Common Core Is a Better Way of Teaching Math Because It Actually IS Math

This is a post for math nerds, and for those who wanted to be math nerds, but sucked at it.  But it's especially a post for those who discovered as adults that they weren't nearly as bad at math as they thought.

Let's pretend for a minute that you're at a store and you only have cash.  You have to buy stuff for dinner, and it has to be under $20 because that's all you've got, so you start keeping a running total.  You don't have a pencil, pen, or paper, and you also left your phone on the kitchen table at home so you don't have a calculator.  The items you choose are $6.99, $5.99, $1.25, and $3.99.  You've always sucked at math, but you somehow manage to add up the prices of the items you want in your head and not wind up mortified at the register or out your proper change.

So how do you do it?  Lay out the numbers in your head and carry from column to column?  Hell no.  That takes too long and it's too hard to keep it all straight without writing it down even if you are a visual learner.  No, instead, you reflexively start counting them.

You round up the first two items to $7 and $6, respectively, so you get $13 so far.  You round up the $3.99 item to $4 and add it, so that gives you $17.  You add the $1.25 item to get $18.25.  Then you count back $0.03 because of the rounding to get $18.22.

Now, to know that the cashier isn't shortchanging you, you figure out how much you're supposed to get back.  You work with the cents first.  From $0.22, it takes $0.08 to get to $0.30, and then $0.70 to get to the next whole dollar, which is $19.  $20 less $19 is $1, so you should get $1.78 back in change.

This is what common core math is.  It's the way we learn to do math as adults after we figure out the way we learned it as kids is bullshit.

The problem with the Old Way of learning math -- in particular, the borrow/carry method of addition and subtraction -- is that it's not math at all.  And because of that, all it does is reinforce bad number theory, and teach kids principles that later turn out to be completely wrong.

See, borrow/carry starts off on a bad foot because it uses the "column" theory of numbers.  That is, separating multi-digit numbers into valued columns: ones, tens, hundreds, etc.  When you're a kid, this kind of visual explanation holds water at first because it's easy to remember and it makes big numbers not seem so scary.  But it falls apart like a bad game of Jenga when you start throwing zeros in there.  Because you're taught that there's nothing in that column, and the zero is just a placeholder.  But we don't do that for every column, because Reasons.  And if you're a kid like me, you start thinking that the difference between 13 and 1,300 is a couple of zeros (rather than "1,287").

Borrow/carry continues on an even worse foot because in the course of learning it, you're taught that the reason you have to "borrow" a 1 from the next column over is because you "can't subtract a larger number from a smaller one."  That very principle turns out to be bullshit later when you start learning integers and negative numbers.  And again, if you're a kid like me, at that point you don't know what to think, because you don't know why the stuff you were taught before is complete lies now, and you're afraid to learn anything new because that will turn out to be more lies in a couple of years.  And while you're passing classes, you're only doing so because you're good at memorizing rules, even if you have no clue what the hell you're even doing and wish somebody would teach you something consistent.  You start to hate math, not because you suck at it, but because you can't trust it.

Common core, on the other hand, uses the far more sound set theory of numbers.  There are no columns; each number is its own set.  It's the exact same way we learned to add and subtract single-digit numbers: by counting.  Counting doesn't change or become irrelevant just because the numbers get bigger.  410 is 410, not 4 hundreds, 1 ten, and no ones (but we put a zero there anyway, because Reasons).  Thus, adding to it or subtracting from it is simply a matter of counting forward or backward (in other words, adding to the set or taking away from it).  Common core simply teaches kids to do so in large, easy blocks since they don't have nearly enough fingers.

This goes for multiplication and division, too.  Take, for example, 26 x 54.  This is the old way you would solve that one:

     26
  x 54
   104
 130  
 1404

Guess what?  This is still using the borrowing/carrying number column nonsense.  And it's still a pain in the ass to keep straight in your head without the ability to write it down (especially having to shift the second row over one, which I always forgot to do as a kid because I never learned why I had to until I was an adult).

Now, let's look at the common core method of solving the same problem.

First, let's remember what we're doing.  We're adding 26 to itself 54 times.  So the easiest way to think of it is to first work with 25 instead of 26 (we'll go back to the leftovers at the end).  So now we're adding 25 to itself 54 times.  And to make 54 easier to work with, we break that down into 50 + 4.

So first we start with the easy stuff:

25 x 10 = 250.

Now, since 10 x 5 = 50, we have to take that 250 and multiply it by 5:

200 x 5 = 1000
50 x 5 = 250

And now the 4:

25 x 4 = 100

So now we add all those neat round numbers together:

1000 + 250 + 100 = 1350

Now, back to the leftovers.  We've done 25 x 54.  So 26 x 54 would just be adding another 54 to what we've already got:

1350 + 54 = 1404

Sure, it's a few more steps.  But those steps are logical.  They make sense, and it's clear why what's being done is being done.  That's the biggest problem with the Old Way; even if it works consistently, you can't really explain the logic behind it (you have to borrow because you can't subtract 8 from 7, but apparently you can borrow 1 from 0 and get 9...somehow).  And if you don't know the logic, it's far easier to miss a step or get the order of steps wrong, and far harder to realize when you've screwed up.  And that's especially bad in math, because like all hard sciences, math functions purely on logic.

So if common core is so great, why are so many schools having trouble with it?

Because even the best teaching methods are going to fail miserably if the execution sucks.  And in this case?  Holy shit does it ever suck.  The point of common core is to simplify these lessons, so adding in counterintuitive steps just creates more confusion.  But that's where you need to come up with better lesson plans that use this method, not pitch the baby out with the bathwater because Everything New Is Bad.

Really, the idea behind common core math is that as adults, we have learned better ways of doing math than what we were taught originally.  So why on earth are we still teaching our children these same cumbersome, nonsense methods that we don't even use anymore?  Why not cut out the floundering and just teach them the better way to start with?

It's taken me way longer than it ever should have to figure out that I don't actually suck at math.  I never did.  I sucked at learning math, because it was being taught to me using faulty, kludged logic.  Using bad logic to teach any kind of science is like using water from the toilet bowl to make your coffee; even the most perfect roast in the world is going to taste like shit if shit is what you start with.