Pam  

Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam. 

Kim  

And I'm Kim. 

Pam  

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 diving in, making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.

Kim  

Today's Halloween. 

Pam  

Whoa! 

Kim  

I know right. 

Pam  

Are you serious? 

Kim  

Yeah, so I...

Pam  

Which really means this is going to launch on Halloween.  Because today is actually not Halloween. 

Kim  

Well, listen. Yeah, it's pretty darn close. But I I have to tell you. Every Halloween, my sister-in-law will send me a picture of one of my kids when he was teeny tiny. 

Pam  

Aww.

Kim  

Maybe not even two. In my very favorite Halloween costume of all time. He was a monster. Little green monster. He had little big fluffy fat. Oh, I mean, just every time like I could squeeze the heck out of it. 

Pam  

Cute cute.

Kim  

What was your kids favorite costume? Did they trick or treat?

Pam  

My children always wanted to be something yucky. Like gory. 

Kim  

Oh. Ew, no. 

Pam  

Violent. 

Kim  

No. 

Pam  

Well, so it's interesting. Should I say this? So, as adults, they're all still... What's the word? Fascinated with the macabre. Like, I have a son that is a vampire.

Kim  

Okay.

Pam  

And I have one that's a werewolf. And I'm not even going to talk about the other two. Not that they're worse. But it's maybe I don't even know what they are. I have a daughter-in-law that's fixated on Halloween. They just... Yeah. In fact, there was talk that the baby shower theme was going to be Halloween. 

Kim  

Oh. Wow. Okay, okay.  

Pam  

Well, what's funny is Yeah.

Kim  

Alright, well...

Pam  

I thought it was a little fantastic that her mom is the one that was like, "No. We're not having a baby shower that was..."

Kim  

It's like a big deal in your family. I had no idea. 

Pam  

it's not for me.

Kim  

Oh, yeah.

Pam  

You know, is it because I... So, as little kids, I would not let them be something violent or gory. And so, it's almost like, "Don't do that, parents. Let them get it out when they're tiny." Because then when they grow up, it's like, "Mmm okay."

Kim  

Well, mine was a green and bright blue dotted, little tiny baby monster with feet. Oh, my. I'll have to send you that picture. (unclear).

Pam  

I let them be ninjas. Oh, yeah I want to see it. Your kids were super cute when they were little. Alright, Happy Halloween, everybody. Whether you like it or not.

Kim  

Yeah. Okay. Alright. So, we are on episode 176. And when you and I started talking about this and decided what we're going to chat about today, I asked... Do you remember I asked you what the number was?

Pam  

Oh, I do remember that now.

Kim  

I was really hoping it was 175. 176 is a stupid number. It is not nearly as fun. 175 is amazing.

Pam  

Did you just call it?

Kim  

I did. 

Pam  

You just called it a stupid number.

Kim  

I did. There's nothing fantastic about 175. Particularly about what we're going to talk about today. So, today...

Pam  

Because I know. why you'd like 175. Because you like quarters so much.

Kim  

I do. I do.

Pam  

Okay, so that's why. But see 176 is just one off of that super nice number.

Kim  

I know, it's just not as good. 

Pam  

Stupid. You called it stupid.

Kim  

Stupid. Well, okay. So, today, we are going to talk about place value.

Pam  

Ah.

Kim  

And I just... I don't know. I like 175 for place value.

Pam  

Well, I think we can play with 176 for place value.

Kim  

Okay, let's do it.

Pam  

So, let's play a little bit. So, alright, if place value, Kim, if I were to say 176, how many... What do people typically do if you hear "place value" and you hear the number 176, what do we typically see in like a third grade classroom?

Kim  

Oh, yeah, yeah. So, they'll talk about how that's 100 and 70 and 6, Oh, yeah. Mmhm. 

Pam  

100 and 70 and 6. And they might say it's 1 hundred, 7 tens, and 6 ones.  Yeah.

Kim  

Absolutely.

Pam  

Either way, either way. And so, they kind of break the number up in place value, and you're like, "Yeah, Pam, that's what place value means."  But let's play a little bit. So, Kim, if I were to say, "How many ones are in 176?" I think we might have some third grade kids that say 6.

Kim  

Yeah, for sure.  Yeah.

Pam  

And I wonder if you agree with that. How many ones are in 176?

Kim  

There are 176 ones. 176.

Pam  

And how would you... Maybe I'll just say it.

Kim  

I mean, if you were going to build that number, you would need 176 ones.

Pam  

Sure enough, and if I was to give you $176.00, could I do that with 176 dollar bills? 

Kim  

I'll take it. 

Pam  

I could, right? Yeah. And then, you might say, "Hey, can we switch that out a little bit? And then, we might do some like bigger bills that represent more numbers. So, hopefully everybody would agree that... So, depending on how you ask the question. If I asked the question, "Given that there's 1 hundred and 7 tens, how many ones are leftover?" then the answer could be 6. But if the question is "How many ones are in 176?" careful teachers that you don't say 6 because that's not correct. There's 176 ones. 

Kim  

Yeah. 

Pam  

Okay, how many tens are in 176? Kim, would you agree with me that many third grade kids would say there are 7 tens in 176? 

Kim  

Yeah, absolutely. Yep.

Pam  

But I think if I were to ask you, you would say? How many tens?

Kim  

17.

Pam  

There's 17?

Kim  

Yeah, 17.6. 

Pam  

Oh, there's 17.6 tens. 

Kim  

Oh, yeah. Sorry, I didn't finish. 17.6.

Pam  

And how do you reason that there's 17.6 tens in 176?

Kim  

Because 17 tens is 170. 

Pam  

170, mmhm.

Kim  

Mmhm. And 0.6 of a ten is six-tenths of a ten, which is 6.

Pam  

Cool. So, one way if I said, "How many tens?" You could say there's 17.6. Another way you could say that is, there's 17 tens and 6 ones left over.

Kim  

Mmhm.

Pam  

It's kind of a remainder feel for that.

Kim  

Mmhm.

Pam  

Cool. How many hundreds? Did I do everything I wanted to do with tens? I think I did. Yeah, I think so. How many hundreds? What do you think an average third grade kid might say?

Kim  

Oh, actually I want to say something with the tens?

Pam  

Yeah, I felt like there was something we didn't say. Yeah.

Kim  

Some, like early third grade, I think I would be okay with there's 17 tens and some leftover.

Pam  

Yeah, yeah. I think we need both of those.

Kim  

I mean, I said 17.6. But what I don't want is seven tens. But I'm happy when they recognize 17 tens.

Pam  

So, if a student were to say, "There's 1 hundred, and I've already taken care of that, and now there's 7 tens, and there's 6 ones." 

Kim  

Yep. 

Pam  

Then, we're okay with that. But be careful how you ask that question. Because if you say, "How many tens?" 

Kim  

17.

Pam  

There's not just 7. There's 17. Yeah. Alright, so how many one hundreds are in 176?

Kim  

There's 1 whole hundred.

Pam  

Okay.

Kim  

But if I wanted to be more precise. 

Pam  

Do the leftover first. There's 1 whole hundred. 

Kim  

There's 1 whole and 0.76 hundred. So 1.76 of hundred. Yeah. And, you know, it's interesting to me because I actually wrote. What I wrote on my paper was different based on the question that you asked me.

Pam  

Oh, interesting. 

Kim  

Yeah. 

Pam  

So, when I said "do the leftover one," I actually meant kind of the remainder thing.

Kim  

Oh, okay. So, 1 hundred and a remainder of 76.

Pam  

And there's like 76 other stuff, other things. But there's 1 one hundred. Yeah.

Kim  

Yep. 

Pam  

Cool. We probably should have had this be like 476, so we could have said there's 4 hundreds, and then there's those remainder kind of things. Or 4.76 hundreds. Yeah, that's kind of hard to say. So, here's a question that we don't usually ask. And I'm not suggesting we ask this in third grade. But maybe in fourth grade. How many tenths are there in 176? Like 0.1. How many 0.1. Tenths.

Kim  

There are 1,760 tenths.

Pam  

How do you know?

Kim  

I actually went back to how many ones there were. And I knew that there were 176 ones. And there are 10 tenths in each one. So, I needed 10 times as many as the 176 ones.

Pam  

And 10 times 176 is?

Kim  

Yeah, 1,760. 

Pam  

Yeah, that makes sense. Nice. Another way that I was kind of thinking about it, because I was wondering how we could think about this, is how many dimes... 

Kim  

Yeah, for sure. 

Pam  

...are in $176.00? Which I would probably do the same kind of reasoning you just did. How many dimes are in $1.00?

Kim  

Yeah. But the context is helpful. 

Pam  

Yeah, I would hope so. And so, then, could we also ask how many hundredths? And could we ask how many pennies? 

Kim  

Yeah, right.

Pam  

Yeah, and then we could go from there. Cool. Yeah. So, in the title, I think we have "Exploding Dots". So, Kim, Exploding Dots. What do you know about Exploding Dots?

Kim  

Well, I...

Pam  

Don't tell us everything. 

Kim  

No, I have just a little bit of experience. And we actually recently did some interviewing with students, and had a student sharing with us about Exploding Dots. And I didn't know tons, but I did some some more looking. I know that it is about thinking about the value of the number. I know that it's about thinking a little bit more about some equivalence, and that the point, perhaps, is to consider. Like, a little bit more reasoning about adding or subtracting, and that it doesn't have to necessarily be traditional. But I also feel like what I've seen from a lot of students is that they get stuck in some drawing. So, you probably know a lot more than I do about Exploding Dots.

Pam  

Yeah, so what you've seen is kind of students interpretations of teachers interpretations of James Tanton's work.

Kim  

Yeah, the after effect, Right. 

Pam  

Yeah. So James Tanton is a mathematician from Australia. He does some amazing things. I think he's kind of funny. He's snarky. So, Kim, I think you would enjoy (unclear).

Kim  

The board he writes on is cool. 

Pam  

The board he writes on his super cool. I want one of those really bad. My team keeps telling me no, I'm like, "Come on!" I want a board like that. So, it's great. I heard him live before he had kind of gotten super big. And now, he has kind of a cool website with lots of Exploding Dots things. And if you listen to his now recorded stuff on his Exploding Dots things on his website, he clearly says at the beginning that, "If this story was true, when I was a boy..." and then he goes off to talk about how he thought about things as a boy. But he clearly says at the beginning, "If the story was true..." I thought that was awesome because the first time I heard him, I actually thought it was true. And he probably said "if" at that point as well. But I definitely heard everything else he said as like, "Wow, he really thought about numbers this way as a kid." But he clearly does some very nice things to help us think about our base 10 place value system. 

Kim  

Mmhm.

Pam  

And maybe I can even say not even base 10. He helps us think about what a place value system is. So, for example, if you were thinking about Roman numerals, that is not a place value system. Roman numerals is a very additive system, where you have to know the symbol for the amount that you're talking about. So, if you're talking about 3, then you draw the three little "I". Like, the letter "I". And you count those, and that stands for 3. But if you want to represent 4, then they had this... Or maybe just go to 5. And that was a "V". And "V" represents 5, and an "X" represents 10. And you had to... Now, that's similar to our number system where we have symbols.

Kim  

Mmhm.

Pam  

Like the numeral "3" represents 3, and the numeral "4" represents 4. But their symbols only represented certain numbers. So, like "3", they didn't have one symbol to represent 3, they used the symbol for 1, the "I", and you had to have three of them.

Kim  

Yeah, three times. 

Pam  

So, that's why it's additive. I had to add those three together. And if I was to represent the number 6, I needed the symbol for 5, the "V", and the symbol for 1, the "I". And I would put those together, and I would add them then. Similarly, if I was to get 12, I would use a symbol for 10, the "X", and then I would put two next to it. People are like, "Why are we talking about Roman numerals?" Because those other numeration systems kept us from being able to develop more mathematics. When the Hindu Arabic numerals came along, and the place value system came along, when we developed that as humanity, all of a sudden, in a relatively quicker fashion, we were able to do much more mathematics. Because of the place value system. So, our place value system is very important. Our place value system says, "If I put a digit, a numeral, in a place, it represents a different magnitude than it does in a different place." So, if I put the numeral "3" in the ones place, it represents 3 ones. But if I put the numeral "3" in the tens place, it represents 3 tens. That's multiplicative. Now, I'm thinking about 3 times 10 because it's in the tens place. That's a multiplication sentence. Then, I add those together. So, what was the number we were messing around with? 176?

Kim  

Mmhm.

Pam  

So, I've got 1 hundred, and I've got 7 tens. 7 is in the tens place, so there's the 70 comes from. That's multiplicative. And then, the 6 in the ones place. And then I add those together. But the multiplicative nature that's happening in that place is cognitively difficult. That takes some understanding. I'll give you an example of when I realized quite how difficult it was. And I was smiling. When my youngest, my daughter, came home one day, and she goes, "Hey, Mom, is 12 a one number or a two number?" 

Kim  

Yes! 

Pam  

I'll never forget that day.

Kim  

Yes! 

Pam  

Oh, man, she is grappling with what it means to have two symbols represent a number, right? Because all the way up through 9 it's one symbol representing a number. And now, all of a sudden, you have two symbols representing a number. 10, 1 and 0. What does that mean? Why do you have two symbols representing a number? So, "Is 12 a one number or a two number?" Ah, that's brilliant.

Kim  

Such a great question. Yeah.

Pam  

She's grappling with our place value system. So, I'm not going to spend too much more time on that today. I am going to say that I think he does an amazing job of... So, Exploding Dots is where he says, "If I have 10 ones, I can stick them together." And he moves them over, and they become 1 ten. And if I have 10 tens, then I stick those together, and move those over, and they become 1 one hundred. Well, how are they exploding? Well, what if I take that 1 hundred, and I move it back into the tens. And he even goes "Pa-pow!", "Bang!" And that 1 hundred explodes into 10 tens. And I could take one of those tens, and if I move it over into the ones, "Boom!", "Bam!", Bing!" it explodes into the number of ones. Sorry, James, if I'm not doing you justice here. I think he does a great job in helping us understand the place value system and our base 10 system. What I'm not as interested in is that then he goes on... I shouldn't even say "then". Also? Because it's not really... I'm not trying to say this in order. He also helps us understand other place value system, in other bases. I shouldn't say other place value systems. Place value systems with other bases. So, our place value system, we use base 10. He, and many other mathy people, have decided that a good way to help us understand our place value system is to do other bases, not just hang around in the base 10 system. And so, he does work with base 2, and base 3, and exchanges. And then, he does work saying, "Alright, with base 10..." And he says, "Now that we understand place value system, and ours is base 10, hey, check out how the algorithms work." And he does quite a bit of work, and I would say way too much work, in understanding the algorithms. Because I would suggest... Maybe not James because I don't know him, and I haven't talked to him about it. But teachers hear that as, "Therefore, our job now is to use Exploding Dots to have kids add, subtract, multiply, and divide." 

Kim  

Yeah, and that's the experience I've been seeing. 

Pam  

Yeah, and we see kids then, "Oh, this is what I must do. I must now draw these dots, and circle them together, and make the higher thing." And it becomes a whole lot of counting by ones, and a whole lot of reading off the answer. And those are two flags for me that say, "Mmm, if we're supposed to be adding, and I'm counting by ones, we're using less sophisticated thinking." Flag. Flag. Red flag. Red flag. If we are reading off the answer... Thank you, Cathy Fosnot, for this tip that if you're reading off the answer, you're probably not mathing. You're probably repeating something rote memorized. And so, two red flags that say, "Mmm, I'm not so interested." So, let me be clear, I think he's does a great job in helping us develop place value systems. I'm not so interested in then taking that into why the algorithms work. And I'm not so interested in taking that into other bases. At least not for most students.

Both Pam and Kim  

Yeah.

Kim  

We've seen that as well in schools, right? Where some of the classes do a couple of weeks of units in other bases. And for me, it's a little bit like I want that time to explore more deeply in our base system. 

Pam  

We have a lot to do in our base 10, right? 

Kim  

Yeah.

Pam  

We have a lot to develop in our base 10. And I think, Kim, you and I would agree. We can do what we need to do with most students in base 10. Spending that time really understanding base 10. That we don't then need to spend all this other time in other bases. And I know I hear people. I can see your face right now, listeners, where you're like, "No, Pam. Like, I really understood blank in my pre-service methods class where we had to mess around in other bases."

Kim  

Yep.

Pam  

May I may invite you to consider what is it exactly that your "aha" was? And I'm going to bet your "aha" might have been about the place value system, but I bet it was more about why the algorithms work the way they do. And if that was your "aha", I'm going to ask you to consider is our goal the algorithms? No. And so, since our goal is not the algorithm... I love that you had that "aha". Yay! Good moment that you had that "aha". We don't need that "aha". We need other more important "aha's".

Kim  

Yeah.

Pam  

To get kids really mathing. Kim, you would agree?

Kim  

Yeah. And well, I would also say that probably their experience is what we're saying is not a rich experience of 176 is 100, and 70, and 6. We need so much more than that.

Pam  

Absolutely. 

Kim  

That that's the richness of place value that we really are going for.

Pam  

Yeah, absolutely. So, I'm not dogging James Tanton at all. I'm just going to de-emphasize a lot of his later... I shouldn't say his later work because he does a lot of work not just in Exploding Dots. But in Exploding Dots. If you want to use Exploding Dots, use it to develop relationships, to talk about what it means to have a place value system. But I would suggest for most students, stay in base 10. If you really want to, and you have a group of students that are ready for an extension, sure. Go ahead, and go play in other bases. But that's not the goal. That's an extension. It's to play with. I think our goal is to get good at base 10.

Kim  

Mmhm.

Pam  

Okay.

Kim  

Yeah.

Pam  

So, let's play a little bit how we would kind of play maybe with some numbers. 

Kim  

Okay. 

Pam  

Kim, how would you play with the number 3,872? What's the first thing that comes to mind for you? And it's okay. I'm not looking for right answer here.

Kim  

Okay, so anything? Because I don't know what you want. So, I'm going to say the first thing that comes to mind for me is that it's 28 away from 3,900.

Pam  

I love it.

Kim  

Did you think I'd say that?

Pam  

No. I wasn't sure what you were going to say. So, I love that you did that. So, 3,872 is 28 from 3900? 

Kim  

Yeah. 

Pam  

Do you also play, then, how far it is from 4,000?

Kim  

I haven't because then, the next thing that I wanted to write down was that it's 8 away from 3,880.

Pam  

Mmhm. 

Kim  

But yes. Then, the next place I would have probably gone is that. How far away from 4,000? And that's 128 away.

Pam  

I'm super curious to let you play a little bit more. Any other playing you want to do?

Kim  

I kind of want to think about 10,000, which would be 6,128. 

Pam  

So, you kind of did it in partners.

Kim  

So, I could change my end goal. Right, like so partners. So, let's see. What else I want to do? We already did the ones. That's 3,872 ones. It's 387.2 tens. I could keep going kind of either direction there.

Pam  

Okay, keep going. How about hundreds? How many hundreds? 

Kim  

So, that's 38.72 hundreds.

Both Pam and Kim  

(unclear).

Pam  

Sorry, didn't mean to interrupt you.

Kim  

So, (unclear) thousands. It's a shift, right?

Pam  

If you don't mind staying with hundreds for a second. 

Kim  

Yeah.

Pam  

So, when you just said 38, and then you went to the... I'm kind of a little less. I'm going to kind of ignore the extras for a second. 

Kim  

Yeah, okay. 

Pam  

I think it's interesting that 3,800 also has the name 

Both Pam and Kim  

38 hundreds.

Kim  

Yeah.

Pam  

Right.  So, go back to tens for a second if you can. I know we're shifting a little bit. So, how many tens did you say? And do it as an extras.

Kim  

Yeah. 387. 387 tens.

Pam  

With 2 extras. Okay. So, in 3,872, there are 387 tens.

Kim  

Mmhm.

Pam  

Where was I going? I just forgot.

Kim  

Because you can say 38 hundreds. But you don't say...

Pam  

Oh, yeah. Why do we call the number? I'll let you finish the sentence.

Kim  

Well, I mean, it's silly to us that we wouldn't say 387 tens as a way to describe that number. We say 38 hundred.

Pam  

Yeah. And you might be like, "Why do we say 38 hundred and 3,800." And I don't know. But I'm suggesting that we can also acknowledge that if we had the number 3-8-7-0, 3,870, that we could call that 387 tens.

Kim  

We do it with thousands too. We say 3 thousands. 3,000.

Pam  

Absolutely. Yeah. Well, so then if we extended that up. Let's say that we have the number 13,872. Do we call that 13 hundreds? We don't. We call that? 13 thousands. Because we're in that place, right? It's 13,872. So, there's a place where we actually... Oh, I don't even know how to say what I'm trying to say. Where the thing that it is, we're calling it that. 

Kim  

It sounds like... Go ahead.  Yeah, it's 13,000.

Pam  

We're not calling it 1 ten thousand, 3,872. That's the break I'm trying to make.

Kim  

It sounds like you're challenging the listeners to join in your movement to start calling tens. 387 tens. I've heard you say that before. 

Pam  

Yeah, absolutely.

Kim  

You might want to start a movement. Let's call things what they are.

Pam  

So, that we get a little bit more flexible. So, that when we say the number 3,872, we get things like all of Kim's partners, but we also get different place values, wnd we don't just get the standard place value separation that we get.

Kim  

Expanded notations.

Pam  

Expanded notation. That we don't just get that, but we play maybe with a little bit more. 

Kim  

Yeah.

Pam  

Yeah. Cool. Alright, Kim, I'm going to suggest that there are some takeaways. What are you thinking?

Kim  

For sure that we need to broaden the question that we're asking students. It's not enough to just say 100, and 70, and 6. 

Pam  

The expanded notation question.

Kim  

Yeah.

Both Pam and Kim  

That's not enough. 

Kim  

No way.

Pam  

So, we're also not saying that's evil. It's just not...

Kim  

That's needed too. It's just not enough.

Pam  

Necessary. Necessary, but not sufficient. Mmhm.

Kim  

Yep. 

Pam  

Yep.

Kim  

And so, broaden the question that the place value... Place labeling. Right? I don't know how much you said about place labeling, but...

Pam  

Oh, yeah. We didn't really talk about that today too much. 

Kim  

That labeling of ones, tens, hundreds. is (unclear).

Pam  

Is definitely not enough to develop place value. 

Kim  

No. For sure.

Pam  

Definitely not. And so, all too often, we have worksheets out there that just have kids label the places. You give them a number, and you just say, "Label the places." Once kids have done that... That's necessary, but it's not sufficient to develop place value. So, those two things together is what we typically see is the most that's happening out there. You label places, you do the expanded notation, and then you're kind of done. 

Kim  

Right.

Pam  

Those are necessary. Don't do so much on them. Do a little bit on them, and then do more work trying to get place value done.  So...

Kim  

Go ahead.  Yeah, because

Pam  

Yeah.

Kim  

it's not doing what people think it's doing, right? It's giving them the names of the places, but it's not creating understanding of value. Yeah. And (unclear).

Pam  

Well, so how do we do it, Kim? 

Kim  

Yeah, well, I think... I mean... 

Pam  

We've told everybody what not to do. What you do?

Kim  

Yes. So, we hope everyone all over will take the place value mini workshop to learn more about place value, and how to develop it in your students. Ya'll, this place value workshop is absolutely amazing. So many understandings. So many things that you can turn around in your classroom. It is a small ask timewise. It's super fantastic.

Pam  

Ya'll we thought as a team. We sat down, and we said to ourselves, "What is something that we could create in a mini workshop? We know teachers are busy. We know they only have so much time. We know they have only so many funds. What could we give that would be high impact that all teachers will go 'Oh! That! Yes! Yes, we know our kids need...'"

Kim  

All the grade levels right? Like, this is not just a "Oh, I teach second grade," place value type of thing. Because it affects everybody.

Pam  

Third grade place value. Sixth grade place value. Yeah. This affects so much of what we do is to get kids really reasoning using our place value system. So, we created Teaching Place Value, and it is a mini workshop. It is intended to take two hours about. We would invite you, dive in to our mini workshop called Teaching Place Value, and learn how to de-emphasize the kinds of things that we talked about today and emphasize building that place value in your students. You and they are going to love it. Alright, 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. Thank you for helping spread the word that Math is Figure-Out-Able!

Kim  

Absolutely.