We can be our own kids' best teachers, even when it comes to gnarly division problems. In this episode Pam and Kim relay their experiences exploring decimal division with their own kids.
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Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris.
And I'm Kim Montague.
And you found a place where math is not about memorizing and mimicking, where you're waiting to be told or shown what to do. But it's about making sense of problems, where we notice patterns and we reason using mathematical relationships. Ya'll, we can mentor students to think and reason like mathematicians do. Not only are algorithms really not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be. Really not particularly helpful. Particularly not really helpful. That's...
I didn't laugh until just now. Oh, my goodness, how are you doing, Pam?
There was a few more. Too too many "ly" words in that sentence. Oh, well. I'm doing fantastic. How are you?
I'm good. Hey, so you know, I'm loving the reviews, which is super, super fun to read. And I wanted to share one with you that we got about a month ago. And this one is from gypsytango. And gypsytango, I don't know if it's a he or she, but they said, "I'm a parent helping my child learn math." Which automatically I love. "I'm learning a lot from the techniques and perspectives discussed. Pam and Kim are a great team and funny too. Thank you for helping me become a PT math teacher for my son." I'm assuming "parent math teacher for my son." "Great podcast!"
I love that because we are our kids first teachers, right? We love parents who get to listen in. So, thank you, gypsytango, for listening and for diving in to do math with your son.
(unclear) a great experience.
Speaking of sons, you know, one of my favorite things to do is talk math with my kids. I'm super fascinated by how they think and what they're learning about. And so (unclear). Oh, I didn't know.
Before you go on, we probably should shout out because you just quoted Christopher Danielson "Talk math with your kids". (unclear) Oh, okay.
Thank you for stopping me. Good.
Yeah. He has a sticker and everything. What is it? "Talk Math With (unclear)".
Oh, yes, yes, yes. I do know what you're talking about. Yep.
Yeah. Yeah. So, and he's fantastic and fun. And hey, another shout out to Christopher Danielson. I went to a thing he did. Where was that? Oh, yeah. It was in New York City. I was in New York City, and I decided to go to the Museum of Math. Or Math Museum? MoMath. Whatever MoMath stands for. And, randomly, he was doing a presentation. He's from Minnesota. I'm from Texas, or at least living in Texas.
And the two of us met up in New York City while he was doing his presentation at MoMath. And I just... The reason I have to say it is, he was doing this thing with tiling shapes. And I remember thinking, "This is fun. Whatever." My son showed me the other day this whole new proof that just came out with this Einstein shape about tiling the plane with this shape. And anyway, I was just... To Christopher Danielson, I had to smile. I had to smile big because I was like, "He is onto something with this tiling thing." Anyway, so there you go. Way to go.
It was really fun. Speaking about "Talking math with your kids," It was me and my son. Yeah, mmhm.
(unclear) That's super cool. Well, and I know that when your kids came home, and you said, "How did school go?" You really meant, "How did math go?" And I kind of feel the same way. So, I try to ask about other subjects, but they're probably knowing that I'm just waiting to hear about what they're doing in math. So, anyway, I'm super fascinated. And, you know, this past summer, my kids had some downtime. They're not quite old enough to get a job, but they have too much time on their hands. So, I, you know, have a younger son who... You know, sometimes the short term memory is not amazing. And so, I wanted to keep things fresh with him, right? And so, we talk math all the time. But this past summer, I said, "Hey, we're going to spend some time this summer doing some math." And so, I did some Problem Strings with them. And we just talk about math in general. But I also decided that for the first time ever, I was going to sign up for an online site. And it really didn't matter which one it was, but I wanted to just have a bank of problems that we could bounce around, and we could think of different things. You know, because I'm working and...
You really just didn't want to come up with your own problems.
Exactly. That is exactly, right.
(unclear) generate some problems for me. So, you could have used ChatGPT, but you decided to whatever. (unclear)
Yeah, yeah, yeah. And so, you know, I sat next to him. Which I think is the crucial part of any of these online programs is that we send kids away and like go do some stuff. But I think what the most important thing you can do in that moment is sit and share and listen, and so that's what we did.
Hey, I'm going to pause you for just a second.
I think somebody might hear that the most important thing you did was sit next to your kid to like enforce that he's doing it.
Oh! Thank you. No, no, no.
(unclear) It's not about like making sure it happens. "Dude, you better get this."
Oh, no, no.
That's not what you meant.
meant I want to have a problem come up and say, "How are you thinking about it?" and listen to what he's saying and how he wants to approach the problem. And it just provides an opportunity where I don't have to come up with every problem for us to have a rich conversation. No, I
So, he was working on some problems one day, and I was sitting next to him. And a particular problem came up. And so, I'm going to give the problem because I want our listeners to think about the problem first before we talk about it. And the problem was 8099.73 divided by 27.
Okay, so I'm writing that down, and I just wrote down 8099.73.
Divided by 27.
Divided by 27. So, I understood 8099.73 correctly.
Okay, good, good.
So, if you're listening, you should pause the podcast and spend some time thinking about that. But we're just going to dive in.
That's an ugly looking problem.
It is. It is pretty ugly. And so, my son Cooper saw the problem, and he literally said, "No way." And he kind of (unclear) back, and I said, "Coop, what do we do if a problem feels maybe a little bit hard?" And it's a division problem, right? And he was like, "Oh. Hmm." He's like, "I can do a ratio table." And I said, "Go. Let me see what you've got." And so, I'm going to tell you what Cooper did.
And he wrote down the ratio table, 1 to 27. He was thinking about how many 27s were in 8099.73. So, he (unclear).
Quotative approach. Mmhm.
So, he wrote 1 to 27.
And then, he wrote 100. Which was 2,700. I need to grab a piece of paper. 100 to 2,700. And then, he wrote 200. Which was 5,400.
And then, he went 300. Which was 8,100.
And so, he had 300 and 8,100. And then, I asked him, "Where are we? How far away are we from where you're headed?" Because he knew he was heading to 8,099.73. So, he got the biggest grin on his face. And I was like, you know, "What are you smiling about?" And he said, Oh, we're only 0.27 away." And so, he looked at the 1 to 27.
That's so nice, right?! (unclear).
I know. It's a pretty yucky looking problem, right?
Yeah. So, 8,099 and some stuff is just a little bit away from 8,100. Yeah, that's really nice.
Yeah. So, at this point, he looked at the 1 to 27, and he grinned really big. And he said, "I know it's going to be 299 and called it... So, 299. Let me see what he wrote. He wrote...I don't have paper... 299, which would put them at 8,073. Is that right?
And then, we had a little bit of a place value conversation, because he knew he was supposed to do something with that. 27 or that 0.27. And so, he did tinker around a little bit at that point about, is it one away? Is it a tenth away? Is it a hundredth away?But I was so pleased with him that he A, like had a model, a tool that he can think with. And he was like, "I'm going to do this. It's going to be great." And then, he was using some relationships that he knew.
And then, we just tinkered around a little bit with place value, and he ended up with the correct answer. Yeah.
so, even when he got to that 300, the ratio of 300 to 8,100, and then could figure it's going to be a little less than that. He's already... The answer is already between 299 and 300. And Yeah. That was nice.
So, then, if I remember correctly, I called you. And I was like, "Hey, Pam, what do you think of this problem?" Because that's what we do, right?
Do you remember what you did?
I think so. Because I think I made a place value mistake off the bat. And so, today, when I knew we were going to record this, I went ahead and solved it fresh.
Okay. Well, tell me what you did today.
I don't know that I remember exactly how I did the place value wrong last time.
Alright, so I also made a ratio table, and I was thinking about the ratio of 1 to 27. And I looked at that 8,099, and I was like, "That's almost 8,100." It's so close to 8,100. And I know there's a relationship between 27 and 81. So, that 8,100 felt like 8,100. And so, I think it has to do with exponents really. I think I've done 3^4. So, 3^2 is 9. 3^3 is 9 times 3, which is 27. So, 3^4 is 27 times 3, which is 81. And there's that nice connection where that 3^4 is the same thing as 3^2 times 3^2. Or 9 times 9, which is 81. Anyway, there's just all these connections now that I kind of have in my head. I said, "I could get from 27 to 81." And so,on my ratio table, I've got 1 to 27. 3 to 81. And then, 30 to 810. And then, 300 to 8,100.
So, Cooper did kind of a little bit of a different relationship to get up to 8,100. I went to 3 to 30 to 300. And then, I had to think, "How is 8,099.73 related to 8,100?" And I literally, then, wrote down 0.27. Because it's just. I played I Have, You Need in my head. I said, If I have 0.73, what do I need to make 1? I need 0.27." And then, I had to ask myself place value. Like, Cooper and I were still. We were both messing with place value. And so, on my ratio table, I've got that 1 to 27. And then, you know, all the other stuff I said. And then, 0.27. So, I'm asking myself, "How do I get from 27 to 0.27?" And, Kim, you know what I did? I feel like I did what my mom does. Next to that I wrote... I said, "I'm going to go from 27 to 2.7. So, 27 to 2.7 is a tenth. So, I divided both by 10. So, 1/10 is to 2.7. But I know I need to get to 0.27 or 27/100. So, then, I divided by 10 again. And so, now, 0.27 is in the same ratio as 0.01 or 1/100.
Okay. So, now, I know that I just need to subtract that 0.27, the 27/100, from the 8,100. And so, I'm going to subtract 1/100, or 0.01, from the 300. And, 300 subtract 0.01 is 299.99. Or 299 and 99/100
Yep. What I love is that you both used a ratio table to keep track of your thinking, right? (unclear)
(unclear) what we know. Mmhm.
And what I wished that I had done when I was working with Cooper is once he called it 299. And then, he was like, "Wait, no, that's not right." He was talking aloud at that point. And I didn't say, "Hey, how about you put those on ratio table?" Because I think that had I encouraged him to do that, I think he would have done the same thing you did, scale down by 10. And he would have been able to hang on to it a little bit. And so, as I'm hearing you talk about, I'm kicking myself a little bit because in that moment, we just started talking, talking, talking. And I think that's a tip for all of us as we're working with our students, both as teachers and parents, that sometimes kids want to hold things in their head a little bit. Maybe because they think it's desirable. I don't know.
Or we're all a little lazy.
In some of my kids case, a little bit lazy.
(laughs). I think we all are a little lazy. Sure. Sure. I mean, I'm messing with a kid... Can I say that?
I was doing math with a kid at church on Sunday. And I asked him a question. And I think it was just I haven't done any math with this kid for a while. And I was like, "You're older. Let me think." And he's not very old. I think this kid is probably 8,9. 9? 10?9 or 10. And so, I said something like, "What's 39 plus 10?" And she said, "49." And I said, "What's 39 plus 9?" And she had to think, think think. And she said, "38". And I said, "What's 47 plus 10?" She said, "57." And I said, "What's 57 plus 9?" And at that moment, I could see her trying to like hang on to numbers. And I said to myself, "Oh, this would be so much better if I could just like just draw a number line and just write that 48 plus 10." Then, I wouldn't have to write anything else. Or maybe just write the 58 down, you know? And then, "Well, how about plus 9?" And she would be like, "Oh, it's just 1 less than that." And then, she wouldn't have to be... When she says, "It's 1 less than that," what she doesn't have to do is then say, "What was the problem before that?"
She can just see it, right? It's not about how much you can hold in your head. It's about what are the relationships you're using? It's totally cool to keep track. Yeah.
Yeah, yeah. So, Cooper ended up with the wrong answer at first. I mean, it was not correct. But what I found in that moment really intriguing, interesting, exciting was that he saw that division problem, and when I just was there to say, "You have a way in," then he was able to relate multiplication and division. We saw him scaling proportionally. And his answer was reasonable. Even though it was incorrect in the beginning, it was reasonable rather than way off. Which...
...if you're having kids do the long division algorithm, this is awful. This particular problem would be awful.
Kim, I have to tell you. Just as you were saying that, I was writing down the long division algorithm. Like, just just the house top. I didn't start doing it or anything. Because I was curious. I was like, "I wonder how bad this would be with the algorithm?" And instantly, I'm like, "27 goes into 8. Eh. 27 goes into 80. Ah, I'm sure it does."
Yeah, yeah, yeah.
And then, after that, you're going to have all these 9s, right? Just over and over. Oh, that's terrible in the long division algorithm.
Anyway. So, I have one other story if we have time for it.
So, you know, I was talking about Cooper and division. And we had some division problems in MathStratChat not too terribly long ago. It comes out in the evening, so sometimes I just kind of like take my phone and shove it in my kids faces. Because they're excited to see it, but they don't know when it comes out. They pay no attention. And so, the MathStratChat problem was 1,176 divided by 48. And I don't know if any listeners saw that problem when it came out. But 1,176 divided by 48. And I walked up to Luke, and I kind of shoved it at him. And his response was, "Mom, I'm tired." And I literally looked at him and said, "Are you serious? How close is it?" And he was like, "Oh. 1,200, so 25. And half, so 24.5." And I was like, "Wait, what?" He was like, "You told me to think about it." And so, he literally... As soon as somebody said, "What's close?" Which is the same question that I had asked Cooper. He needed just a little bit of a nudge to basically say like, "What do you know? What's reasonable?" And he was able to think about 1,200 divided by 48, which he knew was 25 because 2,400 divided by 48 would be 50. So, 1,200 divided by 48 would be 25.
Okay. Cool, cool. MathStratChat!
But it was a little bit too much.
The 1,200 is a little bit more than the 1,176. Mmhm.
It was 24 too much, and so 24 divided by 48 was just a half. So, in that moment, I was like noted. Sometimes even older students look at a division problem. And they're like, "Uh, that's yucky. I don't want to engage," maybe. And so, you know, for my kid, I basically said, "Are you serious?" I wouldn't recommend that with your students. But if we can give them opportunities to think about problems, and nudge them to think about what they know. Yeah, that can be super, super useful.
That's so awesome You know, I do think there are students. Know your content know your kids. When you know your students well, I think there are students who rise to that challenge. You know, where you can look at me an go, "What you got?" You know, "Come on, dude." Like, "Buck up." Like, "Bring it. Bring it. Bring it on."
You know, I do the same kind of trash talk when I'm playing basketball with kids. You know like, "You think you can shoot from out there? Do it." You know, "Pump that thing." For this problem, "Come on, what do you got?" Like, "Bring it." Bring what you've got to this problem, and let's have some fun. That's really cool. I love how you did that. So interesting that you're just like, "Are you serious?"
Yep. So, we actually then talked and I said, "Hey, about you write a Problem String to help build these relationships? And, of course, that's what you do. And so, what is a string... I haven't seen it yet. What is a string that you can do to build these relationships with kids?
Alright, so off the press. Here we go. Ready? What if I were to say, Hey, Kim, I've got a bag of Skittles that have 22 Skittles in it.
And so, that bag, and I just wrote down because I'm going to keep track too. That bag of Skittles. 1 bag, 22. What if I had 66 Skittles? How many bags would I have?
I say that that's 3 bags.
Okay. Because 3 times 22.
Because if 1 bag has 22, then I have 3 times as many Skittles, so 3 times as many bags.
Okay, cool. What if I had 660 Skittles? How many bags would we have?
I would have 30 bags because 660 is 10 times as many Skittles than the 66. And so, 10 times as many bags than the previous 3 bags.
Alright so, 30 bags for 660 Skittles. This is a little weird. But what if I only had 6.6 Skittles?
You'd be sad.
No, I don't really like Skittles, so I'm okay.
I'd be sad.
You'd be sad.
Okay, so on my ratio table I am writing divided by 10 From 66 to 6.6. So, I'm also writing divided by 10 to get from 3 to 0.3. 0.3 bags.
So, point 3 bags of 6.6 Skittles. That's interesting because on my ratio table, I had written down your 30 to 660. But you didn't go from that. You went back up to the 66 and divided by 10. I like that. It's a nice move. It's nice using what you know. What if we had 666.6 Skittles? How many bags?
Goodness. Okay. 666. I'm going to say... Hmm. 666.
Oh, 0.6. Okay. So, I've got 660. I don't... Yeah, I have 660 for the 30 bags,. And I've got 6.6 Skittles for the 0.3 bags. So, I'm going to put those together and get 33.3 bags.
That's a lot of 6s. Okay.
It's a lot of 6s.
I'm going to have some more here for a second.
What if we only had 0.66 of a Skittle? I know, sorry. I probably should have done something that was less discreet. (unclear)
Good gravy! That's okay. I'll just nibble. Nibble on one lone tiny Skittle.
Two-thirds of a Skittle almost.
0.66 of a Skittle. How many bags would you have?
So, if 0.3 bags is 6.6 Skittles, then 0.03 of a bag is 0.66 of a Skittle. So, I'm dividing by 10 again.
Cool. So, 0.03 bags has 0.66 Skittles.
Nice. Next problem. What if I had 660.66. 660 and 66/100.
I'm going to go back to the 660 Skittles, which was 30 bags, and I'm going to add the 0.66, which was three-hundredths of a bag. So, that's 30 and 3/100 bags of Skittles.
So, 30.03 bags of Skittles, for 660.66 Skittles
(unclear) are recording, have already solved.
Yeah, I hope you're not driving.
We should have probably said don't be driving while you're trying to picture this. There's way too many 6s and 3s. All the things. Last problem. Last problem.
How about 659.34 Skittles?
How many bags?
That is really close to 660. It's 66/100 away. So, I'm going to go with 660 Skittles, which is 30 bags. And I'm going to subtract the 0.66 Skittles, which is three-hundredths of of a bag. So, I'm calling that 29 bags, full bags, and 97/100s of another bag. So, 29.97
Bam! And so, you just solved the problem 659.34 divided by 22. So, it was a little similar, right? We were like, "How can we come up with a Problem String that can help kids build relationships, so that they could solve a problem like the one we started with?" Well, so if the problem was 659.34 divided by 22, bam, we just kind of built some relationships that could help you kind of think through not only that problem but several problems, right?
Yeah, well, and I want to point out that, you know, sometimes people will see you do some work on a ratio table, where you do several problems to just go in and out of different relationships. And then, you get to the last problem, and they're like, "Yeah, you had to do all those problems." And I would argue that you didn't. I just sketched out 1 to 22. And if I knew I was heading for 659.34, then I would have a significantly shorter ratio table that would just head me towards where I was going. I wouldn't need all those interim pieces. But it's fun to play with the relationships, kind of in and out of the problems you gave me.
Yeah. And the reason for doing a Problem String is not so that you can answer the last problem.
It's so that you build mental relationships in your head, so that you can answer problems like all of the problems in the string, including the last problem.
Yeah. It's not just a series of, "Here are the steps to get to one problem." No, it's like, "Let's build these relationships, so now you own these kinds of relationship, so that when I give you any of the problems in that string or the problem like we started with, then you own those relationships. You can use those to solve future problems." Kim, I've been thinking a lot about the fact that teachers will say to us, "But, Pam." Like, "My kids own these relationships. They can solve some problems. But what if we give them this ugly problem? Like, the problem you just solved was super. You know, it came out really nice." I'm like, "Nice? You call that nice?" I mean, the answer was 29.97. But it wasn't, you know... I guess it could be uglier. It wasn't repeating. It wasn't nonterminating. Like, there were lots of... But this one's not particularly nice. Sure, there are uglier problems. But the bigger point I kind of want to just bring up here is, I think there's a culture that I'm going to push back against. And I think there's a culture that says, "We must teach students. The end goal is that students can solve any problem of a type of problem. No matter how gnarly. No matter how ugly. No matter how obscure that you'll never hit in life." Whatever. Because what if? You know, we got to teach the kids a way because they might hit this problem on a test, in life, or whatever. To which I would say, "Maybe. Maybe that was true before technology. Maybe." I'll grant you that if you were going to be an engineer in slide rule times, you probably did need to be able to solve any of the gnarly problems. But not anymore.
What we need now... And I might even argue what we've really needed in all of history. But what we for sure need now is students who own the major relationships. Because that means their brain is thinking in a certain way. Their brain is literally thinking more sophisticatedly about relationships. And we can now build more mathematics on that. They are more confident in the math they do. They're using relationships that bleed into a lots of other areas in their lives, where they are now thinking and reasoning like a mathematician. And when they then hit a problem that they maybe don't own some obscure relationship to be able to solve that particular problem? No problem. They can throw it in technology because they know what they're doing. They're confident that they're putting the right problem in technology. And the calculator, computer, ChatGPT, AI can just throw back that obscure. But we've already gotten close, right? Like you said earlier. Cooper's answer was reasonable already. That is so much more important. So, I want to push back against this idea that we have in our culture that when I teach division, boy, I better make sure kids can answer any division problem. No, no, no. Not true. We need them to own the major relationships and be thinking about division. That is so much far more important.
Does that make sense? So much far more? (unclear).
(unclear). It can be.
Okay. It is today.
It's your podcast. You can say whatever you want to say. So, today, we represented our thinking on a ratio table. If those are new to you, or you want to hear more about ratio tables, we have a series starting at episode 129 that you can tune into. If you're also curious about a podcast that we called, "If not the division algorithm, then what?" that is episode 98.
Yeah, check those out. And, ya'll, thank you for tuning in and teaching more and more real math! To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!