Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

And we make the case that mathematizing is not about mimicking steps or rote memorizing facts. But it's about thinking and reasoning; about creating and using mental relationships. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question: if not algorithms and step by step procedures, then what?

Kim Montague  00:41

So Pam, right now we're doing a Challenge, right? We love Challenges. We're right in the middle of a Division Challenge. But today, we're going to talk to Pre-K through second grade teachers, because we just finished a Challenge specifically for them. 

Pam Harris  00:55

Oh, yeah!

Kim Montague  00:55

People commented over and over again, about how this one particular thing that we did resonated so well. So we decided to bring it here to you.

Pam Harris  01:04

Let's do it. And y'all we are doing the "You Can Change Math Class Challenges". If you have not yet joined in one of our "You Can Change Math Class Challenges", stay tuned, we will tell you when the next ones are coming out. Get on our email list to know when they're happening. A lot of wonderful, free, totally free, learning, training, and we're having a blast meeting people from all around the world. 

Kim Montague  01:29

All over.

Pam Harris  01:31

All over. l love meeting so many different mathematicians around the world, who are learning to teach more and more Real Math. Okay, so let's do this young learner thing, this is really fun. I mean... but Kim young learners? All they have to do is like, sing the song of counting, right? To be able to, you know, like, say the counting words in a row, and then they're good to go, we can move on, we can start doing, you know, addition and subtraction - or not, right? If you teach young learners, you are well aware that there is more to counting, and using counting than just knowing the names of the numbers in order, right? Like we know, there's a whole lot more than that. So today, we're gonna try, we're gonna do a really cool thing that tries to give some insight into young learners like, why is it so complicated for them to be able to do anything. "Look, they can count! Listen to my kid." You see, sometimes parents will say, "Hey, honey, go ahead and count to 20, show me how you can count." And the kids start to count. The parents are all proud and it's all exciting. But then does that necessarily translate into being able to use those numbers in some meaningful way? Even if I'm still in Counting Strategies, which as you remember, from the Development of Mathematical Reasoning, that is sort of our initial kind of, we've got to be able to use Counting Strategies to be able to solve problems before I can move on to Additive Reasoning. But just singing a song and counting doesn't mean that I've developed Counting Strategies and that I'm able to use the count. So let's see if we can give some insight into that today. Alright, everybody, ready? So I'm gonna give you a problem. You would do that a lot, right Kim? We do some math? 

Kim Montague  03:13

Yeah. 

Pam Harris  03:13

So today, we're going to do a little problem. Now everybody's kind of relaxed, because you're like, "Counting, you know, alright. Young learners. We can do it. Yeah, here we go." So everybody, just relax right into this problem. Here we go. Ready? C plus M. Yeah, you heard me correct. The letter C, plus the letter M. Pausing, I'm pausing. What? How many of you are looking at your radio right now? You're like tapping your phone. Like thinking to yourself- radio, I just said radio. Okay. You're tapping, whatever you're listening to the podcast on your tapping. You're like, "What does she, what? I must have missed that number that she was saying because she it sounded almost like Pam said C plus M like -" Yes, Yes, I did. Y'all that can be how kids hear three plus four. Like when you say, "Alright, what's three plus four?" They could hear 'C plus M'. In fact, a good friend of mine, when I did a little bit of this alphabet thing with them said, "You mean like CM?" Like they just sort of took the letters - I had written on the board C + M -  and they took the letters and they squished them together. Well, that would be like CM and they were trying to make a sound out of it. "It's like csm. Like what does that even mean?" And the point is that kids if we give them problems, like three plus four out of the air, and we don't have context behind it, we don't have meaning behind it, we don't have Counting Strategies behind it. Then they could just think. "Well I'll just put those numbers together. Like scoot that 3 close to that 4." Just like they would scoot the C plus that M together because there's not a lot of meaning happening.

Kim Montague  04:54

Yeah. So if you even understood what we were asking by saying what is C plus M, I wonder how you thought about that? Is it possible -

Pam Harris  05:04

In fact, I'm sorry, Kim, I'm interrupting. Maybe we should have, like a suggestion. Pause the podcast right now. 

Kim Montague  05:11

Oh, yeah. 

Pam Harris  05:12

Actually figure out C + M.

Kim Montague  05:15

Well, if they even know what you mean.

Pam Harris  05:16

Oh, yeah, sorry. Go ahead and keep going.

Kim Montague  05:18

Yeah. So if you understood what Pam meant C plus M: the correlation between the letter and the number system. So maybe now pause, and I wonder how you're thinking about that? So did you count three times? Did you say, "Okay, A, B, C is 1, 2, 3." And then count all the way one at a time from A to M and then have - gosh, I don't even know what M is Pam. What's M? 13?

Pam Harris  05:53

J is 10. So okay, K, L, M. So it is 13. How did you do that so fast?

Kim Montague  05:58

I actually knew the middle of the alphabet. 

Pam Harris  06:01

Oh, you went from the middle? 

Kim Montague  06:02

I did. 

Pam Harris  06:03

What is the middle? 

Kim Montague  06:04

There's 26 letters. So between M and N. Sidebar. So if you knew what Pam was even asking, then you may have had some sort of strategy. But I wonder if you counted three times, if you counted to find out what C was, counted to find out what M was, and then recounted all of those numbers together.

Pam Harris  06:26

Like, literally, like count up to 13. 

Kim Montague  06:28

Yeah, yeah. 

Pam Harris  06:29

Or then say to yourself. Let's see 13 plus three is 16. And then started over from A and counted up to the 16th letter? 

Kim Montague  06:39

Uh-huh. So what does that look like with numbers? Well, we just described that right, like 1, 2, 3, 4, for young learners might say 1, 2, 3, 4 plus three, if it's four plus 3. 1, 2, 3, and then recount 1, 2, 3, 4, 5, 6, 7.

Pam Harris  06:58

Just like you might have just done for the C plus M, you're saying little kids might do that for a problem like four plus three? 

Kim Montague  07:06

Yeah.

Pam Harris  07:07

They would have to recount, like, find out where the C is, find out where the M is. And then find out where the 16th letter is, right? And sing the song up there. And kids might actually do that with numbers. That's interesting. Now, we might have somebody on here going, "I didn't do that. I just thought about C's that's obviously three. So I just started from M. Like I just counted, I just felt went well, M, M, N, O, P. It's clearly P. M, and then I'm gonna go three more. So M, N, O, P, it's clearly C plus M, (or you're really thinking about it as M plus C.) Clearly, that's P." So you might have counted on from the M. but y'all when we work with teachers, often they'll count on from the C, they'll be like, "Okay, C," and then they'll go, "Sso D is like A and then let's see, E would be B and F would be C," and they're literally start counting on until they get up to having started from C until they then put M more letters on. And then they happen to land on P. So that's interesting. You might be like, "Pam, why would they start at C?" Well, I don't know y'all, but they do. Well, and they do. And all of us do. And many of us did with numbers, because we don't really own C yet, or M yet. What is even happening with numbers as a correlation?

Kim Montague  08:38

Well, and I would maybe add that when it's confusing, when you're not sure. Then you read left to right. And so if you're not sure what the questions even asking for you to do, then for some people, they might say, "Okay, let me start at the first, which was C? And then add on the rest." Right? It's a little bit elevated to start with the second or the larger.

Pam Harris  09:03

Oh, yeah. And the larger. So Kim, are you suggesting that if I would have said M plus C, then everybody would have added on from quote unquote, the larger. Because then if I'm reading left to right, I would have done M plus C. And if I'm thinking about adding on the larger, I also would have added M plus C. So maybe we gave you C plus M on purpose, because we wanted to see - oh, and I'm using the sound C in both places. We wanted to view? We wanted to experience whether y'all would think about starting from the first number and adding on or starting from the bigger number and adding on.

Kim Montague  09:43

And in either case, whether you started with the C or the M that's a more sophisticated strategy, then counting three times - finding where the C is, finding where the M is, putting those together - because knowing where the M is and knowing what comes before or after is cognitively more difficult, you have to keep track of when to stop. So like, for numbers, if we're thinking about four plus three, and you hold the four, and you have to count three more, then you're simultaneously keeping track of five, and one, six, and the two, seven, and the three, knowing to stop at counting three more, is much more difficult.

Pam Harris  10:32

Because you have to keep track of those sort of simultaneously. You can almost picture a kid go four, and I'm like, holding my fist. 

Kim Montague  10:39

Yeah, that's exactly what I did. 

Pam Harris  10:42

So I'm like holding on to that four. I own four enough to go "four." And then I also own what comes after far enough to go, "5, 6, 7." But then I stopped, because I have three fingers up. 

Kim Montague  10:54

Yep. 

Pam Harris  10:55

And knowing that those three fingers is what I'm supposed to add. And then I stopped when I see those three fingers. But I'm literally saying, "5, 6 , 7," that is cognitively difficult. Keeping all of that stuff simultaneously happening is difficult. Hey, Kim, what's the best way to do that? We just should tell kids to do that, right? Just tell them that, like, we just, "stop counting three times, stop counting out the four" because we'll see kids count out four objects, count out the three, we'll see them count out three objects, then they shove the two piles together, and then they recount all the objects. We should just tell them. We should say, hey, don't count out four just start at four. How well is that gonna work?

Kim Montague  11:35

Yeah, telling is really helpful. So much more experience, right? They need a lot of experience. And you have commented before that when I'm working with a really young learner, and they're kind of stuck in this idea, or are doing a lot of counting three times that one of the things that I'll do, especially if they're counting on their fingers, is if I see them go to make that first count, let's say again, that they're working with four plus three, and they're about to count out four on their first hand, I will lightly put my hand over theirs like in a fist, and we'll say the number four. And then we'll count on together 5, 6, 7. So just that gentle reminder that you know four, you've already got that first part of the problem that it's four. Sometimes I think kids continue to do the count three times, even when they already - 

Pam Harris  12:29

Could be counting on.

Kim Montague  12:30

Could be counting on because they get really fast at it, right? It's just what they do. And they think that that means - let me solve a problem means - to count three times. And so we need to just encourage and nudge them when we see and we feel like that's something that they can do. 

Pam Harris  12:46

They're on the cusp of being able to do it. That's the moment where you kind of go, you say in your head, I can see you're about to count those four. 

Kim Montague  12:54

Yep.

Pam Harris  12:54

You just sort of go, "4? 4? 4." And they're like, "Oh, yeah, five, six, seven," you kind of do that counting on with them just a little bit. And if you do it and they look at you blank, then you're like, oh, we might need some more experience before we then try it again. You said that, especially if they're counting on their fingers. And I think what you meant by that is if they're counting on your fingers, you'll sort of put your hand over theirs.

Kim Montague  13:15

That's when I put my hand on, right.

Pam Harris  13:17

But I've also seen you if they're counting out cubes, or counters or sharpie's, or something, I've seen you've kind of cover up those counters, like not even four counters, but you just sort of like cover, like take their hand and go, four. And then you drag in 5, 6, 7, right. Like I've seen you sort of like, it's okay, I just didn't want anybody to hear that it's only when they're doing their fingers. It's when it's their fingers, you use your hand over their hand, when it's something else use your hand over sort of that counter to kind of go ooh, let's like unitize. Let's like make that first thing, this thing. And then we can kind of count on from there. And then slightly more sophisticated would be if they're not noticing, then I would want to make sure I give students three plus four or even a bigger difference between the addends. I want to give them three plus six, and then say, "Okay, three plus six." And when I start noticing the student going three, four, five. Their counting on from the three and they're doing that successfully. They've got the hang of that. Now's the moment where I can go, "Oh, that's interesting. I noticed over here that so and so started with six. Can you do that? Hey, so and so tell us how you did that." And have that students share their thinking about how they started with a larger number. Notice that the totals are the same. "Huh!" Maybe model that thinking by actually drawing the tally marks or the fingers or something on the board where they can go, "Oh, check it out. I have this set and I have that set. What if I moved the sets like I had the bigger set first. Does that change the total number of objects I'm counting?" Then we have that conversation. And then I might notice that same student counting on from the first number again, and I might then go, "Oh, hey, remember the conversation?" Like I'm going to nudge that student. Know your content, know your kids, I'm going to get in there. I'm going to be listening to students solve problems. And I'm going to be nudging that. "Oh, remember, you know, to count on from the larger. We've had that conversation. You made that realization. Let me remind you." "Oh, yeah, right." And then and then keep encouraging that student.

Kim Montague  15:21

Yeah, well, and it has everything to do with the commutative property, right? But this is kind of a side note, but I worked with a sweet boy for quite some time and he never counted on from the larger number. And I found that so interesting, because we've had some conversations and what I realized was that he did not trust that four plus three and three plus four was the same thing. And so we did a lot of work on a number rack a rekenrek, modeling that when you flipped it over, that it was indeed going to be the same total. And once he trusted the commutative property, then he was like, "Oh, it makes sense to start with a bigger number, I'm going to have to add less on to it." It was  such a beautiful moment when I worked with him because I thought, "Oh, my gosh, this is what happens when kids don't trust the commutative property. They want to start with the first number every time no matter what it is."

Pam Harris  16:14

Flip the number rack over. Oh, that is an amazing teacher move nicely done to help that student trust the commutative property of addition, it gives us an equivalent quantity. That's amazing. Nicely done. I'm so glad that you shared that. So Kim, kind of what I hear you saying is reading is more than singing the alphabet. And so counting is more than singing the song of the numbers in order. 

Kim Montague  16:41

Absolutely.

Pam Harris  16:42

Nicely done. Alright, young learners who knew? All the math for young learners. We think that it is so important to deal with the intricacies of teaching young learners that we've just created a brand new workshop we call Building Addition for Young Learners. It is debuting this fall. So Fall of 2021. Whenever you're listening to the podcast, check out Building Addition for Young Learners, our new workshop in our selection of workshops for teaching math k-12. We're really excited to help teachers who teach young learners really get into the details about how we can help young learners really make progress from nothing all the way up, nudging into Additive Reasoning. So you might want to check that out. So if you want to learn more mathematics and refine your math teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able!