Why "common core" maths isn't as stupid as you think

Every so often, I see a post show up in my news feed about how "common core" maths is new, and thus, terrible. Well, not quite. It's mostly people who don't understand it because, for one thing, they've been shown an example that's deliberately cherry picked to show it looking dumb, but for another, they haven't had the method behind it explained.

And, well, if there's one thing I love doing, it's explaining maths things like a nerd!



Now, I want to ignore for a start, the fact that the new method of maths isn't "common core". I'm going to get this horribly wrong, because I'm not American and the name "common core" comes from the new American teaching standards. The fact that it includes this new style of doing maths is kinda just a coincidence. America isn't the only country teaching students this method, my younger sister was being taught it a year or so ago.
But to me, this is kinda pointless semantics. I want to explain the method behind this, so to do this, let's start with the example constantly being shown.

32 - 12

Now, hopefully anyone can look at this and figure the answer is 20. The problem comes from trying to actually properly figure it out by hand. The old method and the new method side-by-side look something like this:

You get the same answer either way, but the old way on the left looks much easier. There's only one step, and the way on the left just adds in more ways you can make errors. Why would anyone use it? Where do those numbers even come from?

I'll answer the second first. What you're never shown in this example is what's actually being done. You just get shown some weird numbers and, if you don't understand what the method is, of course they'll look strange.
The point behind the new method is that, well, the human brain is really awful at maths. Seriously. Our brains are simply not wired that way. Our brains evolved in an environment where the only useful numbers were "one enemy" and "lots and lots of enemies". That's why we invented the calculator.
So the new method is designed to get the numbers to something nice and friendly. Specifically, multiples of 5 and 10, because multiples of 5 and 10 are pretty much the easiest numbers to deal with. 12 + 3 = 15, 15 is a nicer number than 12, 15 + 5 = 20, and 20 is a much nicer number than 12.
The other thing it's designed to do is to skip subtraction entirely. Our brains are pretty awful at that too, and pretty much any time anyone subtracts anything in their head, they're actually just thinking "now what number adds to make it up to the full amount?" Subtraction is, at its core, reverse addition, and that's the way our brain subtracts too. By leaving out any actually subtraction, we make it easier for ourselves to actually subtract in our head. Keep in mind that, this also poorly represents how the second method actually works. You don't do all those additions, then do that last addition at the end. You keep a running total as you go. You go, "oh 12 + 3 = 15, 15 + 5 = 20, 5 + 3 = 8 and I'll keep that in the back of my head, 20 + 10 = 30, 8 + 10 = 18, good thing I didn't forget that 8 I said I'd remember, 30 + 2 = 32, 18 + 2 = 20".

Because that's the main point behind this new method of maths. To teach students how to subtract in their head in a way that is easy, actually works with out brains and not against and, most importantly, has a smaller chance of errors. Let's face it, pretty much any time you need to do some subtraction, or any other complicated maths problem, and you've got a pen and paper handy, you've almost certainly also got a calculator handy, either on your computer or on your cellphone. Nobody really needs to know how to do pen-and-paper maths except as an absolute last resort. Being able to do it in your head, however, is pretty handy.

Let's look at a different example.

31 - 12

Oh dear. That old method isn't looking quite so good any more. It still kinda looks better though, right? Also, note I skipped a step in the second one there. While the example shows adding to multiples of 5, I don't really see the point in that. To me, it's easier to go straight to multiples of 10. That way, until the final step, you're only ever adding a one multiple of ten to a different multiple of different.

But also remember what I said. Half the point is being able to do this in your head. How do you carry the 1 in your head? It's doable, but it's easy to slip up.

The difference becomes more obvious when you start having 4 or so digit numbers. Like, say... counting out change!

Let's say, you're buying something for $3.45 and you're paying with a $20 and you want to know how much change you get. Let's look at both methods again!

20 - 3.45

Look at the convoluted way the old method takes to figure out change, whereas the new method is much, much easier to do in your head. A lot of cashiers already know this method for counting out change, but suddenly, when it's labelled "common core", it's terrible? I know I'd used it a few times before I'd heard the phrase "common core", because it's easier to do in your head.

Lastly, though, I want to use a nicely cherry picked example to, well, demonstrate why cherry picking is stupid but also to illustrate my point again.

1000 - 999

Wow! That old method is so slow and terrible! Why would anyone use it?

BEMA, or how to not fail at troll maths

Okay, if you've done any form of maths in school, you've probably heard of BEMA, BEDMAS, PODMAS, PIMDAS or some variation of that acronym. Basically, it means the order of operations and is the cornerstone of basic arithmatic. Some very successful trolls out there have decided to post basic maths equations and trick people who have a poor understanding of it.

And so, I'm writing a brief note, explaining the order, why it's in that order, and hopefully clearing up some misconceptions.

Firstly, the acronym itself. The acronym isn't important, but I'm most in favour of BEMA because it cuts out division and subtraction, no longer implying that they come before or after addition or multiplication.

So!

1. Brackets (Or, Parentheses) ()

Brackets always come first. Why? Because mathematicians need a symbol that means "Do this first". Plenty of useful situations, such as 2×(x+3) and so forth. The (x + 3) always comes first. I should note that things outside the brackets are not part of it. This seems obvious, but apparently people seem to think that 2(x+3) means the 2 is part of the brackets. I'm not sure why. It's not. There's an invisible multiplication sign (This is identical to  what I wrote earlier) and that means you do it in the multiplication step.
You can nest brackets and stuff too. Anything inside the brackets follows the normal order of operations so you do the innermost brackets first and move out.
There are also implied brackets, where the brackets aren't written because mathematicians are lazy. These mostly come in two situations.
Firstly, square roots. Anything inside the square root symbol is considered bracketed even though the bracket is nearly never written. I'd give an example but facebook notes don't let me.
Secondly, things inside the power. Say, 2^-1, the -1 is considered all part of the power.
Secondly, division written as fractions. The numerator (ie the top line) and the denominator (ie the bottom line) are both in seperate brackets. So, for example:

2x + 3
3x - 1

This is the same as

(2x + 3)
(3x - 1)

I should point out now that 2x + 3 ÷ 3x - 1 does not have any implied brackets, and is, in fact, the same as 2 × x + 3 ÷ 3 × x - 1. This is important but I'll get into why later.

2. Exponents (Or, orders or indices) ^

These are normally written as a superscriptor the alternative symbol is ^. It's just powers and square roots. 2³ or x² or √4 come second. Why is this?
Basically, it's because of what a power is. It's repeated multiplication. 2^3 means 2 × 2 × 2. If you have, say, 3×2³, what you're doing is 3 × 2 × 2 × 2, which is very different from (3×2)³ or 6 × 6 × 6.
Now here, we get problems from -1² because is this -1 × -1 or -(1 × 1). The answer is the latter. Exponents come before the inverse sign. If we meant -1 × -1, we'd write (-1)².
Square root happens at this time too because the square root is just something to the power of a fraction.

3. Multiplication (And divison) ×÷

This seems to cause a lot of issue because of the whole implied brackets thing above. But, first things first, multiplication and division happen at the same time. One doesn't happen before the other, you do them both at exactly the same time. Why? Because multiplication and division are identical operations. The only difference is when you divide, you multiply by the inverse of a number. What's the inverse of a number? If your number is x, the inverse is either x^-1 or 1÷x. Both are the same thing.
Go on, try it. Get a decent calculator and do, say 6 ÷ 2. You'll get 3. Then do 6 × 0.5 (0.5 being the inverse of 2). You'll get 3 again. They're both identical operations and that's why I don't like having BEMA explicitly state division. Division is just a funny way of doing multiplication.
Now then, why do these come in this bit? Because, like exponents, multiplication is just repeated addition. 6 × 3 is just 6 + 6 + 6. 2³ is just 2 × 2 × 2 is just (2 × 2) × 2 = (2 × 2) × (2 × 2) = 2 + 2 + 2 + 2. Again, 2 + 6 × 3 is 2 + 6 + 6 + 6, which is different to 8 + 8 + 8.
Now, I mentioned 2 × x + 3 ÷ 3 × x - 1 earlier. Now, order of operations says that the middle bit is all done at the same time. So this isn't 3 ÷ (3 × x), it's 3 ÷ 3 × x. The first is equal to 1/x, the second is equal to x. That's a huge difference and the original troll maths was based around this distinction.

4. Addition (and Subtraction) +-

Okay, this is fairly obvious. Last, you do all addition and subtraction. Again, these happen at the same time because subtraction is just adding the negative of a number. 5 - 3 is the same as 5 + (-3). I don't think I need to go in to more detail here.

Doing things at the same time.

I've mentioned this a few times. Doing multiplication and division at the same time, and addition and subtraction at the same time. Obviously, that's a rather tricky thing to do. So, what I mean when I say this is you work from left to right. If you have 5 + 3 - 2, going from left to right, you do the addition first, 5 + 3 = 8, then subtract 8 - 2 = 6. For 5 - 3 + 2, you do the subtraction first, 5 - 3 = 2, then the addition, 2 + 2 = 4. Same with division and multiplication. 12 ÷ 4 × 2, do the division first to get 3, then multiply by 2 to get 6. Or, 12 × 4 ÷ 2, you do the multiplication first to get 48, then divide by 2 to get 24.

Now, for something more advanced (Feel free to ignore this if you aren't a nerd), the truth is, the order doesn't actually matter. To use a fancy maths term, addition and multiplication are commutative. Which means the order doesn't matter. If you remember that subtraction is adding by the negative of a number and division is multiplying by the inverse of a number, the order becomes unimportant. 5 + 3 + (-2) is the same as (-2) + 5 + 3 is the same as 3 + 5 + (-2). Likewise, 12 × (1/4) × 2 is the same as (1/4) × 2 × 12.


Aaand that's about all I have to say. Remember people, don't feed the troll and remember the order to do things!