Socratic educator Michael Strong with six year old Alana
Michael Strong asks six-year-old Alana how she thinks and solves math problems. She describes thinking as building brain muscle, counts down mentally for subtraction, and uses her fingers to track numbers. The conversation shows how she moves from using external tools to recognizing patterns on her own.
Chapters
Good, how are you?
I'm well, where are you?
I'm in the car right now.
Okay, it's kind of dark, it's darker than usual.
Yeah.
So, oh, there you go, brilliant, brilliant. Um, so where are you going?
I think I'm going to breakfast at the golf course.
That sounds nice. And do you know what you're going to do after that?
No, I don't.
Okay, so I'm wondering, what do you think that thinking is? What is thinking?
Thinking or thinking?
Thinking.
Thinking. It's like probably the process.
I missed you, you froze. You said a process and I missed what you said after that.
I think it's probably like a process of giving your brain more muscle and helping it exercise.
I see. So when I ask you a question, you come up with an answer. Are you saying that's kind of like giving your brain more muscle somehow?
Yes, I think so.
Does it feel like your brain is getting exercise?
Yes, because it gives me more like ideas and people give me more feedback and my brain thinks of, yeah, that's right.
Okay, let's play around with some different kinds of thinking to see how they feel, okay? Okay. So let's do, let's start with math because in a way math is simple. You know basic— let's see, I always forget where you are in math, but let's say you know what 10 10 is, right?
Yeah.
What is 10 10?
20.
Do you know what 20 20 is?
40.
All right. Um, those— these sound so easy, it feels like you don't even need to think about that, is correct?
No.
So you just— mind— your mind goes instantly, you know, 20 plus 20 is 40.
Yep, it's because 2 plus 2 is 4.
Excellent. Um, how about 100 plus 100?
200.
Ah, how about 200 plus 200?
400.
How about 1,000 plus 1,000? 2,000. Sounds like you know this stuff so fast you don't even need to think. So I need to give you something trickier so that your mind has to slow down and think. How about, let's see, do you know, how about 20 20 20?
20 20 20? 60.
That was pretty fast too. How about 24 minus 12.
24 minus 12?
Um, 12. Whoa, good job! And I noticed that that time you did have to think, right? Yeah, the other ones your mind just almost knew it automatically. I think that your mind knew the patterns of like 10 10 and 100 100. We had talked about patterns before. Do you remember talking about patterns?
Yes.
And so it sounds like certain mathematical patterns you know, and so you did not need to think. But with 24 12, you had to stop and think. Can you explain how you figured out that 24 12 is 12? What went through your mind when you did that?
First, I used subtraction in my head, and I also used my hands to subtract. I took away my fingers.
Did you do all 12? Did you kind of count from 24? It's hard because you don't have 24 fingers. So how did— how exactly did you use your fingers if you don't have 24 fingers?
I did 24. Without any fingers, then I did 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12.
Ah, you use your 10 fingers and then 2 more.
Mm-hmm.
Okay, and so what's the relationship between thinking in your head and using your fingers? Are your fingers doing the thinking, or is your brain doing the thinking, or how does that work?
So my brain is figuring out what, which number goes next so that I can reveal the answer, but my fingers are doing the same to keep track of them, I think.
Ah, the keeping track piece. Because it sounds like, do you think you could do it without your fingers or you pretty much have to use your fingers?
You can do it without.
You can. So can we— I give you one with where you do— where you don't use your fingers?
Yep.
Okay, we'll start with— how about 28 minus 4? What's 28 minus 4?
That's 24.
That was pretty easy. You didn't need to— you didn't need to think very long nor use your fingers, right?
No.
Okay, how about 36 minus 8.
36 minus 8. Maybe I don't know that one.
Ah, is it hard without your fingers?
Yeah, it's hard.
And do you think would be easy with your fingers?
Yeah, it might be easy with my fingers.
And so I want to stop and see how our fingers help us. You said, um, you said I think that your fingers help you keep track. Is that how it works?
Yes.
So let's do a littler one. Let's see, what's 36 minus 1? Can you do that without your fingers?
36 minus 1 is 35.
Okay, that's easy without your fingers. How about 36 minus 2?
36 minus 2 is 34.
Is that pretty easy without your fingers?
Yep.
How about 36 minus 3?
36 minus 3 is 33.
You got that one right away without your fingers. Why don't you use your— why don't you need your fingers anymore?
It's because, um, it's just going down and down and down, so like 36 minus 1, 35. 36 minus 2, 34. 36 minus 3 is 33. But also on 33, it's because 3 plus 3 is 6, and if you add— and if you subtract 3 more, it's 3.
It sounds like you're using patterns again. You know what I mean by mathematical patterns? Yep. So going down 1 is a kind of pattern, and then the 3 3 6 is a kind of pattern. And it sounds like if you have patterns, then maybe you don't need to use your fingers anymore. That sound right?
Yes.
How about what's 36 6 without your fingers?
36 6? 30.
There you go. And you did that without your fingers, right? How did you do it? What was going on in your mind when you were thinking about that?
When I was thinking about that, I realized that, um, 6 minus 6 is just, it's just like 1 or something.
Minus 6, go ahead.
So, um, it might— so I thought it might be 30 because, um, because 6 minus 6 equals, um, 6 minus 6 equals probably 1 or 2 or 0.
It's 0, it's 0. Go ahead.
And then, and then when we add it like that, it, it And 36 plus 6, I mean, um, subtracting 6, it'll be 30.
So it sounds like you were figuring out a pattern because, yes, 6 minus 6 is 0, and you saw that 36 minus 6 is probably like the similar, it's a similar pattern, and it sounds like that's how you figured it out without fingers. Is that accurate?
Yes.
So is— would you say in general subtraction is hard?
Maybe a little bit hard.
You think addition is easier than subtraction?
Addition, that you mean like adding?
Yeah.
Yeah, probably easier.
At school, do you mostly addition or do you just do some subtraction at school?
At school we don't At school, I think we do probably equal both.
So do you do big numbers like 36 minus 30? I mean, 36 minus 6 at school?
Yeah, some big numbers, some small numbers.
At school, do you write it out? Do you use your— do you use pencil to figure it out?
We do it in our journals.
Is that— does that help using pencil?
Yes.
Okay, so in general, one of the things that I'm going back to thinking, it seems like our minds can hold some things well and some things not so well. So, you know, with pencil, one of the reasons I like doing math with pencil is that helps me keep track, just like fingers can help keep track. Does that make sense?
Yes.
But I would also say knowing patterns in math helps us keep track. Because a pattern is easy to remember. Like, imagine if you had to use your fingers to do 1,000 plus 1,000. There's no way you could do that with your fingers, I think. So do you see how patterns help our minds figure stuff out, even when, um, we couldn't do it with our fingers?
Yes.
So, um, I, I am interested in how you figure things out, and so I wanted to explore thinking and how you think about different things today. Do you feel like you are a little bit more aware of how you think?
Well, maybe a little bit more.
Does it do— when we talk about mathematical patterns, does that make sense?
Yes.
So I'm going to do one more time without fingers if you can, Now can you figure out what 36 minus 8 is?
28.
Whoa! How did you do that? Tell me how you did that.
So, um, if 38 plus, I mean subtracting 8 if that's what you said, it might be 28 because 8 minus 8 is, um, probably 0 or 1.
8 minus 8 is 0, that's correct. But how do you get from 36 minus 8 is 28? How did you get, because it wasn't 8 minus 8, it was 36 minus 8.
36 minus 8.
Yeah, how did that— how did you figure out that the correct answer was 28?
Well, mostly because I use my thinking to take some numbers away to reveal the answer.
Did you do it kind of one by one? Did you go 36, 35, like that, or did you use some other way?
I did it like that.
Oh, and so without using your fingers, you kept track of everything in your head? Mm-hmm. Wow, that's a lot to keep track of 8 different, kind of 8 numbers going down in your head. Another way I thought you might do it is if you knew that 36 minus 6 is 30. Remember that one?
Yes.
8 is 2 more than 6, right? Mm-hmm, it is. So if you knew that 36 minus 6 is 30, I thought maybe you'd figure out that, okay, just 2 more down and we get to 28. Does that make sense? Yes. So I'm trying to get you lots of different ways to use mathematical patterns in your head. That seem possible?
Mm-hmm, yes.
Do you like thinking about math?
I I love math, so yeah, I like to think about math.
Even when it's hard?
Mm-hmm, because that's how I learn.
Absolutely. No, I think it's really fun to think about math, and I think it's fun to think about how we figure math out too. So I'm gonna let you guys have a fabulous time for the rest of the day, and hopefully we see you next week, okay?
Okay.
Bye-bye. Bye.
