Math is Figure-Out-Able with Pam Harris

Ep 67: Revisiting the Development of Mathematical Reasoning

September 28, 2021 Pam Harris Episode 67
Math is Figure-Out-Able with Pam Harris
Ep 67: Revisiting the Development of Mathematical Reasoning
Show Notes Transcript

It's been over a year since Pam and Kim first introduced the DMR (Development of Mathematical Reasoning) and in that time they've refined their ideas even further. In this episode Pam and Kim dive deeper into the development and why it's so important. 
Talking Points:

  • It's not about answer getting, or students doing it 'their way'. 
  • Our goal is more sophisticated thinking and reasoning
  • Added Statistical Reasoning to DMR
  • Where do subtraction and division live on the DMR graphic?
  • We don't leave a level of reasoning behind
  • Examples of Counting, Additive and Multiplicative Reasoning
  • How can you build your own mathematical reasoning?
  • Free DMR workshop for anyone at anytime!

Pam Harris  00:02

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

 

Kim Montague  00:09

And I'm Kim.

 

Pam Harris  00:10

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 mathematical relationships. We take a strong stance, that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being a mathematicians they can be, we answered the question, if not algorithms and step by step procedures then what?

 

Kim Montague  00:42

So if you've been around for a while, or you've been really good about going back and listening to older episodes, you know that in Episode Five and Six, those are the episodes where we talked about your thing, right? The Development of Mathematical Reasoning.

 

Pam Harris  00:58

Yeah. 

 

Kim Montague  00:59

In those episodes, we describe the progression. And we talked about the graphic in detail and kind of highlighted some important points for each of those domains. We also shared that Pam created a workshop all about her progression, and it is absolutely free. 

 

Pam Harris  01:17

Dah-tah-dah.

 

Kim Montague  01:17

So we decided that we, it's been a year, right? So we decided that we wanted to revisit and dive deeper into what we call the DMR. The Development of Mathematical Reasoning. So right now, I'm going to say that if you have not listened to Episode Five, you might want to pause now. Go check out Episode Five to get a little bit of intro, and then come back and join us. 

 

Pam Harris  01:41

Yeah.

 

Kim Montague  01:41

Okay. 

 

Pam Harris  01:42

It's true, right. But it will give you some important things that we're not going to repeat today. 

 

Kim Montague  01:46

Yep. 

 

Pam Harris  01:46

So that would be kind of important. 

 

Kim Montague  01:48

Okay. So, Pam, you talked briefly before about why this progression? Just really briefly. And so I'm going to ask you again to share briefly, what brought this up for you?

 

Pam Harris  02:01

Well, right now, I want to talk about what brought this up just recently, so why are we thinking about it right now? In a huge way, it's fascinating to me, I talked to a lot of teachers. I work with a lot of adults that are working with kids learning more math. And often teachers will say things like, "Ah, yes, it's so great to let students solve the problem in their way. I've got students who struggle, and it's wonderful that they can find the answer in their way, the way that makes sense to them. Because Pam, it's okay, if they don't do it the right way, quote, unquote, because if it makes sense to that student, then that students should be allowed to do it that way, their way." So yes, and no. I'm always a little bit like, I want to keep listening when teachers say that, because I have this major concern. It might not surprise you that I'm kind of about a little yes and no, because often, I'm a little bit, "Yes, but let's like, let's really think." But it also might surprise you that I'm not just carte blanche. I'm not just like, absolutely saying. "Yeah, like that kid can do it that way, if that's the way they understand it. Absolutely. That's their way and let him do it." Because I am a huge advocate for student identity and making sure the kids really feel like they're mathematizing and using what they know. But I want to parse it out. So, to be clear, it's not about, it cannot be about getting answers any old way. And that that's good enough. It must be about helping students develop as more and more sophisticated thinkers, using more and more sophisticated relationships. Solving problems, using what a student knows the way that comes naturally to them is so important. It is a necessary first step, but it cannot be the end result. We need to find out how kids are thinking, that has to be, we have to do that. We have to hear where they are, what they're thinking, but then plan to help them develop from there. 

 

Kim Montague  04:13

Yeah, that's so good. So we both listened to the previous episodes and realized that how you speak about the progression, the development is a little different a year later, right? Like the big stuff, it still remains but you've added to it and clarify your thinking a little bit. So, start us off a little bit about what you would want to add on today to that work.

 

Pam Harris  04:36

Excellent. So, one of the major points of that development is that there are these., there's a progression. And that levels or domains of thinking get progressively more and more sophisticated. And in order for kids to think in those most sophisticated domains, like functional reasoning and proportional reasoning, then we need to build on that we need to have kids reasoning successively more sophisticatedly. 

 

Kim Montague  05:00

Right. 

 

Pam Harris  05:01

And so there are those levels that build on each other. There are also some kinds of reasoning that we put along the bottom of that graphic, that are more longitudinal, in, what's the word I want? By nature, the nature of them is that they're more longitudinal, they don't, they're not necessarily dependent on the other levels of reasoning. We want to start developing Spatial Reasoning, that's one of them, we want to start develop Spatial Reasoning early. And we want to progressively get more sophisticated Spatial Reasoning. And we want to develop, so we had Spatial Reasoning, Statistical Reasoning,  and what was the other one?

 

Kim Montague  05:39

Algebraic Reasoning, 

 

Pam Harris  05:40

And Algebraic Reasoning. Ah, come on, Pam. And we want to develop Algebraic Reasoning early, and then get progressively more sophisticated in that kind of reasoning.

 

Kim Montague  05:49

Right. 

 

Pam Harris  05:50

Early, when I put out the development graphic, I only had the two, Algebraic Reasoning and Spatial Reasoning. And I didn't have Statistical Reasoning yet. And then Jo Boaler, not too long ago came out with a lot of discussion about number and about data, and how we can use data to really help us think, and reason more sophisticated about a lot of things. And I was like, "Oh, yeah, Statistical Reasoning." That's longitudinal as well, I think that also belongs with Algebraic Reasoning and Spatial Reasoning along the bottom. So we added that to the graphic. So if you've ever seen one of the earlier versions of the graphic, just add that little bit of Statistical Reasoning along the bottom, and that's a helpful addition. A couple other poignant things that we could add some extra details that it didn't actually change the graphic. But some details that I hear teachers sort of talking about, and I just kind of wanted to add in today's episode, is that we have Counting Strategies, counting thinking, Counting Reasoning is kind of the first domain that kids learn about and then that moves to we want to get more sophisticated with Additive Reasoning. And then we want to get more sophisticated with Multiplicative Reasoning. Sometimes teachers will look at Additive Reasoning, and they'll say, "Where's subtraction?" Oh, yeah. Subtraction lives in Additive Reasoning. Similarly, teachers will say, "Where's division?" Well, division lives in Multiplicative Reasoning. If I'm really reasoning about division, then I'm reasoning multiplicatively about division. If I'm reasoning about subtraction, then I'm reasoning what we would call additively about subtraction. So those operations live in those domains. So just so that you haven't, if you didn't see them, and you're wondering where they are, they sort of live in those domains. 

 

Kim Montague  07:39

Yeah. 

 

Pam Harris  07:40

So one other detail is, and I actually mentioned this in Episode Five, that it has come up enough since then, that maybe it wasn't quite clear, people need to hear it again, that you don't necessarily leave a level of reasoning behind. So what I mean by that is, if you are solving, say, a multiplication problem. Inside that multiplication problem, while you are thinking and reasoning, multiplicatively, you're probably going to do some addition, subtraction. You're probably going to use some Additive Reasoning. Well, I want you to when you're doing that addition and subtraction inside that multiplication problem, I want you to absolutely use Additive Reasoning to do that addition and subtraction. I don't want you using Counting Strategies to solve that addition part or that subtraction part within that multiplication problem. But the multiplication part of the problem, I don't want you using Additive Reasoning to solve. I want you using Multiplicative Reasoning. Are you thinking in terms of bigger chunks of numbers than one group at a time. But I might do an over strategy for that multiplication problem. I might think about something where I'm thinking about, I don't know, nine times something, instead of thinking about that, I think about 10 times that thing. Well, then I need one less of the thing. So I've got to do a little subtraction. Well in that subtraction, think as sophisticatedly, about that subtraction, as you can. Think additively about it. Don't count by ones. Don't use a Counting Strategy, counting by ones. But at that idea of using 10 things to subtract just one of those things in order to find nine of those things. That's Multiplicative Reasoning. So I'm using Multiplicative Reasoning. When it's called for. That doesn't mean that I leave Additive Reasoning behind. 

 

Kim Montague  09:25

Yeah. 

 

Pam Harris  09:26

Similarly if I'm thinking about Proportional Reasoning, a Proportional Reasoning problem, I'm probably going to use multiplicative or multiplication in that problem. But I don't want to just use sort of simple multiplication in order to solve a Proportional Reasoning problem. I want to be thinking and reasoning proportionally. And then say to myself, "Ooh, how do I use that to solve this? I might do some multiplication to do that." I don't want, as I think about that, I don't want to go back and then skip count to solve that multiplication problem inside that Proportional Reasoning problem that I'm doing. I want to think as sophisticated as the problem calls for. Why? Because it's all about building reasoning. Building more and more sophistication in kid's brains. Literally making their brains more dense, having more relationships, being able to grapple with more simultaneity. And the big point that I just wanted to add is, so you don't leave reasoning behind. You don't then never add again, just because you've moved on to a more sophisticated reason, right? Yeah.

 

Kim Montague  10:28

So I'm glad that you brought up a couple of specific examples just now. And I want to take a minute to do that some more, because it's something that we left out of the earlier episodes, the first two episodes.

 

Pam Harris  10:38

Kim and I were laughing after we went back and listened to Episode Five and Six. We were like, "Oh, we had given him any specific examples. Huh. Whoops."

 

Kim Montague  10:44

So let's do that. Now. Right? Let's do it now. Let's get specific. And give some examples.

 

Pam Harris  10:50

Okay, cool. So let's just a random addition problem. Kim, if I were to ask a kid, "I've got five gumdrops, and I'm going to give you seven more gumdrops. How many gumdrops do you have?" What might we see kids do or think about?

 

Kim Montague  11:08

So if they're using a Counting Strategy, well, first of all. First, we often think about the answer, right, as teachers. And hopefully, your listeners are moving away from that, but a lot of times, we're focused on the answer. So we would hear 12. But we're suggesting that we need to ask how students are thinking about that problem. We need to ask them, "What were you thinking?" And so if the problem was five plus seven,

 

Pam Harris  11:35

To be really clear, it's not that we don't need the answer 12.  Sure, of course, right.  WE don't want that to be the only focus.

 

Kim Montague  11:42

Yeah, absolutely. Thanks for that. So we might see or hear kids verbalize how they thought about that. Or maybe we start seeing some fingers or there, like a head bob. Right? So if a kid says 5, 6, 7, 8, 9, 10, 11, 12, maybe whisper or fingers. Or even if they start with seven, starting with a bigger number 7, 8, 9, 10, 11, 12, that counting by ones is a Counting Strategy. Because that's kind of the definition of a Counting Strategy. Counting by ones.

 

Pam Harris  12:13

But, Kim, they were solving an addition problem? They were solving five plus seven, they're not, that's not a Counting Strategy. They are solving five plus seven. 

 

Kim Montague  12:21

Right. 

 

Pam Harris  12:21

Or is it?

 

Kim Montague  12:23

Right, so we could have kids using an Additive Strategy to solve five plus seven. 

 

Pam Harris  12:29

What would that look like differently? 

 

Kim Montague  12:31

Okay, so that might be a kid, knowing five plus five is 10. And thinking of the seven, breaking up the seven into a five and two. And then knowing embedded in that seven, I've got a five and five, which is 10. And then two more, which is 12. That's a way they can be thinking additively.

 

Pam Harris  12:51

If they're thinking about five plus seven additively, instead of counting by ones using that less sophisticated strategy of Counting Strategies. Give me another one, I'm curious, five plus seven. 

 

Kim Montague  13:00

Okay. 

 

Pam Harris  13:00

Give me one more Additive Strategy. 

 

Kim Montague  13:02

So they might start with seven, and know that that other five has a two and three in it. So start with seven and add three more to make 10. And then that leftover two to make 12.

 

Pam Harris  13:16

Totally cool. And it's kind of a Get to a Friendly 10 strategy. Nice. And so those two strategies, thinking about five plus five plus two, or thinking about seven plus three, plus the extra two, those are Additive Strategies that we would want to nudge kids towards. We want to help students develop those Additive Strategies to think about a problem like five plus seven. Because then they'd be thinking more sophisticated than counting by once. And that's our goal. Our goal is to develop all of us into more sophisticated thinkers. Cool. So what would that look like with subtraction? Like, let's just throw out a subtraction problem, because I just said earlier, subtraction lives within that Additive Reasoning sort of oval on the graphic of Development of Mathematical Reasoning. So if I gave you a problem, like 15 minus eight. 

 

Kim Montague  14:09

Okay. 

 

Pam Harris  14:10

15 minus eight. What would we expect to see if a kid is solving that using a Counting Strategy?

 

Kim Montague  14:16

So 15 minus eight, they could start with 15 and simply count back eight, so 15, 14, 13, 12,  11, 10, 9, 8, 7, knowing that the number they landed on was kind of their end count, but still counting by one's. 

 

Pam Harris  14:32

Totally cool. And another strategy might be a kid who instead of starts, who started with 15 and lands on the seven might say, 15, 14. Like you sort of said 15 and then kind of subtracted from there and landed on the seven. Sure.  Kid also might go 15, 14, 13, 12, 11, 10, 9, so the answer's eight. So there's kind of two different ways they can count back. Sometimes people make a big deal about those two different ways of counting back. We kind of don't so much. As long as a kid is sort of counting back successfully. And they kind of, well or unsuccessful, at that point, like when a kid is using that counting strategy to solve something like 15 minus eight, we want to then help nudge them on to using Additive Thinking. How can they use bigger jumps of numbers than counting by one? So, Kim.

 

Kim Montague  15:22

Yeah, I'm actually gonna dive in, because I think you said so the answer is eight. And I think what you meant was, if they start with 15, they're counting the counts and not where they land. I was with you. I knew what you meant.

 

Pam Harris  15:33

Well, they didn't really give that. 

 

Kim Montague  15:35

Yeah. 

 

Pam Harris  15:36

So what I should have said was 15, 14, 13, 12, 11, 10, 9, 8. So the answer's is seven. 

 

Kim Montague  15:45

Yeah. 

 

Pam Harris  15:45

And so do you know what I did that time? Y'all can't see, but the time before I did it all in my head. This time, I literally put up fingers until I had eight fingers.

 

Kim Montague  15:53

And kids actually do that. Right? 

 

Pam Harris  15:55

Yeah. 

 

Kim Montague  15:55

But sometimes teachers say, "No, no, you have to count this particular way." Which is, which is not true.

 

Pam Harris  16:02

Even though there are two fine Counting Strategies for solving 15 minus eight. And obviously, I don't do one of them well in my head, but I did. Hey, I did great once I had my eight fingers out that I could see the seven leftover anyway, fine. Okay. So those would be Counting Strategies. Hey, it's real folks. This is a real podcast, us really talking. So we could use a Counting Strategy to 15 minus 8. Kim, what would it look like, what would we see a student do, where we would go, :Ah, this student is using Additive Reasoning to solve 15 minus 8?"

 

Kim Montague  16:33

Sure. So they could know that the eight is composed of a five and three. And so somebody who's thinking additively, might start with the 15, remove the five to get to a friendly number of 10. And then remove the last three to get to seven. That's a way to think about removal with an Additive Strategy.

 

Pam Harris  16:59

Yeah. And you've removed bigger jumps of numbers than one at a time. You removed that whole five all at once. And then you knew 10 minus the leftover three. And those big jumps, that Additive Reasoning. And so we're trying to acknowledge that we want to help students move to using more bigger jumps of numbers and less jumps one at a time.

 

Kim Montague  17:19

Yeah, Can I give you one more, because those, all always examples that we've been talking about or removal. But a kid with some strong Additive Thinking might also think about the distance between those two numbers and might say, "I'm going to start with eight. And I'm going to think about two more is 10. And then five more is 15."

 

Pam Harris  17:42

Yeah, and the idea that subtraction can have two different interpretations, that I can either remove the eight or I can think about the difference from the eight up to 15. That's an important thing that we need to develop in students so that we have a robust sense of subtraction as kids develop Additive Reasoning. Yeah, nicely done. I'm glad you brought that up. So do you reason additively, when given a problem? So could I give, listeners, can I give you a problem like, here's care comes a problem ready? If I give you a problem, like 57 plus five. Go ahead and solve that. Maybe pause a little bit. 57 plus five? What are you thinking? Are you kind of nodding your head saying to yourself, "57, 58,  59, 60, 61, 62." Are you counting those five by ones? Okay, if you are you're thinking additively? Alright. No harm. No foul. Okay. 

 

Kim Montague  18:35

Counting counting. 

 

Pam Harris  18:36

Did I just say additive?

 

Kim Montague  18:37

Yeah. 

 

Pam Harris  18:38

I'm losing my mind.

 

Kim Montague  18:39

It's ok. I'm listening to you. 

 

Pam Harris  18:41

You're thinking in Counting Strategies if you were counting by ones. Okay, so that's, no harm, no foul. That's where you are on the Development of Mathematical Reasoning. Keep listening to the podcast. Let's get you thinking more additively. So Kim, if I say 57 plus five, what would a more Additive Strategy look like for 57 plus five?

 

Kim Montague  19:02

Yeah, so we're thinking about bigger chunks of number, right? So if I'm at 57, I might want to get to a friendly number and add three at a time. So 57 plus three is 60. Plus the remaining two is 62.

 

Pam Harris  19:18

Nice Additive Reasoning. Alright. Let's give it another one. What if I said 42 minus 6. 42 subtract six. Everybody solve it. Listeners, if you're thinking about backing up by ones, then you're thinking in Counting Strategies. Kim, what would an Additive Strategy look like? Just one Additive Strategy for 42 minus six.

 

Kim Montague  19:44

Let's go with 42 minus two is 40. And then 40 minus 4, knowing that group of four back would be 36.

 

Pam Harris  19:55

Nice, bigger jumps of numbers, then counting one at a time. And remember listeners, that the upshot is you might be like, "Pam, I keep count so fast by ones, it's I get the answer. I'm good."

 

Kim Montague  20:07

Right. 

 

Pam Harris  20:08

Right. But if you're not thinking in terms of those bigger jumps, it's going to affect your ability to think about more and more sophisticated mathematical concepts as we go. And then you're going to be stuck with only being able to do less sophisticated thinking, like mimicking rules and procedures.

 

Kim Montague  20:26

Can I pipe in for just a second? Because we have a colleague that has said to us before, that she does some counting a little bit, because that's what she grew up in. And that every once in a while, she will find herself counting. And she has to stop herself because she knows that the goal is to build her reasoning. And so if you find yourself continuing to allow yourself to count, because you're fast at it, or because you've become good at it, then you're not developing your own reasoning, right. So when you catch yourself, stop, say, "I'm not doing that." It reminds me of you in the grocery store. Long ago, you've mentioned the story that you would start to line numbers up and then you almost pulled the numbers apart because you had a goal that was different than just getting an answer.

 

Pam Harris  21:15

Exactly. I wanted to build my brain into being able to think and reason about more and more sophisticated things. Y'all, it took longer for me to solve problems for a while. I'm not even gonna tell you how long, until my brain got more, until I strengthen those neural connections and they got stronger, and then they became my go to. So it might take you a hot minute to get your brain to have the connections be your go to strategy. That's okay. Like, we would recommend that you put in that effort to get your brain thinking and to spend the time to actually, "Okay, what would I do? What are bigger jumps of number?" So that you can build your Additive Reasoning so that then more and more reasoning can become accessible to you. Because your brains ready for it. Your brain is, you have more neural connections to build on? Yeah, totally cool.

 

Kim Montague  22:08

So in Episode Five, you delivered this fabulous gift to the world of a free workshop. And I'm super excited that now this workshop is available anytime, right? 

 

Pam Harris  22:21

Anytime, wahoo. 

 

Kim Montague  22:22

Anytime. So you can register. You get six weeks of access. Anyone can register. But when your six weeks is up, you can just register again anytime you want to. There used to be periods of time where it was available. Now. It's anytime. So if you're a pre-service teacher, absolutely for you. If you lead book studies or PD in your schools, your PLCs could study it. That's absolutely for you. In fact, there have been some interesting ways that schools have used the DMR workshop, haven't there been?

 

Pam Harris  22:53

Yeah, check it out. So when we first put it out for free, we had people come to us and say, "Okay, so we totally want to use this free workshop. But could you, like, help us spark people's interest in our district before we just say, Hey, everybody go do this thing? Could you spread it." So we offer actually some things around the Development of Mathematical Reasoning free workshop where you can have me do a virtual, we call a Spark Session now because that's literally what the first group asked for, "Hey can you help us spark some interest?" We could do a Spark Session virtually on zoom, where I meet with your people, and I give them some ideas about what's going to happen in the free workshop. I get them excited. Use the energy and the Pam Woo-woo, kind of thing. And people are like, "Whoa, this is interesting. And she's actually not gonna bore us to death." And then you have your people dive into the free workshop. Again, because now you can register any time, We can also do a live Q&A at the end. So we have little packages that we can sort of put around the Development of Mathematical Reasoning workshop. If you're interested, feel free to contact us. Go to the website and go to the book me link. And then you can just sort of ask, "Hey, we want to do some stuff to use that free workshop to its highest advantage and all those ways that can just sort of talked about." 

 

Kim Montague  24:04

Absolutely. If you want to know about the Development of Mathematical Reasoning, then you should check out the free workshop. You can find that at mathisFigureOutAble.com/workshop.

 

Pam Harris  24:15

Yeah, we'll put that link in the show notes. 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.