will cost money and how much money do you want to spend on analyzing how much money you will spend?, b) how much math do they know or are they just punching a bunch of numbers into a software program that some team of (likely math guys and gals) developed?, and c) how flexible is that software program if she wanted to explore other scenarios (maybe setting up a home office). If I wanted to do this analysis, I could just carve out a Saturday morning and write the appropriate programs catered to my specific needs. The only cost to me is 4 hours on a Saturday morning that I probably would’ve burned on Twitter edu-chats. But what math would I be using? Algebra!!! and heavy amounts of it. And maybe if I spent more time I can make it into a proper software program and sell it.
Let’s take history for example, and specifically doing research into the past, not just reading about someone else’s research. I encourage any student who wants to go into a field that is “not technical” to peruse through this book: “Alexander The Great And The Logistics Of The Macedonian Army” (I am not getting any revenue for the link to Amazon) … the author convincingly demonstrates that certain events in Alexander’s campaign which were thought to be true based on historical records were actually not even possible or grossly exaggerated — and he uses a decent amount of Algebra. For example, he shows that certain marches were impossible because it would have required the army to move at a stupendous pace, or that certain figures for army size weren’t possible because it would have required more pack animals than ever had existed, or that certain routes that were taken weren’t possible. And this is simply awesome because it gives us a more accurate understanding of what actually happened. History is written by the victors and perhaps we want to know how much of that history is true, how much is true but embellished, and how much is simply a lie for propaganda’s sake. If we don’t understand our past, then it’s our future.
In “Sum”mary
What we’ve gained in structure of knowledge and content by category as taught in school, we’ve lost in connectivity of knowledge. This in turn, forces us to think about “Can we teach creativity?” because we see first hand the automatons that are being produced through school. We don’t have to teach creativity, we just have to teach in a more unified way! There’s nothing wrong in having an Algebra class, for example, but at some point teachers from different disciplines ought to come together and mix the subject topics together.
Anyway, I can drone on endlessly about this. But in a nutshell, “Is School Useless?” Yes. and No. It comes down to necessity vs sufficiency. Is school necessary for us to gain useful knowledge that we can apply? No, it is not necessary. One doesn’t have to go to school at all to acquire theoretical or applied knowledge. But with that said, school offers a certain structure that, for the most part, can be sufficient if we bend school a little bit towards our needs. We as students and teachers have to recognize that schools, as they currently exist, are structures not necessarily designed for the individual but for the masses. We have to take that giant piece of pie we’re being force fed and cut into bite-size pieces, eat it one piece at a time, and let it digest before we overstuff ourselves and feel sick.
Ok metaphor gone a little too far. End post.