Math is Figure-Out-Able with Pam Harris

Ep 94: If Not Algorithms, Then What?

April 05, 2022 Pam Harris Episode 94
Math is Figure-Out-Able with Pam Harris
Ep 94: If Not Algorithms, Then What?
Show Notes Transcript

We want students to be real problem solvers, and we know that means much more than helping them get correct answers. But we never want to put students in a position where they are disadvantaged because they don't know the traditional algorithms. We actually want them to have a much deeper and richer understanding of mathematical relationships. In this episode Pam and Kim begin a series exploring the major strategies for each operation that students need so that they can solve ANY problem that's reasonable to solve without a calculator. 
Talking Points:

  • Pam's journey from learning only step by step procedures to mathematizing
  • How this series can take Pam's published books a step further
  • How Kim, and many other mathematicians approach problems
  • Ownership of the major relationships empowers students with choice
  • Honor students thinking, but don't leave students where they are


Get the download!: mathisfigureoutable.com/big

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 y'all, 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 keep students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what? And y'all, we're gonna answer that question more today than possibly we ever have.

Kim Montague:

Hey, before we start, though, I want to say thank you so much for all the reviews. Pam, I'm gonna have you check this one out. We got a review from Math Count. And it says that the subject is not just for teachers, and it says, "Everyone should listen to this podcast to improve their own understanding of quantitative reasoning and doing math. I enjoy every single podcast, and I feel like I'm in the room participating in the conversation." Isn't that so great?

Pam Harris:

Bam! Nice!

Kim Montague:

Yep. Yay.

Pam Harris:

That's exciting. Thanks. We love reading the comments. And we really appreciate you leaving ratings. It helps others find the podcast and frankly, it makes us feel good. So we appreciate that.

Kim Montague:

Okay, so in this episode, we're going to tackle something that we have hemmed and hawed about for, gosh, a really, really long time.

Pam Harris:

Back and forth and forth and back.

Kim Montague:

And that's really because of what we believe about teaching and learning. So here we are.

Pam Harris:

So what are we crying uncle just a little bit? We're about to... tell. So if you've been listening to the podcast at all, if you know our work, then you know that we believe strongly that learning is about experiencing. It's about making mental connections based on like literally diving in and doing the work and experiencing something. It doesn't work well for someone to just tell you something. Like if you just like say the thing, then that doesn't help your brain make that neural connection, and it's not as strong, and you don't own it. It's more of like a fact that goes in. It's not really something that you live and breathe and own and then can use well. But -

Kim Montague:

Well, that's also one of the reasons that we do problem strings.

Pam Harris:

Absolutely.

Kim Montague:

To build those experiences.

Pam Harris:

Yeah. Like when people told us you can't do that

Kim Montague:

Yeah. Yeah. on a podcast, and we were like, well, then we're not going to do a podcast. Like we feel so strongly that we really want to

Pam Harris:

We want to take people who were like me, and help people develop as mathematicians, not just telling you a bunch of stuff. It won't work, like the podcast wouldn't go over well, if we did what people were telling us to do, because we'd just be telling,. I mean, some people would listen, but we wouldn't actually have the potential to change, to really influence the change that we hope to really empower people. That's our whole gig, right? turn them into more like I am now. Like, not the rule follower, that I believed what math was. Oh, we want to open the world to what math actually is. So, Kim, sometimes I will hear you talk about how it's not for teachers to, how do you say that? Sometimes people will ask us questions on like, in the Facebook group, so the Math is Figure-Out-Able Teacher Facebook group, they'll say, hey, you know, what about this? And you and I will kind of get on behind the scenes were like, this isn't what we want teachers to print out and put on their wall. It's not, help me, it's not something that you give them in their take home folder. It's not some student notes. It's like -

Kim Montague:

We're about to share some things today that we don't think that you can just whip out on a piece of paper and put up on the wall or share with your students and say, "Okay, now you got it." Because it's not. It's not how learning occurs.

Pam Harris:

Here it is. Here it is. Woo! We own it. Good. Moving on. Yeah it doesn't work that way. In our introduction, I say, we answer the question 'If not algorithms, then what?'. Well, we're gonna answer that question more than perhaps ever before. Because we want to today, lay out what are the strategies students need to own down deep so that they can use to solve any problem that's reasonable to solve without a calculator?

Kim Montague:

I'm gonna interrupt you. You said today, but we think it's so important, and there's so much to share that we're going to share like over the next couple of weeks.

Pam Harris:

Yeah. So this is the beginning of a series. We've planned a series where we're going to dive deep into each operation, to answer the question, if you are a teacher and you're like, well, my students are supposed to be able to add. Like that's a thing. And it might be whole numbers, it might be decimals, my students are supposed to be able to add. And maybe all we've done or all we've known in the past - for sure all I had ever seen was a traditional algorithm where I line stuff up and it was all very digit oriented. If kids don't get that, if that's what you're suggesting, Pam and Kim that we don't show them the steps and give them you know, step one, step two, step three, here's what you do, you line these digits - if we don't do that with students, then what? Like, are they left to their own devices? Like they're just supposed to fuss around and fumble and look at a problem and have no idea what to do? And they're just supposed to, like reinvent something every time they hit another problem? No, no, no, like, not at all. What we want to do is create students who can, for example, add fluently, like a mathematician would add, not mimicking, not like a monkey would add, not like pushing a bunch of buttons on some sort of machine that nobody knows what it is, which is kind of what the algorithms are. We just put them in our brains, and we just follow the steps. What do mathematicians do when they add? And so we will have an episode where we'll dive deep into addition, we will dive deep into subtraction. Today, we kind of want to lay out what are we talking about? Why? Why this idea of what - how do I say this? We've done a lot of that work on the podcast, right? Like we've already done podcast episodes where we played with fractions. We've played with fraction multiplication. We've done podcast episodes where we played with the partners of 10, or partners of 100. And played I Have You Need. Like we've done all sorts of things, where we've given you listeners ideas of what you can do to help students think and reason more like mathematicians. What we haven't done is a little bit of the unique part of my work, which is to lay out for an operation: what are the major strategies kids need to know/to own so that they can solve any addition problem without reaching for technology? Or they can solve any multiplication problem without a calculator? What are the major strategies? If you listen to a lot of really good people that talk about numeracy and talk about building number sense, from my perspective, most of them do a great job of throwing out some ideas. Like, it'd be great if kids could do this, it'd be great if kids could do that. But what very few people have ever done is lay

out:

these are the major ones. Let's identify if I have a kid who doesn't have an algorithm, doesn't have some memorized step by step procedure, then what do they need? What are the relationships that we need to ensure they own that then we know they're good? Like we can give them any problem that's reasonable to solve that a calculator, and they're good to go. We're gonna lay those out for the first time.

Kim Montague:

Which is a big deal!

Pam Harris:

It's a big deal, yeah. It's actually funny. I'm gonna tell you a little bit. We believe in being real. Yeah, y'all? Being genuine. So we were talking with the team this morning about the fact that we were about to record these podcasts. And somebody said, as we said, you know, we're gonna lay out what the major strategies are. Somebody on my team said, "Yeah, like, when I read one of your books, I was like, okay, but what are the major ones? Like, we just played around with a bunch of them? Like, what are the-?" And I was like, "Yeah, I know, I know. Do you know why I didn't put that in my book?" And she kind of looked at me. And I was like, "Because I didn't know what they were." Like, early in my journey I knew that there was a collection of things. I knew there was a bunch of relationships I was using to solve problems. But I had not yet really hammered down what were the major important ones, what were the ones that we've got to work with students on, or we've left them with a hole. We've left them with some problems that they will be really inefficient to solve. I don't want to do that. I don't want to have this place where students are going to have to do too many little steps and they'll be so inefficient that then somebody could say, see, that's too inefficient. We've got to give them step by step procedures so they're not so inefficient. No, no, no, no, no. I wanted to make sure that we thought through for each operation, give me any problem that's reasonable to solve without a calculator, do I have a relationship that leads to a strategy? That yeah, bam. I can do that one. Yep. Give me another. Yep. Yep. I've got a relationship for that one. It took me building my own numeracy. It took lots of conversations with Kim. It took lots of thinking about all of the kinds of problems that are out there. And then about several years ago, I was like these. These are it. And we put it to the test. And yeah, like we've hammered on these are the more important ones that we want students to own.

Kim Montague:

And now we're ready to share.

Pam Harris:

Hey, um, one of the things that we've talked about is that Kim, I just said, one of the important things for me was to work with you and I would say, "Hey, Kim, here's a problem. How would you solve that one?" I would listen. Now, not just you, I threw it out to lots of people. I mean, everybody I worked with I threw out problems, and I listened. We took note and really, you know, how can I make those visible and all the things. One day, I will never forget this day. I don't know if you remember this day. But one day you and I were working together, and you said something. And I was like, "Wait, what?" And you were like, "Yeah." And I said, "Whoa, whoa, whoa, you're telling me..?" I don't know. I'm putting words in your mouth. Do you want to tell me? Do you remember what you said?

Kim Montague:

Oh, about not necessarily automatically knowing which one was best?

Pam Harris:

Yeah.

Kim Montague:

Yeah. So I think in that moment was a moment - I don't know exactly what I said. But it was something along the lines of oh, it's not that I know what the best strategy is, every time for every problem. It's that I play with them. And then if I, if I don't like the strategy that I'm about to use, I realize that's not really what I want to do, then I back out of it and I try a different strategy. And I think that was noteworthy to you. Because up until that moment, you were like, "Oh, my gosh, how do you know which strategy is the best to use every single time?"

Pam Harris:

Well, and I don't think I ever said that out loud. It was more of a mindset for me. Because I had come through learning math as fake math. And so it was step by step procedures. I was really clear in that atmosphere of just rote memorizing facts and steps and procedures that I needed to know for that problem. It's this step by step procedure. But for that one, nope, nope, that's a different. See that little change in there? Nope, that's got to be this other procedure. And then I had to be really clear what were the different steps in those procedures. And to me, it was so ingrained in me that that's what math was, because I was good at it. Like I could tell, oo for that problem, it's got to be that thing you told me how to do. But for those other ones, no, it's that other one that you've sort of told me how to do. So since I had that perspective I think what happened, if I remember correctly, I was actually listening to you talk out loud. And I said something like, "Hey, for this problem," and you said, "Well, I would, oh, no, actually, let me get back to my first." And in that moment, I was like, what? Did you just like? What? I mean, I was speechless. And you go, "Pam, no, that's what I do. Like, I play with relationships. And absolutely, sometimes I'll go back to the first one that I tried, because it's gonna turn out to be the one that is kind of the most fluid for those numbers, or the one that strikes me that day." I tell you what, I walked around with my jaw on the floor for I don't know how long after that, where I was like, okay, let me make sure, how do I make sense of this? How do I integrate this? Into what - Because I was so in that moment clear that it was not what I thought math was originally, even though I had been already changing and developing numeracy and everything. It just hadn't occurred to me that doing mathematics is far more about playing with relationships until you find one that's sort of slick, or, frankly, until you find one that illuminates something about the problem that's interesting.

Kim Montague:

Yeah. It's so much about preference.

Pam Harris:

Yeah. And when you said that, it was like, oh! That just freed me up. In fact, I think you said something about, it's more about preference, and not about rightness, not about perfect, not about choosing the exact correct. And it freed me, it freed me to go, "I can just play. I can just try stuff. And then decide which one I liked the best?" And all of a sudden, the idea of playing with math took on a completely different bent. The pressure was gone for me to know the one and only one right way to know the best right way. And without that pressure, I think that helped my brain work better. It somehow took some of the time pressure off, because all of a sudden it was like, oh, I don't need to know it. I mean, I was already saying we don't need to be fast. But somehow I still thought I needed to be fast at choosing the strategy. Isn't that crazy? It's crazy how our minds are. How we you know, if you have this sort of background, understanding or experience that kind of colors everything. And I'm so grateful that you help me color that.

Kim Montague:

You're welcome. And how much fun is it now that we both will be in a conversation and we'll solve a problem. It may be the same way, maybe a different way from each other, but then we continue to look at it and we go, "Oh, like that would have been a cool strategy to use."

Pam Harris:

There's been some more complicated problems that we've solved. And then as we talk, we actually come up with a third that we both -. And it illuminates relationships in the problem that builds us, that makes us more complete thinkers, more robust problem solvers, because now we own even more relationships. It's not just about getting an answer. It's about playing with those relationships. So fun. And Kim, I'm reminded, we've quoted the study on the podcast before. Ann Dowker did a study where she asked mathematicians, "Hey, here's some problems to solve, if you'll solve those, and I'm going to kind of listen in and you know, tell me how you solved them." And of all of those problems that mathematician solved, 4% of the time those mathematicians used a traditional algorithm. So 96% of the time they were playing around with relationships. The reason I bring it back up today is that then I also found it fascinating that part of her study was that she went back to some of those same mathematicians with the exact same problems on a different day, and said, "Hey, I have some more problems. If you don't mind, I'm just going to listen to it again, you know like how are you solving these?" And didn't tell him they were the same problems, and had the crazy fun outcome that on that different day, mathematicians often used a different strategy. Because it was all - go ahead.

Kim Montague:

I was gonna say we experienced that here. A couple episodes ago, with the three fourths of two fifths. You had heard me solve the problem one way before and thought, oh, that was Kim's favorite. And she'll do it again, and I didn't.

Pam Harris:

Totally true. It was not what I was expecting. I was like, "Okay, there we go. Podcasts are live recording." It's a good thing we don't mind looking slightly silly.

Kim Montague:

So that's what we want for kids, right? For students, for anyone, really, we want them to be empowered to have choice.

Pam Harris:

Yeah. Because if you have only one way to do something, and I've handed it to you, it's this rote memorized way. And that's all you've got. Or even if you've developed a strategy for an operation, you've got a strategy that you can use. But that's all you have. That's not empowering. That's not choice. We need kids to own these major strategies. Because once they own them that's like deep in their heart and their soul. And they can look at numbers and they can like, ooh, now they can let the numbers influence which of those they choose. That's being empowered. It's not choice, it's not power, if you don't have a choice. Kim, the reason I bring this to a point here is all too often I will hear teachers say, "Yeah, yeah, it's whatever a kid can do. You know, like, for so long, we've like forced them to do stuff, but some of my kids I'm just gonna let them solve it the way they do. And I'm going to honor their strategy and honor their thinking." Well, yes, we absolutely want to honor kids' thinking. We must start where they are. But we can't leave them there. If we leave them there, that's not choice. That's not being empowered. We've got to help them own these major relationships. Now they have power. That's empowering to go, "Hmm, I'm gonna look at this problem. And I'm gonna let these numbers, this structure of the problem, and let that influence how I solve that problem today." That's being empowered. That's what mathematicians do.

Kim Montague:

Okay, so you listeners are not going to want to miss the next several episodes, where we share what we believe are the major strategies for every operation. And as a bonus, we are so excited to share with you for free, the ultimate download that you're going to want.

Pam Harris:

Y'all you're gonna want this download!

Kim Montague:

Absolutely. We've outlined all the major strategies with examples and you're going to love it. You can find that at www.mathisFigureOutAble.com/big.

Pam Harris:

Yeah, because it's big, we had a nice conversation about what that URL was gonna be. And we decided to land on 'big'. Big, because this is big. Like we are putting out these strategies. Now you might be like, "Pam, why did you take so long to do it?" Well, I mean for several reasons, like we already said, because we don't like to just tell, we like to develop it with people. But I will be honest with you also, because I wanted to make sure that I was set. Like I wanted to make sure that yeah, I do think these are the major relationships. So we welcome your comments. We welcome you to dive in, dig into those relationships and wonder, do you have one that we didn't list? Is there one, that you're like, "Oh, we don't need that one because we have this one." I'd love your feedback on what I've decided are those major relationships in order to solve any problem that's reasonable to solve without a calculator. Okay. Alright. 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!