Sebastian: Daddy, do you want to know what happened in school today?
Me: Of course I do.
Sebastian: My teacher was subtracting and said, we can't do 2 minus 8. So I raised my hand and said, "yes we can; it's negative 6." My teacher said, "you're ridiculously smart."*
I made a quick decision to ignore the fixed-mindset compliment and instead focus on the positive by focusing on the negative (so to speak):
How did you know 2 minus 8 was negative 6?
Not the best explanation, but the thinking is there.
At this point, you're likely thinking that I'm a math teacher, so I'm probably patting myself on the back for having taught Sebastian how to subtract in this case. I have not. About a month ago, he started asking me what negative numbers were, and I talked him through a conceptual explanation (something like counting down from three, then beyond zero). I was excited not because he was doing 'advanced math' but because he had applied his conceptual knowledge to this new situation.
So I thought to myself, I wonder what this kid can do. I remembered a story my parents told me (about asking about square numbers at an early age) and wrote this on a piece of paper:
0^2 = 0
1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = ___
10^2 = 100
...and I read it aloud:"Zero squared equals zero; one squared equals one; two squared equals four," and so on, ending with "what do you think is the value of five squared?"
When I returned a minute later, this is what the paper looked like.
1. The kid got it. Nice!
2. His explanation focused on how he knew the answer was 25, not on how he knew what "squared" means. We call this a "what-how" explanation...not nearly as strong as a "what-why" explanation.
3. He decided to go one step further and write 20^2 = 200. I love everything about this, even though it's not correct.
I asked him about the 20^2=200. How did you figure that out? He explained: If 10 squared is 100, then 20 squared is 20 times 20, which is 200. I asked how he knew and he said that 20 was twice as much as 10 so it was 200 instead of 100. I asked him how he could check it and he said, "I could count by 20s." I stayed silent. He started counting by 20s and, when he finished ("...320, 340, 360, 380, 400. Four hundred!") he said, "I don't want to cross it out, so I'll just write 200 plus 200."
My heart sang. And then things got even better when I thought: Let's see if you can do these, too. These might look like your typical exercises, but, given where Sebastian is mathematically, they're at least one step above your average "now that you've seen one solution, try five identical problems."
He tried it out. He got halfway through and stopped. "Wait," he said, "first I need to add six dots and then seven dots, so I need to add thirteen dots."
How did you get thirteen?
His response was somewhat messy. There was a lot of what-how talk thrown in there ("I need to add seven and six, so I need to add 13, so I know that four more gets us to 40, and then...") and a lot of false starts where he was trying to get to the answer while talking through his method, but his answer boiled down to this: "It was six groups of six but if I add six dots it will be six groups of seven. Then I need to add seven more dots so it will be seven groups of seven."
There are so many beautiful things about this response. The kid is seeing groups in his mind. He's understanding the nature of multiplication. He's using the concept of multiplication to make connections between two multiplication problems. He's using a pattern he discovered to determine that he even needs to multiply in the first place. He's up at 9:30 at night doing math because he (and his dad) just can't get enough.
He then spent about 10 minutes going down a rabbit hole, coming up with various answers that were not the actual value of 7 squared, partly because he was accidentally adding 13 to 30 (not 36) and partly because he was thinking really hard and his working memory was filled with a lot of things and he kept trying to decompose 13 different ways to make the addition easier.
The only helpful nudge I gave was to point to 43, point to 36, and say, "if it were 10 more than 36, what would it be?...but you added more than 10, so it should be more than that..."
At one point, I made what I consider a critical mistake. I said something like, "you're doing it right, but 36 plus 13 is not 43." The second half is fine, but the first half reinforces the idea that there is a 'right' strategy. I should have instead validated that his strategy made sense or that his strategy seemed like it was based on solid reasoning. That's OK. I like Sebastian's mistakes here, and I like my mistake too - if I had not said this, I don't know that I would have reflected on this idea.
He figured it out from there: He laughed at his handwriting-based mistake, used his tens and ones chart, and quickly got 49.
I asked him how he could check to see if he was right and he said, "I could count by sevens, but I don't know how to count by sevens." Then he counted by sevens anyway, and when he landed on 49, his face lit up.
Feeling confident, we strapped on some wax-and-feather wings and tried to see how high we could fly: 14^2. Here are some action shots:
That last picture is his sticks-and-dots representation of 14+14+14....+14 (14 times), and his tens-and-ones chart that he used to figure out what 14 tens and 56 ones added up to.
Here are a handful of brief takeaways from all of this:
1. Sebastian was able to do all of this math because he has been taught everything in a highly conceptual way. From the concept of subtraction to the concept of multiplication to the visual representation of multiplication that allowed him to find the link between 6x6 and 7x7, everything that he has been taught set him up for success here.
2. To take #2 a step further, I would argue that he was able to do math he hadn't yet learned explicitly precisely because he has not been taught a series of procedures. Teaching kids a bunch of procedures means they know those procedures but don't know how to approach procedures they have not been taught. "But his tens and ones are an example of a procedure!", you say. You're right, but this procedure is backed up by his conceptual understanding. When he learns the standard addition algorithm (with the corresponding conceptual understanding), it will be a natural extension of what he has already figured out.
3. To all you engineers out there complaining about how your child's common core math homework is far too complicated, a. in many cases, you're probably right, and b. you're looking at otherwise simple problems seemingly made complex by nonstandard strategies; it may be worth it to also check if your kids can do something you wouldn't have been able to do at their age because of their developing conceptual framework.
4. I'm grateful that Sebastian has teachers who have embraced the Common Core math standards and have committed themselves to teaching things conceptually.
5. Hearing and pushing kids' thinking is a lot of fun.
6. Kids, in general, are capable of more than we often give them credit for.
7. Notice how this "real-world" math was so engaging!
8. More than a handful.
Special thanks go to Sebastian for his hard work, to his teachers at Elm City College Prep Elementary for doing such a great job teaching him mathematical concepts, and to Joseph "Compadre" Yrigollen for his encouragement via text message while all of this was going down. Thanks also to you, loyal readers, for indulging me in a long-winded monologue about the virtues of talking math with your kids.
*The use of "you're smart", while clearly well-intentioned here, is dangerous because it discourages mathematical risk-taking, particularly in relatively high performers who want to preserve their image of being "smart", and discourages relatively low performers from trying - if another kid is just "smart" then no amount of effort is going to get me there anyway. Beware the small phrases that reinforce fixed mindsets!
"We can't do X" is troublesome because it implies that, in math, there are things we can and can't do. The less arbitrary rules we impose on math, the better. After all, even the famous "can't divide by zero" edict really just means that dividing by zero is problematic. I mean, of course you can take a pile of money and try to split it equally among no people. Of course you can take a pack of M&Ms and put them into piles of zero M&Ms. It's just problematic, which is not always a bad thing. Every case of something you "can't" do in math seems to come down to constraints anyhow.