Pam Harris  00:02

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

Kim Montague  00:09

And I'm Kim Montague.

Pam Harris  00:11

And we are here to suggest that mathematizing is not about mimicking or rote memorizing. But it's about thinking, about reasoning, about creating and using mental relationships, that math class can be less like it has been for so many of us and more like mathematicians learning and working together. We answer the question: if not algorithms, then what are you teaching? Alright, so happy, happy anniversary to the Math is Figure-Out-Able podcast. Wahoo! The podcast is one year old because this is the 52nd edition. Yay!

Kim Montague  00:51

So fun. Alright, so we have decided that we're going to do a highlight reel for this episode. And we've gone over our most listened to episodes and a couple of our favorite episodes and pulled a few highlights. So the first up today is where you set the stage for how you're different. Let's listen to a clip from Episode Two, where you make that clear.

Pam Harris  01:13

Yeah, sure. So there's a lot of really good people out there. And I think most of us, when I talk to colleagues at the university, when I go to conferences, most of us agree on that sort of first point that I made, that math teacher is about knowing your content, knowing your kids that we need to listen to students. We need to notice what students are thinking. We need to notice the way that they're attacking problems. We need to notice the strategies that they're using, We need to notice the way that they are thinking about relationships. There's a whole thing out there that the Drexel Institute came up with, the notice and wonder that's wonderful. A lot of people have picked up on it. And so there's people out there that are doing rich tasks and three act tasks. But for the most part, all of that, all of those really good people are agreeing that we need to listen carefully to kids' thinking. But their end goal is still the algorithm. So they do a lot of really good work towards getting students to be able to estimate and use reasonableness, and get some number sense, and all that stuff. Like, of course we should do that. But their end goal is the algorithm, is getting kids good at repeating these steps in order to get an answer. Their goal is to get kids good at mimicking all the steps all the time. They're headed towards the algorithm. Not me. I'm not headed towards having kids just do this one thing all the time to solve every problem in that class.

Kim Montague  02:40

That's a pretty bold statement you're making that algorithms are not your goal.

Pam Harris  02:43

Yeah. And it's where I differ. So it's one of the reasons that we're talking about this on this podcast so early is to sort of say, "My work isn't just about getting kids kind of good, as you know, like some number sense and some being able to, you know, kind of reason a little." Like, that's actually my goal. My goal is number sense. Number sense isn't just this little thing about being able to estimate or being able to look at your answer and make sure that it's reasonable. Number sense is a huge part of what it means to be a mathematician, that I use relationships and connections I know. That instead of looking at a problem and saying, "Oh, this is an addition problem, I must now do these steps." Or, "Oh, this is solving a proportion. I must now cross multiply and divide." Instead, it's looking at the problem and saying, "Ooo, how are these numbers affecting me? What relationships do I see here? And how do I want to then let those relationships influence how I'm going to solve the problem?" I let the numbers and the structure dictate the strategy. It's not about one set of solution steps all the time. It's about, "Ooo, based on what I see in this multiplication problem. Ooo, I see these numbers, this relationship. I'm going to use those relationships to solve this problem."

Kim Montague  03:56

And you still advocate using relationships to solve problems rather than algorithms a year later. Even more boldly now, I'd say.

Pam Harris  04:03

Yeah, because I'm trying to be more clear about that really, very important point about what mathematics really is about. It's not about mimicking. It's about using what you know, using relationships to solve problems. Okay, cool. So what's next? What clip are we listening to next?

Kim Montague  04:23

Okay, so probably one of the most helpful, where we parse out something that's super confusing to a lot of people. And that's models versus strategies from episode nine. So important.

Pam Harris  04:35

We've been posting for a while on social media, some graphics that asked people - you might have seen him on Twitter or Facebook or Instagram - that ask people whether the examples they see are two models, two strategies, or one of each. And what we found is that not only are parents confused about the difference between a model and a strategy, lots of teachers are too. Even though the standards call for different models and strategies, we've got a lot of confusion out there about the difference between them and how we can use them to better teach Real Math.

Kim Montague  05:08

So how can we help teachers and parents?

Pam Harris  05:12

Yeah. So in today's podcast, we're gonna define models and strategies. And let's start there. And then we'll talk a bit more about how we can use them better. So let's start with definitions. So first of all, Kim, what is a strategy?

Kim Montague  05:26

Okay, right. So a strategy is going to be the way that you mess with numbers. It's how you solve the problem, the relationships that you use in your head.

Pam Harris  05:35

Okay, so if a strategy is how you mess with the numbers, then what's a model? 

Kim Montague  05:40

The model is the way that you represent those relationships that you've used. It's a way to represent to others what's happening in your head so that you can communicate the mathematics.

Pam Harris  05:50

It's kind of the picture of what's going on. Yeah. Alright. So we'll do a podcast later on the word model in math, because there are a lot of different ways that word is used. But today, to parse out the difference between strategy and model, we're going to really focus that the strategy is how you mess with the numbers. And the way we're talking about model today is that it's what your strategy looks like, the way you've represented how you mess with the numbers.

Kim Montague  06:18

And I think what's so confusing for so many people is that given a problem, you can solve it with a few different strategies. And each of those strategies can be represented with a couple different models. And so it's kind of like what's going on?

Pam Harris  06:31

Yeah, exactly. So let's give some examples. For example, if you're going to solve an addition problem, and y'all if you want to get a paper and pencil out to sort of follow along with the relationships, this might be a time to do it. So if you're solving the addition problem, like 48 plus 36. 48, plus 36. You could - in fact, you might want to pause the podcast and actually solve 48 plus 36, and then come back and hear some different strategies. We always like to have people mess with the numbers before we ever superimpose someone else's strategy. So for the problem 48 plus 36, you could think of the strategy that we describe as Add a Friendly Number, you could start with the 48. And say to yourself, "Instead of adding the 36, I'm just going to add a friendly 30. So 48 and 30 that's 78. And now I've still got that six hanging around. So 78 and that six, let's see, that's at 84." So you could think about that 48 plus 36 as adding that friendly 30 first. That's the strategy: Add a Friendly Number. I can say that with words. I can write it with equations. I can represent that relationship on an open number line, which is the one we prefer, especially with beginning learners. Because the open number line is so nice to show those relationships that makes that thinking really visible. What would that look like, with the strategy I just used? I'd write down the 48. And then I would draw a big jump of 30. And I would write down that we got to 78, right? And then I would say, "Okay, I've still got left over six." And so then I might add the two to get to 80, and then the leftover four to get to 84. But the big point of that strategy is that we added a friendly number. And the model that we used was an open number line. 

Kim Montague  08:12

Okay, so we defined model and strategy. And you and I are clear that a problem can be solved with different strategies. And then also they can be recorded with different models. So I'm going to ask you a big question now. Which of those is more important? Which should teachers and parents be more concerned with? Strategies or models?

Pam Harris  08:33

So in a nutshell, if you force me to answer this question, teachers should care about both. But if I have to make a choice, it's more about strategy. Like you just told us earlier about your husband, who has some strategy. That's so important. He's not just mimicking a bunch of steps, regurgitating a procedure that he learned. Like, he's got some ways of dealing with relationships. That's so important. Now, if he was my student, ideally, I would want to build on that. I would want to represent what he was doing in such a way that he could then build other strategies. That would just help him gain more relationships and have a more dense brain structure. So models can help bring the thinking forward, make it visible, so more students can pick up on that particular strategy. More students get more dense, they own more relationships. But it's those relationships, that's the strategy. So models are important in communicating what the strategies are doing.  Yes, great episode. So y'all check out episode nine, if you want to hear more on that. Also, MathStratChat is one way to help parse out the difference between strategies and models. I throw out a question to the world. And we get back great strategies, how people are reasoning their way through the problem. And people represent those strategies, most often with equations because they're typing. But sometimes, and we love it when this happens, sometimes people will draw models, take a picture and then submit that picture. 

Kim Montague  10:05

Yeah. 

Pam Harris  10:05

Those are models that represent the relationships, the strategies that they're using to solve the problem. So there's another way that you can help parse that out if you're interested. Alright, Kim, what's up next?

Kim Montague  10:18

Okay, so probably one of our hottest topics: Multiplication facts.

Pam Harris  10:24

Yeah, wahoo!

Kim Montague  10:24

Listen in to hear what Pam really thinks about basic facts and her ultimate goal for students. Here's a clip from Episode 34.  Let's be honest, memory fails for every student. Everyone's memory fails at some point.

Pam Harris  10:41

Everybody's fails at some point, and we don't want to leave kids without something in that moment. We need them to have the relationships that they can then get that fact quickly using relationships. Yeah.

Kim Montague  10:50

And we do a lot of work on campuses, right. And I have been on campuses where kids do the exact same thing in third grade, then in fourth grade, then the same rote memory practice and time tests in fifth grade over and over and over and get the same results. And at some point, for me, that's the definition of insanity. It just is.

Pam Harris  11:10

Right, doing the same thing over and over and expecting different results. That is the definition of insanity. You might find it interesting that I often work with high school math teachers, calculus teachers, and when I ask them if there's a fact they don't know, but they refigure in a mathy, sophisticated way, so it doesn't bog them down in what they're doing. Every single workshop at least one teacher, usually more, admits it. Y'all that's calculus teachers, every single one admits there are facts they don't know. But it's not like they're like, "Woe is me. I should have known it. Maybe fire me from my job." No, no, no, no, no, they confidently tell me. And it's almost like, "Well, of course, there are some facts that I refigure." Their attitude is very much like, "I'm busy thinking about all these higher level math things. Every mathematician knows that you don't have to fill your brain with that stuff. Just find it when you need it." That's huge! That speaks to the fact that when we're thinking about higher math, we're really clear we don't necessarily have to have rote memorized all of these facts. What we want are relationships and connections. And then we want to deal with those facts a lot. We want to do things where it demands that we have those facts, that we use them, that we play with them, that we are thinking about the relationships between the facts. And then if one of them doesn't pop out when we need it, we can refigure it quickly in a multiplicative, mathy, kind of way. There's no detriment to that. We're not at a loss because we had to refigure it quickly, because we had these relationships at our fingertips so that we can refigure them. So these calculus teachers are really clear that mathematizing is about using relationships. What they didn't do was fret about the fact that they didn't know some of the single digit facts. They're not stressed about it. Oh, if we could give that unstress, that de-stressing - I don't know if unstress is a word - to our students, so many of them would be able to relax into just using the facts enough that they become then automatic. Of course, we want students to know their facts. Of course, we want them at their fingertips. But we don't want all of the stress that comes when we ask kids to rote memorizing them without meaning. In fact, Kim, I'm going to share a quick story with you. We were at the table, I was talking about the fact that we were going to do the series pretty soon. And my husband, bless his heart who struggled in math, all of his life, said to me, "If my teachers would have helped me understand that there were relationships," he said, "I literally thought I was memorizing a bunch of disconnected facts, that they had nothing to do with each other. It was like if you would have said apple plus orange is banana, and then I was supposed to remember that apple plus, or times whatever, times pineapple was a mango, that it was that disconnected; that there was no rhyme or reason, there was no sense." He's like, "I just couldn't make sense of it. And I was horrible at the rote memorizing and so I failed all that stuff, and then over, you know it just kinda escalated." So let's be clear. If your perspective has been that the goal of your math class is for students to get to say the traditional long multiplication algorithm, then we get why it would make sense that you would think, "Okay, in order for students to be successful in this multiplication algorithm, they must be able to recall the single digit facts." Because the only thing you do in that multiplication algorithm is do single digit multiplication facts over and over and over. There's a magic zero that shows up and then you do a bunch of addition at the end. So if you're gonna be successful at it, yeah, you want those single digit facts at your fingertips. So I get that. I understand why there would be this emphasis on knowing the facts. However, even if that was your goal, what if you could get to having kids have those multiplication facts at their fingertips, but without all the stress. If we do without all the stress then we could get kids that don't get further and further behind because the stress is impeding them. We get kids who can actually use those single digit multiplication facts in that algorithm. But we're suggesting that's not the goal. It's not the goal to just have kids get answers by not thinking, recalling from rote memory a bunch of single digit facts, and then the magic zero and then they add the stuff up at the end. That's not our goal. Our goal is mathematizing. Our goal is helping students develop the way they think and reason and use relationships. Therefore, we recommend helping students learn the facts through relationships.  Ahh! Relationships, that's what matters in math, and in life, right? 

Kim Montague  15:51

Yep. Alright. So speaking of relationships, the last clip we want to share is from the episode called the Y perspective. And you first realized this perspective working with your daughter, Abby, let's listen in to a bit of Episode 27.

Pam Harris  16:08

If she can't understand the rule, the procedure, the thing to mimic, then she can't do it. It's like a part of her psyche just decides, "I should be able to understand that because it looks like all the rest of them are understanding and I can't." She just feels like she can't just perform steps if she doesn't understand them. It's so interesting, right? Because so many of the rest of us were just happy to perform the steps and she just feels like it's giving up a part of her soul to do that. So one day she came home, I'll never forget, she goes, "Mom, fractions -" Oh, I totally just threw my pen across the room. I'm getting excited here. She came home. She's like, "Mom, division of fractions are Figure-Out-Able, right? Division is figure-out-able of fractions?" And I was like, "Yes, yes it is." She was so like, "Aaah!" And I was like, "Yes, it sure is." So then we talked about division of fractions. We made sense of what it means to divide fractions. One day, I was at a presentation actually. So doing a workshop for teachers. I got a text on my phone. This is my daughter. So this is when she's in 10th grade algebra two, no ninth grade algebra two. And she says to me - 10th grade algebra two. It doesn't matter. And she says to me, "This is me right now. 'Wait, that rule doesn't make any sense.' Teacher and other students: 'Well, that's just the way the rules are. That's how it works.' Me: 'But that doesn't make sense. Why is it that way?' They respond: 'That's just how it is.'" And she puts that emoji with the little angry guy with smoke coming out of his nose like, grr.  "They expect us to mesmerize all these rules." There's the dyslexia coming out as the autocorrect fixes things. "They expect us to mesmerize all these rules, instead of understanding and reasoning through it." That emoji again. I'm like cracking up. I'm like, whoa. So I respond back to her, "Goodness. What's the topic today?" Y'all if it was going to be a topic in algebra two that you maybe are not sure how to teach without teaching rules and procedures, what would it be? So she responds back to me, "Logs." Yeah, it's logarithms. I mean, logarithms are complicated. There's a lot going on. And if you don't understand exponent relationships, you're not going to understand logarithms. And so it was interesting. So she then sends me a picture of the board. And the board has kind of the subtraction division relationship with logarithms. And she said, "Why do you subtract? Divide? Makes no sense. Logs are stupid." I'll never forget that day. So I responded back to her, "You make me smile. Let's talk all things logs when you get home, or at the DPS while we wait in line." Because literally she was getting her license that day. So y'all, that day, when we got home, we talked about the exponent relationships she built the week before. We built on that, that's why it's a log. She growled at me again. And she's like, "What is wrong with all these high school math teachers that won't help me understand the why". And I have to just keep reminding her, it's because they don't know why. But let's change that. Let's get to the point where we can explain or help students develop the why. So that people like Kim's husband and my daughter don't have to go through life. so frustrated that we're not letting them in on the secret of what's really going on.

Kim Montague  19:04

You have these students, if you haven't taken our perspectives quiz that we mentioned, head over to www.mathisFigureOutAble.com/XYZ to take that quiz.

Pam Harris  19:17

It's a fun quiz. Super fun. You're gonna love it. Yep,

Kim Montague  19:19

Yep, we were gonna play a little bit from the Development of Mathematical Reasoning, but they were all a little bit too long. So if you like this and want more, check out episodes, five and six.

Pam Harris  19:29

Yeah, because that's really important part of what we do. But again, a little too long. So check out episodes five and six to hear more about the Development of Mathematical Reasoning. Y'all what a super, crazy, pandemic year it has been. We appreciate your support this year for the Math is Figure-Out-Able Podcast. Thank you. Thank you for listening, for mathematizing, for changing the world of learning mathematics and opening up opportunities for more students to do more mathematics. Hopefully we've been able to add a little sanity in the craziness. Because if you're interested to learn more math, and you want to help yourself and students develop as mathematicians than the Math is Figure-Out-Able Podcast is for you because Math is Figure-Out-Able!