# Ep 68: Revisiting the Development of Mathematical Reasoning Pt 2

October 05, 2021 Pam Harris Episode 68
Math is Figure-Out-Able with Pam Harris
Ep 68: Revisiting the Development of Mathematical Reasoning Pt 2

We've described how students can get away with solving multiplication problems with counting strategies. What about division problems? In this episode Pam and Kim explore the Development of Mathematical Reasoning even further, and discuss how important it is that we help students become more sophisticated. It's more than just answer getting.
Talking points

• Solving division problems with counting strategies

Find the infographics here: https://www.mathisfigureoutable.com/blog/development-4

Pam Harris:

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

Kim Montague:

And I'm Kim.

Pam Harris:

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. Y'all, 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:

So last week, we revisited the Development of Mathematical Reasoning, we had-

Pam Harris:

In a little bit of a less precise way. I misspoke several times . Kim saved me. Anyway, sorry. Kim keep going.

Kim Montague:

No, you're good. So we tackled that a little bit in episodes five and six, and we decided to have a revisit. And so you got to add on a couple of things that you felt like were really important to share. And we did a couple of examples of counting and additive thinking for some different problems.

Pam Harris:

first step:

Kim Montague:

Um, 36 plus 99. I would add 36 plus 100, which would be 136. But I added one too much. So then I'm going to remove one at the end to get 195 .

Pam Harris:

100 and...?

Kim Montague:

I'm sorry, 135.

Pam Harris:

Kim Montague:

So a kindergartener might say, I have four people. So they might draw four circles or four people even. And they might deal 1, 2, 3 to the first person 1, 2, 3 to the second person 1, 2, 3 for the third person, when we got three, were the fourth person, right, and then go back and count 1, 2, 3, 4 or 5, 6, 7, 8, 9, 10, 11, 12, and end with 12 cupcakes.

Pam Harris:

Bam, they just solved a multiplication problem. They must be reasoning multiplicatively. Or were they just using counting strategies to solve that problem? So young students can solve multiplication problems using counting strategies. That's great. For a kindergartner. When we get to third grade, fourth grade, we need to start nudging them to well third grade, we need to start nudging them to multiplicative additive reasoning for sure, like repeated addition. And then multiplicative reasoning. I may have just given away what I wanted just a little bit. Kim, what would it look like for a student to solve that problem? You just exemplified counting, what if they were using additive reasoning to solve that cupcake problem?

Kim Montague:

So additive reasoning, is going to look like skip counting. And so they might say, three for the first person, and then six, 9, 12. 12.

Pam Harris:

Yeah, they're sort of adding three every time. And they're able to do that using additive reasoning. And they've solved a multiplication problem additively, right. That's super for a kindergarten or first grade kid. I think it'd be okay for a second grade student. Although I'd kind of maybe help them I would hope some of them were thinking about three, six, that's two people, we could double that to get four people. Those are some sort of interim strategies that are kind of half of the additive half multiplicative. But by the time we get to where we want kids to be thinking multiplicatively we need kids thinking multiplicatively, about those three cupcakes for four people. What would that look like?

Kim Montague:

So if they don't know three times four is 12, a multiplicative thinker might think of, I know three threes is nine, and another three is 12. They might think about two threes is six, double that to get four threes is 12.

Pam Harris:

If they happen to know something crazy, like no, that's not a good example.

Kim Montague:

Ooo, if they happen to know five threes is 15. Then four threes would be 12.

Pam Harris:

Cool. So depending on what they know, if they sort of reason from there, then we would hope that would be an example of multiplicative reasoning. So let's think about division. Because division lives in multiplicative reasoning. Can we have a division problem that kids could solve at all of the different levels? Again, with our emphasis on, let them start where they are, but then we want to help them progress to think more and more sophisticatedly. So let's say random division problem. What if I said 36 divided by nine?

Kim Montague:

Why don't you give us a counting strategy for this?

Pam Harris:

Okay, I'll do this one. 36 divided by nine. We actually interviewed a student not too long ago, with 36 divided by nine, and the student drew nine circles on her paper. And she said, Okay, we got nine circles, and I need to divvy out and deal out 36. And so in each of the circles, she drew, like, one tick marks. A one. And then like one in each of the Nine Circles, to in each of the night, like she's literally drawing tally marks. So now each circle has two tally marks. And as she's doing that, she's counting. So she's, I've got nine, and then 10, 11, 12. And she did it all. Now I've got eighteen, and then she keeps doing it. Now I've got 27 and that she keeps dealing out and now I've got 36. And then she said, okay, I've dealt out all 36 because she counted each of those tally marks. I dealt out all 36. Then she looked back at each of the circles, and each of the circles had four tally marks, it was almost as if it's a surprise at that point. She's like aha! 4. 36 divided by nine is four. Who knew? That would be a counting strategy, for division, a multiplicative problem by nature by using a counting by one strategy. We've also seen kids where they're bigger numbers, when they've done that. But they don't actually when they see the four in there, they actually then have to count. She didn't have to count the four because she could just recognize, she's subitize those four. But if a quotient is a bigger number, than they might have to actually count what's left in that circle so that's one more time where they're sort of counting. Okay, so 36 divided by nine. Kim, what would be an additive strategy for 36 divided 9?

Kim Montague:

So additive, they might be thinking about 36, subtract nine, to get 27, subtract nine to get 18. Or maybe skip counting nine, if they skip count by nines. Well, they might say 9, 18, 27, 36, and then the number of counts would be their quotient.

Pam Harris:

So either the number of subtractions or the number of sort of skip count up, those would be additive reasoning ways of solving that division problem. Yeah, totally cool. Okay, so then what would a multiplicative reason, like how do we hope kids think about 36 divided by nine? Okay, I want to go first. So ya'll, we hope students for both the multiplication and the division problems - so I don't know if you heard when, when I asked Kim the three times four cupcake problem. The very first thing she said is, well, if a student doesn't just know four times three, then we would hope they would think multiplicatively this way, well, I would say the similar thing for division, right? If a student doesn't just know, 36 divided by nine is four, or doesn't just know, well let's see I'm dealing with nines. What do I know about nines? Oh, well, I know nine times four is 36. Therefore 36 divided by nine is four, right? We would kind of hope that eventually we get there. That's like our ideal. But we can also reason if we don't know that. And we need this reasoning for all of the myriad facts that we don't just know. So of course, we hope kids know 36 divided by nine, but boy, there's a lot of division problems that no one's gonna just know that we need them to be able to think similarly about. So what would be a multiplicative reasoning way? I did one Kim, now you.

Kim Montague:

So I know that a lot of kids can think about 18 divided by two, or 18 divided by nine, sorry, is two. 18 divided by 9 is two. So knowing that they can also think about that twice. So 18 divided by nine twice, would be 36 divided by nine, so the answer will be four.

Pam Harris:

So kind of like if I'm thinking about, I know that nine times two is eightteen, if I know that, and I double the 18 to get 36, then it's gonna have to be nine times four is 36. And I can use that reasoning about multiplication to think about division, you kind of said it in a more division way, which is totally cool. But I could also think about in a more multiplicative way, or multiplication way to find that division problem.

Kim Montague:

So this is helpful to parse out the difference between counting, additive, multiplicative thinking, but I also don't want to leave today without a little bit of application. So how can teachers help their students if they're stuck in counting or additive, and they want them to be in multiplicative thinking?

Pam Harris:

Kim Montague:

Absolutely. So Pam, one more thing. Before we go, we got the chance to interview a lot of students recently, right? Lots of kids to do this special project. Tell us about that a little bit.

Pam Harris:

I'm so glad the REL South East - the Regional Education Laboratory, Southeast Florida State University - asked us to do a project where we created some professional learning around the Development of Mathematical Reasoning. And then we created these infographics. We created one page infographics to really exemplify and make it visual, each of these levels or these domains of reasoning. And they've just come out and we're so excited, they're totally free, you can absolutely download them. And the most exciting part of all of it that we have is graphical way of making the thinking visible. But also, we put in these videos of real students solving real problems where you can see exemplified kind of what we just tried to do in this podcast and the last episode, where you actually get to see us students solving a multiplication problem using counting strategies, using additive reasoning, or using multiplicative reasoning. Get to see it in action, as we interviewed all these students, and then we studied how they solved particular problems. And then we sort of sorted them out and we gave you this really experiential way of learning more and more about these different levels of reasons. Now, in the last episode, and in this episode, we only got to counting strategies additive reasoning, multiplicative reasoning, y'all we exempl fied proportional reasoning. Yo can see real students so ving proportional reasoning pro lems using proportional reasoni g or maybe using multiplicative easoning, maybe using additive easoning. Same with functional reason. Secondary teachers, i you're like Pam explain better like more, I need to see fun tional reasoning in action. am! We've got it for you. C eck out these wonderful resour es. I'm so glad again that they sked us to create them. Kim, tell us about where they c

Kim Montague:

Oh, yeah. So you can check out the blog at mathisfigureoutable.com, and cli k on learn now and you'll ind the blog. And then if you ant the direct link, you can jus check out the show notes h re.

Pam Harris:

Yeah, absolutely. So download those infographics, you can watch the video of those ki s, you're gonna love learning m re and more about develop ng mathematical reasoning. So if you want to learn m re mathematics and refine your m th teaching, so that you nd students are mathematizing m re and more, then join the Mat is Figure-Out-Able Movement and help us spread the word ma is Figure