Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim. 

Pam Harris  00:10

And we're here to suggest that mathematizing is not about mimicking or rote memorizing. But it's about thinking and reasoning; about creating and using mental relationships: empowering teachers and students. We answer the question, if not algorithms, then what? In today's episode, we talk about the problem solving process with a twist.

Kim Montague  00:35

Recently, Pam got a question that sparked a lot of conversation between the two of us. And since that's kind of the point of the podcast, we thought we talked about it today. So here's the message: "Hi, Pam, I have a question for you about providing all classes from kindergarten to year six, with a visual on how to solve word problems. So if a student didn't know where to begin to solve such a problem, they would look at the chart and then work out what they might use to do or do to solve their task. What are your thoughts? The one I was given had step one, understand the problem. Step two, make a plan and so on. It was designed by George Polya. Is there any research out there about giving students a chart to work out problems?"

Pam Harris  01:17

Okay, so that's the message that I got. And here's sort of my reply. The short answer is, sure. But I wouldn't spend a lot of time or money on it, and I wouldn't expect great results from it. Okay, so, Kim, when we were first talking about this, we went back and forth a little bit, and I asked you if you knew who George Polya is, and you weren't sure. So tell us about George Polya.

Kim Montague  01:42

Yeah, I learned so much from you. Dr. Polyais a Hungarian mathematician. And he taught in Zurich and Stanford. And he had a book that he was known for called How to Solve It. And he was also known for his efforts in the wake of Sputnik in 1957, to teach math teachers how to teach math. So apparently, he was regarded as the father of the modern emphasis in math education on problem solving. And he had a four step problem solving process. And here it is, step one, understand the problem. Step two, devise a plan. Step three, carry out the plan. And step four, look back. So as you were telling me about this guy, it reminded me of a time when I was in the classroom, and we had a six hour staff development, where we brought in this woman, and she did a training for us all about her specific problem solving process. And she introduced it, she walked us through examples, we all did some math together. And we left that day with a set of, a class set of cards, that we were to hand our students. And every time they solved any kind of problem, they were apparently supposed to walk through this process. And they were required to use it every time and I struggled, I struggled with that quite a bit.

Pam Harris  03:09

You loved it, you love spending six hours and making the cards. So, I'm also around the same time, there was a school in the district that said, "Pam, we need you. We have this thing and you know, can you please?" So I booked out some time, blocked out some time, and went to the school and met the principal, all the people were there, the math coach, heads of grade levels, you know, we sat down and everybody was really serious. And they kind of ask for what that professional development person did for you. They said, "We want you to come in. And we want you to set up a problem solving process. We really think that our kids are struggling solving math problems. And so we need to help them like with this, these steps, we need to give them the steps that they can clearly." And the steps they really had decided that they wanted to underline and highlight certain words and pick out different things, like do things to the numbers, and then off to the side, they were going to rewrite part of the problem. Anyway, they had this whole system set up and they wanted me to come in and work with the teachers to make sure that the teachers really understood the whole process on and on and on and on. And I was a little stymied. Not because I didn't know what to say, well, kind of didn't know how to say it... Because I wanted to be respectful of the fact that they were, you know, very honestly coming at the fact that their students were having a hard time solving math problems. So you and I began to continue to talk about like, what do we do in those situations? Is giving kids this problem solving process? Is that the be and end all? Is that the answer to helping kids be able to solve math problems better?

Kim Montague  04:55

Yeah. So, actually I gotta tell you since we're on the topic today, of Problem Solving methods, if you will. I searched, just google searched, some acronyms you want to hear? 

Pam Harris  05:06

Yeah, go for it. 

Kim Montague  05:07

All right. All right. So you might be doing the STEP method or the PAWS method like, animal paws. You could do the ROCKS method because math rocks. You could be doing CUBES or FUSE or CLEAR or GRASS.

Pam Harris  05:26

Hey, at least that one's sort of mathy. You know, like graphs, like x and y axis.

Kim Montague  05:31

No GRASS, like green grass outside.

Pam Harris  05:34

Ah, it's not even mathY. Okay.

Kim Montague  05:36

RIDES because a problem solver rides through problem solving. 

Pam Harris  05:40

Ah.

Kim Montague  05:42

LOVE.

Pam Harris  05:43

Yah, we need more love.

Kim Montague  05:45

RICE.

Pam Harris  05:46

Rice? Rice, like rest ice compression elevation.

Kim Montague  05:50

Yeah, I didn't really get what the steps were. 

Pam Harris  05:55

I've sprained my ankle a few times. I know that one. 

Kim Montague  05:57

Rucksack

Pam Harris  05:57

Oh, rucksack ok okay, that's good. Yeah.

Kim Montague  05:59

Isn't rucksack like a carrying thing?

Pam Harris  06:00

I think it's like a backpack yah.

Kim Montague  06:01

Yeah. And, oh, gee, like 11 million other options. So here's the question I have: how are kids supposed to know what to do, when they like RISE or STEP one year, and then they rucksack the next? Right? That's a problem.

Pam Harris  06:18

I mean so that's why the whole school should get on the same bandwagon. And they should all use this - I'm kidding. Okay, so is the four step thing bad or five to eight step, with the circle and underlining, is it bad? Are any of these bad? They're not bad. We just don't find them particularly helpful. So if anything, I think you could have a discussion with students about Polya's problem solving steps, I think you could hang it on the wall, and you'd still get kind of all you're going to get out of it. So why? If kids are already successful at problem solving, here's the upshot. If they're already successful at problem solving, they don't actually need this. And so if we take a lot of time, or spending money to create individual charts, it's not worth it for those kids. They basically don't need it. So it doesn't help. And then it becomes drudgery. So we have basically have a group of students who are already thinking, they read the problem, they're already really clear with it, they need to figure out what the problem is asking. And so when we say to them, in order to be successful, you must follow these steps. They're like in the head going, "I don't think so. Like I've been solving these problems just fine." And so it becomes drudgery for those kids. They don't want to solve math problems, because they know that they're going to have to do all the things, like all the underlining, circling, and ...instead of just focusing on the math. So I remember my personal son came home one day, and he said, "Mom, look at this." And he had probably the chart that you learned how to do. 

Kim Montague  07:48

Oh I'm sure.

Pam Harris  07:49

Yeah, because he was at your school, and he came home and he said, 'Mom, can I just do the math." Like, what is all this hoopla that I have to sort of go and jump through all these steps, and it's so boring, and it's tedious. And it basically became drudgery. And I just looked at him and smiled. I said, "Yeah, just just do the math. Just you focus on the math." "But mom, I'm gonna get it wrong." And I was like, "You know, what, sweetheart, do the math. I'll talk to your teacher, whatever." Like at that point, those marks, the grade, neither he or I care about that grade. We really care about the math. We care about him thinking and reasoning about what's happening in the math problems. And he was having fun doing that. I didn't want to turn math into drudgery for him. Okay, so you might say, "But Pam, what about the kids who aren't already successful at problem solving? Like they need it? Right?" Well, here's the upshot. It actually doesn't seem to help them either. So it's not really worth it for them either. What we find is that students spend all of their time and mental energy doing the things, underlining and highlighting and picking out, rewriting the problem in their own words. And what they don't do is spend their time and energy actually thinking about the math

Kim Montague  09:02

Right and the best you'll ever get. Is that now that they've made sense of the problem, but they're exhausted, because they've had to do all these extra things to get to that point, right? They haven't even begun to solve the problem. And by giving them these cards, we're in a sense saying that we have to tell them that a part of the process is to actually solve the problem, or could they just solve the problem?

Pam Harris  09:24

Yeah, exactly.

Kim Montague  09:25

So, Pam, what does help?

Pam Harris  09:29

Yeah, it's a good question, because we definitely have students that are struggling solving math problems. And we have honest teachers and administrators that really want to help students in the problem solving process. We really want to help students be able to understand math problems and have confidence in attacking and solving them. So what does help with all students? Number one is teaching Real Math so that, when students are using what they know to solve problems. They're not trying to reach into rote memory and mimic memorize steps. Then they're actually just confidently, like attacking the problem. They know, "Hey, I can read this problem, I can think about it. And I can actually use Real Math, what I know, to solve the problem." But two, it's also helpful to solve problems with students periodically. Get the group together, and make the thinking visible. So have class discussions about the thinking that's happening as students are solving the problems. Make the process come alive for your students. Spend time figuring out the questions you've been answering. Again, make it out loud with students and have the students do most of the talking. What are you guys thinking about? What are you realizing? So that as the students are doing this out loud together, students realize how important it is to understand what the question is asking. They sort of understand that the steps in the problem solving process, that it comes alive for them, becomes a part of what they're actually doing. Instead of putting all the emphasis on this memorizing these steps to do. While you're doing that, use important models to represent students strategies, like open number lines and arrays, ratio tables, so that the thinking is visible, that it's more discussable and more take-up-able. In other words, it's much more helpful to help students' experience the problem solving process organically than it is to just give them a chart and another series of steps for them to memorize and things to do.

Kim Montague  11:20

Yeah, absolutely. If you haven't already heard of our friend, Brian Bushart, you should check out his numberless word problems, where he suggests removing the numbers and the question in the problems, and then doing a slow reveal to help students make sense of the situation.

Pam Harris  11:35

Yeah, it's a super way to, one of the ways that we think that you can help students organically kind of understand the problem solving process, instead of them just kind of number plucking, where they might just look at a problem and kind of grab the numbers, flip a coin and decide on which operation to use. Instead, we want students to actually understand the problem. So Brian suggests, for example, that you would start the problem like: Some fifth graders entered a school art competition. Fewer fourth graders than fifth graders entered the competition. And then have kids talk about that. What's going on? What are the fourth graders doing? What are the fifth graders doing, and who has more? Who has less? talk about the art competition, talk about art a little bit, and then bring in some numbers. "Okay, guys, here's some more information. 135 fifth graders entered a school art competition. Fewer fourth graders than fifth graders entered the competition. Alright, what do you guys know? Does this change what we know?" Now they can start to talk about ranges of what's happening, then give them some more information. "Alright, guys. So we've got that 135 fifth graders enter the school art competition. Fewer fourth graders than fifth graders enter the competition. What question could I ask you about this situation?" Now you're actually asking students to come up with the questions. Brilliant. So you're really getting kids to understand what's happening in the situation. You're getting kids to then guess what the question at the end of the situation might be. You know what, in this one particular situation, all of the kids that we asked this answer, gave the same question. They all said, "Oh, we know what you're gonna ask us. You're gonna ask us how many fourth graders enter the art competition?" So that was kind of the obvious question. So instead, the teacher in this situation decided to ask: How many total children enter the art competition? Brilliant, and the kids are, like totally thinking and reasoning. And the idea is that you can give kids just enough information that they have to sort of talk about what's happening. And then gradually put the numbers in and then you can even ask kids at the end of it, "What do you think I'm going to ask you. What do you think the question is?" Now kids are really thinking about the relationships. They are really diving into the problem solving process without just grabbing some numbers, flipping a coin and doing some operation. Hey, y'all we'll put Brian's info in the show notes. So you can go look at his numberless word problems. He's got discussions about it and sample problems and the whole thing. Great job. Alright, so today, we've talked about the problem solving process. We hope that you can help kids live the problem solving process, help it come alive for students, but don't make it one more thing for them to memorize. Make it something that becomes internalized for kids, so that they can focus on the math and understanding what's actually happening in the problems. Alright, guys, just so you know, we have been running the Developing Mathematical Reasoning workshop. It is about to start again. The first time that we ran it, we had over 1300 registrants and we were so excited. It went so well. But registration is open again. So head on over to mathisFigure-Out-Able.com and find the Developing Mathematical Reasoning workshop. We'd love for you to take our free workshop out there about developing mathematical reasoning. And don't forget MathStratChat on your favorite social media on Wednesday evenings. We'd love to have you join us there. If you haven't yet, please head on over. Give us a rating and a comment for the podcast because it helps more people find the Math is Figure-Out-Able Podcast. So, if you're interested to learn more math, and you want to help students develop as mathematicians, then the Math is Figure-Out-Able Podcast is for you because math is Figure-Out-Able!