Monday, November 11, 2013

Best Shortcut First or How the Order of Operations SHOULD Be Taught

I once taught a class that used a book that defined the order of operations as "an agreed upon" order to do math problems!


As though there was some dark cabal of mathematicians meeting secretly across the world to decide how math works.

I cannot begin to describe to you how frustrated this made me.

...oh wait, that was sort of the point of this.

You see, the thing about math, with some obvious exceptions like vocabulary and notation, is that it is not some arbitrary set of rules that are handed down from on high that we must follow or be shot.

The rules of mathematics are inviolate based on their assumptions. They are universally true.

If we meet aliens, the first language we will be able to communicate to them with is mathematics. It will be a common ground for all thinking species because all thinking species are capable of learning math and this math will be the same for everyone.

In science, a science theory is something that is accepted as fact based on all known evidence. It's pretty much fact, but part of the great thing in science is that you know you don't have all the facts, so you can't ever say something like, "This is how it is." You always say, "This is how it is based on what we know."

That's great and all, but in mathematics, we have theorems, not theories. These are facts. Universally true facts that so long as the assumptions to the theorem are true, will always be true.

Period. The end.

The mathematicians of the world didn't get together and decide, "This is how we'll really screw with those algebra students". Math is the way it is because that is the way it is.

The order of operations is not the order of operations because we agreed upon that order. It's not like the order of who gets to go at an all-way stop sign. It works like this:
Counting (or succession) is the simplest mathematical operation we can describe.
Addition is repeated counting. Subtraction is its inverse.
Multiplication is repeated addition. Division is its inverse.
Powers are repeated multiplication. Roots and logarithms are their inverses.
So, if you were to come up with an order of operations for our notation of how we write mathematics, wouldn't you come up with the same one that we did? That is,
Best shortcut first.
Now doesn't that make sense? Which, after all, is the point...

One of the nice things about mathematics, as opposed to, say, English, is that it makes sense. There are not arbitrary rules that exist just to screw with you.

You might be wondering why I don't have "parentheses" included, because parentheses are insufficient. Here, let me pose a problem to you:

You have the order of operations, which is "best shortcut first", but sometimes you want to go out of order, how do you take care of that?
I'm hoping that you said something along the lines of, "why not just call out the ones that we want to do first by like, grouping them together and separating them from the other operations?"

Well, there you go. Groups should be taken care of first. That handles pretty much anything that you want it to handle, from the basic parentheses or brackets which exist only to define groups, to functions, absolute value bars, and a plethora of other math groups.

Now you know how the order of operations works, go forth and multiply.

Wait a second though, one more criticism of the order of operations you were all taught in primary school and one fantastic application of what I just taught you.

Criticism: "Please Excuse My Dear Aunt Sally" or "PEMDAS" results in people thinking that multiplication comes before division and addition comes before subtraction.

If you're thinking, "Well of course it does, that's how it works," guess what? You're totally incorrect. First of all, multiplication and division are pretty much the same thing, so they wouldn't be done separately. Don't believe me? Change every division ever into multiplication by the reciprocal of the divisor and you won't ever have to do division again. Thereby removing any controversy.

But wait, there's more. Change every subtraction ever into addition by the opposite of the subtrahend and you won't ever have to do subtraction again! Oh, and just as an added benefit, by changing both of these they will become commutative and associative (which division and subtraction are not).

If you insist on using PEMDAS, please write it vertically like so:
M or D
A or S
Application of what we just learned: You might be thinking, "Why bother learning how something works so long as I know that it works and I've memorized it?" (If you're thinking that, please try to find the curiosity that you had as a child and apply it to everything in life).

Well, because knowing how something works means that you will be sooooo much better equipped. First, you're not going to make the division/subtraction error I described above, second, you will have no issue whatsoever incorporating OTHER operations.

"Other operations?" You may be asking, well yeah. You didn't think we have to stop at exponents, did you?

We can have repeated exponentiation, called tetration (which is a stupid name, but sounds cool, so whatever). Heck, we can also incorporate succession!

And in fact, we could have repeated tetration, repeated whatever that is (pentation I think), repeated whatever that is, repeated whatever that is, repeated whatever that is, etc, etc, etc.

So if you wanted to memorize the order of operations up through pentation, then you would do the following:

Parentheses and other grouping symbols, pentation, tetration, exponentiation, multiplication, division, addition, subtraction, succession (or counting).
At what point in your memorization exercises do you find that it is easier to understand what you're doing than it is to memorize?

Now, I agree that you'll rarely see tetrations, but you see succession all the time in any more than basic level of math. It's the principle that our summation notation works on after all, and you see it all the time in computer programming, and you see it all the time in anything that ever needs to be iterated. 

Can you imagine learning how to ride a bike by memorizing all of the different aspects of riding a bike? Two years after the last time you tested your memory, do you think you would still remember it all? Would we have a phrase like, "You never forget how to ride a bike"?