Math is Figure-Out-Able with Pam Harris

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
Show Notes Transcript

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
  • Brand new free info-graphics and videos to help you use the development in your classroom

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:

And we talked a lot about how we could be solving an addition problem. But if students are actually using counting to solve it, then even though they're solving an addition problem, they're using reasoning in the level before, in the less sophisticated level. We don't want that, like we, we have to accept where kids are, but that we want to move them on. We want to help them develop as more and more sophisticated reasoners using additive reasoning. So like you said, We gave some examples of how we could give an addition problem or a subtraction problem, both of those live in additive reasoning, or should, and how kids might solve them using additive reasoning. Or they might be solving them using less sophisticated counting strategies and how we can sort of differentiate between those two. Well, today I want to talk about an outcome of that. So often, in our education system, we say to kids, here are some problems, solve these problems, and we give them single digit addition and subtraction problems, lower number addition, subtraction. And then once kids get answers, we kind of move on, we say, All right, now you've got that, we're gonna teach you how to add and subtract bigger numbers. And all too often in our education system, when we do that, okay, it's about bigger numbers. Then we sort of force a step by step procedure, we force an algorithm and we say, Okay, here's what addition of bigger numbers means. And that's a fallacy. It's not what it means. It's a way to find answers, using a step by step procedure that could solve any problem in the class. We call those algorithms. They are step by step procedures, that I could give you any numbers, and you could follow those procedures to find the answers. Computers need those algorithms. Humans do not. But all too often, in our education system, we say, all right, line those numbers up. And so if I give you a random problem, like 36 plus 99 - y'all, if you're listening to the podcast, you might just scribble that number down, just scribble the problem 36 plus 99, down and line them up like we traditionally do when we give kids from those bigger number addition problems. And then we typically in American, in lots of education, we typically say to kids, okay, you're supposed to add 36 and 99. But now, don't think about 36 and 99. anymore. Now I want you to focus just on the digits just on the ones. Just only focus on the six and the nine. And in that first step of the algorithm, we actually work against kids' intuition. We work against average intuition. Because kids are at that point, thinking about 36 and 99. They're grappling with what those numbers mean, they've counted to those numbers probably at this point. But they don't really have a vast network of connections about them, they're starting to think about 36 as 30, and six, maybe as four away from 40, they might be thinking about like, it's not 20, but it's 30. And then it's sort of in between 30 and 40, a little bit further than 35. They're developing kind of a network of connections about what 36 means. Similarly, for 99, they might be thinking about 90 and nine or ooh, it's almost 99- 100. It's almost 100. They might be thinking about those. In that algorithm, step by step procedure, when we say okay, don't think about the big numbers anymore. Just focus on the six in the nine. You could conceivably have a kid go, Whoa, that feels weird. I'm thinking about 30 and 90, not six and nine. Okay, okay. Wow, I guess I'm not a math person, because I'm not - It's not my intuition. It's not my gut instinct to think about those small numbers. That's weird. Okay. All right. I'll do what my teacher tells me to do. I guess I'm not a math person. I guess in this moment. I'm proving to myself I'm not a math person, because that wasn't my first inclination. Darn. Okay. All right. I'll do what my teacher says. Y'all, we want to avoid those moments at all costs, the moments where we say to kids inadvertently, and nobody, of course, is doing intentionally, but those moments where unintentionally, we say to kids, you're not a math person. Just do what I say don't go with what your natural inclination is. So let's keep going. We want to avoid those moments. But notice if I'm doing the algorithm, I'm stuck there in that moment. Now, could you agree with me that a kid could say I'm supposed to add six plus nine? That's what my teacher just told me to do. That's the

first step:

six plus nine, could a kid go 6, 7, 8, 9, 10, 11, 12, 13, 14, 15? And count all the way up to 15 and put down the five and carry the one over into the next column. Could you agree with me that they are now looking like they're an additive reasoner, but notice that that kid was just counting. Now, that kid to add six plus nine might have thought about 9, 10, 11, 12, 13, 14, 15. They might have started with a bigger number. Ok. We prefer that. That's a better counting strategy. Better meaning it's more sophisticated. I'm thinking about adding on or counting on from the larger, that's great. But I'm still using a counting strategy. If I'm following that step by step procedure. Now I'm over in the 10s column, then I might think about okay, now I have a one, a three and a nine. I've got to add those together. It's so important. How does that kid now add that carried one that carried from over. That regrouped one, which really represents a 10. Now, as they're thinking about that, one, three, and nine, how do they do that? Could you agree with me that a kid could conceivably count again, they could be like, okay, 9, 10, 11, 12, like they could add those together. They could - or excuse me, count those one by one, get that correct answer, write it down. And now we think, Oh, good. Now that you've written that down, you've gotten that - those two numbers if I add, by the way, that's 13. So if I add those together, now I've got one, three, next to that five, I wrote down earlier. And so I'm saying to myself, okay, that's 135. And the kid writes that down, and we're like, Yay, you're an additive reasoner, because they got the answer to an addition problem correct. So we're like yay, you're an additive reasoner, when in reality, that kid was thinking using counting strategies the whole time. And if we were doing this live right now, I would draw a frowny face on the board. Because too often we focus on the fact that the kid got that answer correct. Cause we're like, good job, you're thinking additively because they had an addition answer correct. But how were they doing it? If they were doing the algorithm, they may have been using counting strategies. Not sophisticated enough. Now, what we would suggest, Kim is that in this moment, if a kid is using those counting strategies, we do some more work with smaller numbers to get that kid thinking additively with smaller numbers, before we maybe move on to try to get them to think additively with bigger numbers. Now, you might be thinking, Pam, Kim, my kids are doing the algorithm. But when they do it, they're not using counting by one strategies. They're using additive reasoning. So again, back to our 36 plus 99, they might be looking at that six plus nine, my kids Pam, my kids are thinking about six plus nine as, let's see, I could do six plus 10, that 16. I'll just backup one to get 15. Or they might be thinking about, I can think about six plus nine as nine plus six. And I can go from nine to 10 and get to that friendly 10. Nine to 10 is one I've got five leftover 10 and five is 15. Pam, they're thinking additively while they're doing the algorithm. Okay, okay, I could agree with you. I could submit that you could be correct, that your kids are thinking additively as they're doing the traditional algorithms. However, how big of numbers will that student ever think about additively, if they are doing the traditional algorithm? Now think about the algorithm, even if I have a long number, lots of digits, added to another long number, lots of digits. Every time I'm thinking about columns of digits, columns of single digits. The biggest I'll ever be able to think additively about those numbers is with those single digits. And what we're suggesting is that's not big enough, it's not sophisticated enough, it's not enough work in getting kids thinking additively about those big numbers. Now, somebody's probably thinking, but Pam, you can't think additively about numbers like 36 and 99. You have to line those up. Like you have to line those up and then add the columns which can't think additively about 36 ad 99. Or can you? Now if you've listened to the podcast at all, you've heard us do this problem or one like it lots because we wanted to sort of exemplify additive reasoning. But Kim, I'm just gonna ask you, how would you add 36 and 99? What's one additive way of adding 36 plus 99?

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:

Sweet! It's catchy. I'm not the only one that has spaced. Oh that's hilarious. Okay, so Kim is an over strategy girl, and I had a feeling that that's what she was going to do. So I was glad I asked her. So she's like, I'm gonna add 36 plus too much and then backup. Totally, totally cool. Could you also do a little give and take if I've got six plus 99? I could think about Well, 99 is almost 100. So I could add just 100 plus what's left over 35 135 is 135 cannot hurt 90. Okay, cool. So those would be some additive ways of thinking about bigger numbers. And we need students to think about bigger numbers. additively not not using counting strategies. Alright, so we thought we'd spend some time on this podcast thinking, are exemplifying how that can happen with multiplication as well. If multiplicative reasoning is more advanced, more sophisticated, I'm grappling with more and more things simultaneously, multiplicative reasoning than I am in additive reasoning. How do we build that multiplicative reasoning? What does it look like to solve a multiplication problem? Using multiplicative reasoning? And what does it look like to solve a multiplication problem using less sophisticated additive reasoning, or less sophisticated counting strategies? Is that even a thing? So here's an example. multiplication problem. Let's say I'm baking cupcakes. And I know that I want to give three cupcakes. For the four people. I'm baking cupcakes for four people. Each of them needs three cupcakes. How many cupcakes do we need? Could you picture a kindergarten kid solving that problem? Kim, what might we see a kindergarten kid do to solve that problem?

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:

Yeah, absolutely. Let's talk about that. Because our big upshot is how do we help students progress? Let's acknowledge where they are. It is a necessary starting point. We need kids to solve the problems the way they naturally do it. But then we can't rest on our laurels. We can't stay there. We have to help them progress. So in the little bit of time that we have to finish out, let's mention a few things. So what we are not suggesting? Not suggesting is traditional review. Traditional review where we say to kids, you should have learned this already. And we do the same thing that has been done before, go memorize your multiplication tables, like whatever it is, that hasn't worked in the past, if we try to redo that, it doesn't work, like redoing stuff that didn't work in the past is not now going to magically work this time. So we're not recommending traditional review. We do recommend that we acknowledge where they are, we know our kids. Of our mantra, know your content, know your kids, you got to know your kids. So you got to figure out where they are in the landscape. And then know your content means how do we then expose them and help them construct the more sophisticated things? Well, that's why we love Problem Strings, because Problem Strings are all about multiple access problems, problems that give all kids access. No matter where the kids are, we give them access to the problem, they can solve it using what they know. But then in the share, we expose them to more sophisticated strategies, we help them connect what they were doing to something that's more sophisticated, that's higher in that level of reasoning that is using something more and more simultaneous. So we give them a problem, students solve it however they can, then we ask certain strategies to be shared. And we say, how were you thinking about that? We help connect less sophisticated strategies to more sophisticated strategies. And then so important. Next problem. And the next problem is a chance for that student to then say, oh, what, how can I use what I just was exposed to? How can I use that on this next problem? And that is the process of construction, where they get a chance to go, I'm going to grapple with these relationships, using what I just sort of experienced using these things that are pinging in my head from the problem before, I'm going to now attempt to do that in the next problem. And then beautifully, we share strategies and students go, Oh yeah, that's how I could have. Even if they weren't successful in that next problem, but they were grappling with it, they now get a chance to go Oh, like that. And then the teacher expertly helps connect strategies, and then brilliantly the next problem. So you might have heard oh, so let's go do problem talks. Let's just go do number talks, because then they'll see what everybody else is doing. It's not enough to just see and experience once what someone else is doing. We need that next purposely planned problem for the chance to grapple with trying it. And then we need that next purposely planned problem with the chance to grapple with trying it, then we need that next purposely planned problem. Now maybe we're grappling well with it, we're using it well, now we get a chance to Oh, oh wait, that next purposely planned problem gives us a different way of applying it or maybe it's a push back problem where that strategy doesn't work so well. Like Problem Strings are designed on purpose, to give us those opportunities to construct those relationships in kids' heads. That is a huge way, it can be your go to strategy teachers to help students progress from where they are to that next more sophisticated thinking.

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