I must be a teacher at heart. I’ve been out of the profession for almost two years now, and I still can’t help sticking my nose in teaching discussions… or gripes, as it were.
By now, you’ve probably heard of Common Core, the new education standards that people all over the country are hating on. Here’s a pic that’s been making the rounds of social media, as an example of how baffling common core math is:
“Look at how simple the old way of doing subtraction is,” the caption invariably reads. “And compare that to the ridiculous new way. Does anyone even understand what’s going on?”
Well, yes, actually. I totally understand, and here’s a secret:
This new way is exactly how you handle subtraction in your head, on an intuitive level. You just don’t realize it.
The reason the new way looks silly is that they’re using a simple subtraction problem. It’s easy to solve in your head, so you don’t need to use the actual process they’re trying to teach. But hey, that’s a good way to teach. You use simple examples to illustrate the underlying process, and then once the student understands it, you have the student apply the process to more difficult problems.
Imagine you’re learning to juggle. You’d probably start by throwing and catching a single tennis ball twenty thousand times. Yeah, you’d feel pretty dumb doing that, but isn’t that what you need to do to someday be able to keep six flaming chainsaws in the air at the same time?
That’s what this example problem is. It’s starting you with a single ball, instead of throwing a bunch of chainsaws at you. You should be glad for that.
So let’s skip ahead to an actual problem where you would need to use this new subtraction process. Let’s say I ask you to subtract 1,789 from 3,024, and I make you do the problem in your head. You can’t write anything down. What would you do?
Well, many of you would probably start by doing this:
But assuming you get past that, what would you do next?
In your head, you’d probably go through a process similar to this:
- Okay, I need to subtract 1,789 from 3,024.
- Well, this is way too complicated to do in my head, so I’ll break the problem down into chunks.
- So, the difference between 1,789 and 1,800 is 11.
- The difference between 1,800 and 2,000 is 200.
- The difference between 2,000 and 3,000 is 1,000.
- The difference between 3,000 and 3,024 is 24.
- So, to get from 1,789 to 3,024, I add up all these differences: 11 + 200 + 1,000 + 24, which is 1,235.
- Therefore, 3,024 minus 1,789 equals 1,235.
Does that make more sense now? The problem is too complicated to perform in your head, so you break it down into a bunch of simpler subtraction problems, and then you add up all the individual pieces. If a maniacal mugger cornered you in a dark alley and pointed a gun at your head and demanded that you hand over the difference between $3,024 and $1,789, this is the exact process you’d have to follow. You know, assuming you’t not Rain Man.
Here’s what it comes down to: Subtraction is ridiculously difficult to perform in your head. Addition is way easier. So if we can convert a subtraction problem into an addition problem, then we turn an impossible process into one that’s entirely doable. We’re taking an ingrained algorithm — the “old fashion way” [sic] — that we’ve been doing wrong for, what, centuries (?), and we’re illustrating the way our brains will naturally process it instead.
I think that counts for something. And that’s why I’m not ready to give up on Common Core yet.
Addendum, 3/31/15: To clarify, I’m not saying that my example is the only way to solve the problem. I’m saying that we’re breaking down a complex problem into simpler steps. Yes, there are many ways to break it down, and ultimately, what matters is that we understand the entire process, not just memorize an algorithm. That’s what Common Core teaches.
Dennis Hong's Attic 
That’s a better explanation than I’ve seen anywhere else, but it still doesn’t jive with the way I (and I assumed, everyone else) do subtraction mentally:
* Okay, I need to subtract 1,789 from 3,024.
* Well, this is way too complicated to do in my head, so I’ll break the problem down into chunks.
* 3,024 minus 1,000 is 2,024. That leaves 2,024 minus 789.
* 2,024 minus 700 is 1,324. That leaves 1,324 minus 89.
* 1,324 minus 80 is 1,244. That leaves 1,244 minus 9.
* 1,244 minus 9 is 1,235.
Less guessing and fewer steps.
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Fair enough. To me, your way and my way are both pretty much the same, actually. It’s breaking down the process into individual chunks that can be calculated in your head.
I guess I should have clarified that my specific way wasn’t the *only* way, but what I was illustrating is the process of breaking down a complicated problem into simpler problems. My bad.
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My thinking was:
– 3000 – 1,800 = 1200
– 1800 – 1789 = 11
– 3024 – 3000 = 24
– 11 + 24 = 35
– 35 + 1200 = 1235
On a different day I might have done a different order. There are a lot of different ways to get there, but some process like this is always how I do math. And I think how most people who are “good” at math do it. Its also much more closely aligned to how I do algebra and calculus. Find subproblems that are easy to solve and break them down.
I think actually teaching this is great.
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Yep, exactly. Take a complex problem, and break it down into simpler steps. And understanding how to break it down gives us a deeper understanding of math.
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Reminds me of how I convert pounds into kilograms using a multiplier of 2.2 (it’s actually closer to 2.205, but I don’t care about that level of accuracy). I double the pounds, then take 10% of that new number and add it. So 90lbs is 180+(180*.1)=198.
As far as Chris’s example, I’d quickly know than I need 1,211 (due to the fact I learned how to subtract properly on paper and know how carryovers work, so it’s easy to reverse that process for addition) to get to 3,000, then add the other 24 for 1,235.
Do the kids not need to show work in this new way of teaching? “I did it in my head like I’m supposed to.” 😉
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Haha, yeah. That was a problem I always had in grade school. I’d do stuff in my head, but not be able to articulate how I did it.So, I guess this newfangled stuff is good in that respect. 🙂
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“I need to subtract 1,789 from 3,024.”
The thinking process you describe is nothing like how I do the math in my head. Just goes to show that brains are diverse and we all learn and process information differently. I think having a common core across the nation is good — I’ve long wanted more standardization for educational baselines. However, the “show your work” part being so regimented is going to (I think) leave a lot of students behind — students whose brains just don’t work that way. When I took in my brother’s twin teen nieces and had to tutor them in math, I got a hard lesson in just how different brains are. Things that made perfect sense to me made no sense to them. Things that made sense to one twin made no sense to the other. Having the flexibility to find and use learning systems that work for each kid is important, and I think that’s the danger of the way they’re approaching common core. Having standardized goals isn’t the same as having standardized kids.
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I understand what you’re saying, but just to clarify, I don’t claim that my example is the correct or only way to solve the problem. The point is that we’re breaking down a complex problem into simpler components, and it’s important to understand how we’re breaking it down. I’d argue that it’s also important to understand how other people might break it down, as well.
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My problem with this is that it has NOTHING to do with Common Core. Why do people want to blame or -in this case- credit Common Core for the teaching of strategies in Math?! When I was 16 (I’m 41) I worked the register at Burger King. When they broke down, my boss taught me to “count up” to make change quickly. Common Core was definitely not around then. It’s called common sense.
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