When dealing with word problems it can be tricky for students to decide which operation to use, so we need a way to help them memorize what to do in any given situation... right? In this episode Pam and Kim discuss how reasoning empowers students to skip rotely memorized steps to choosing operations and get straight to problem solving!
Pam Harris 00:00
Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
Kim Montague 00:07
And I'm Kim.
Pam Harris 00:08
And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But y'all it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be. So podcast listeners, did you hear our first MathStratChat edition last week? Ah?
Kim Montague 00:43
Pam Harris 00:43
Whoa. Okay. So Kim and I are having a blast doing those. We just take the MathStratChat question that comes out that week. And we hope you all solve it, and post your solution, chat about other people's strategies, and then pop on and listen to the second Math is Figure-Out-Able podcast of the week where we share each other's strategy. So Kim will solve it and share what she's thinking about. I'll solve it, share what I'm thinking about. Double the fun. The math is Figure-Out-Able podcast, just doubled. Bam! Dah-ta-dah.
Kim Montague 01:18
Very cool. So make sure you subscribe to the podcast, so you don't miss those too.
Pam Harris 01:22
Absolutely. So if you missed it, there's your clue to go click that button, subscribe to the podcast. So you'll get the notifications when those come out as well. Totally fun.
Kim Montague 01:31
Very fun. So last week, we answered Amanda's question and talked about how we think that there are very specific major strategies that everyone should own for each operation. But that there's some other cool strategies as well, even ones that we use, but they're just not the most important ones. So in this episode, we're going to answer some more questions or comments that we received from people who have tuned in. And actually Pam, the first place that I went to look for some questions and comments was our teacher Facebook group.
Pam Harris 02:03
Kim Montague 02:04
It's a really great place for some conversations. Teachers leave their thoughts, their questions, and then you know, we you and I go in there and we leave some thoughts and we get some conversation going with other teachers who are also trying to build numeracy in their students.
Pam Harris 02:18
Yeah. So if you're interested in conversation with other Math is Figure-Out-Able teachers, join the Math is Figure-Out-Able Teacher Facebook group. Love to have you there. Yep.
Kim Montague 02:28
Yep. So in that group, Jessica said, "Teaching kids in a math is Figure-Out-Able way in fourth grade, after they've been taught in a sit and get, mimic the teacher way since kindergarten." That was a struggle. That was like, "Oh, this is what I'm dealing with right now." And Carolyn said, "How do you help middle school kids who are comfortable with algorithms to do Real Math? If they've been taught the traditional way all along it's really hard for them to switch."
Pam Harris 02:54
And boy, do I appreciate that as a former high school teacher and a university instructor. I mean, I just get them where they've been at it longer, right? Like they've just been in this place, where math is about rotely repeating steps and not thinking and reasoning. Remember the story I tell sometimes, Kim, have I told that on the podcast? I'm going to retell it. So if I haven't, I'm saying it again. We were doing some videoing. I think I have but I'll say it again. We did some videoing in classrooms where we go in and we teach Problem Strings with students, we video those, we put those in our Problem String hub in Journey, which is our online implementation support system. And in that Problem String hub, you can go watch these videos. But when we shoot those videos, Kim taught me this, very wise Kim, you always go in the day before, and you have kids create the name tents, so you can say their names correctly. Names are important, totally agree. And you do a short Problem String with them. So they kind of get a feel for what's going to happen. And then you go back the next day with the camera crew, and then we shoot everything. So I wish the cameras were on this first day, because it's a group of seniors. So 12th grade students, fairly successful 12th grade students. They've made it to this specific class in Texas, so they had to have gotten decent grades before that in classes. They're in this class, 12th grade students, I do this Problem String, I'm halfway through it. It's not really a time for kids to ask question, but this kid right in the middle, the room raises his hand, has this look on his face like, "What?" And he raised his hand. I'm like, not really time to ask a question, but okay, "Yes, yes. What do you got?" And he goes, "It's almost like you want us to use what we know to solve the problem." And every other kid in the room was like, "Yeah." And I'm in the front like cracking up like, "Oh, yeah." Part of the funny story for that is, it was actually my daughter's senior class. So she was sitting in the front of the class and she looked at me and just was like, (snicker) because if you've listened to anything on the podcast, you know my daughter is very figure, she absolutely figures out math, super mathematician. But she doesn't memorize math well at all. So she's always figured it all out. So she's in this classroom where she's been figuring it out. So when he's like, "It's almost telling us like you want to see is what we know to solve the problem." She's just like, "Yeah, like you could have it all along." Anyway, it was great. So as you read these comments from teachers who are in these places where, wherever they are, the students have come up to them, the students, they have now have been in kind of a fake math atmosphere. They've been in a mimic me. I'll do it, then we'll do it. And then you'll do it kind of a perspective where they're mimicking the teacher. It is difficult, then like, what do we do? What are some hints? What are some thoughts about that? So those were great, thanks. When you were reading those, Kim, it made me think about a tweet that had come out recently. I think that's what it was. Is that where I heard from Tad, where Tad Watanabe, great guy, he's a professor, and he will often ping us back after a podcast episode. We super appreciate the fact that he listens. He's very thoughtful. Love his pushed back. He has a really unique perspective. He was raised in Japan and spoke Japanese is his native language. I hope I'm getting all this right Tad. And is now in the United States and from that has a very unique perspective and shares a lot of really interesting things. So Tad said, "I noticed that you do a lot of like, sort of (these are my words, but like naked number problems). But isn't it really important to help students decide which operation to do?" So Tad, correct me if I'm wrong, but what I hear implicit in that question is, "Sure, sure, we can, it's important to help kids do computation. Yes, it's important to help kids, you know, like, understand and feel and have numeracy and all. But, Pam, there are teachers out there that are telling us that one of their major sticking points, one of their major pain points, is how to help students know when they read a word problem, which operation to do." So Kim, would you agree that that is a, that teachers would tell us that is a major pain point?
Kim Montague 07:02
Pam Harris 07:04
Okay. So I want to get under that a little bit. This is not a trivial answer. This is in fact, Tad, sorry to answer you on Twitter. I've been thinking about it ever since you asked it. And I wanted to put some time and thought into it. And I really wanted to like chat with you about it. So let's chat today on. So how are these? How are these related? What Jessica, and Carolyn was saying about this difficulty of having students that have been in this mimic atmosphere and now we want to do real math? And Tad saying, how do we help students decide in a word problem which operation to choose? If I may, if students have been in that situation that Jessica and Carolyn just described, where math is about rote memorizing, it's about mimicking the teacher, and they get to a word problem. In that moment, they have not been in thinking land, they have not been reasoning using relationships, they have been rote memorizing and mimicking. When they get to a word problem, then they're bent, their mindset is, "All right. I'm supposed to, like guess what's in the teacher's head this time. I'm supposed to pick from an operation. The goal of a word problem is to pluck the numbers out of the word problem, flip a coin and choose, (I just flipped a coin that was me grabbing, can you hear that?) So I'm looking at that like, okay, it's head. So I'm going to choose this operation, this time." It becomes this guessing game, or Tad wants him to reason. I don't think Tad's suggesting that we should be guessing. But he's like, "How do you help students reason about which operation?" But I'm going to suggest that question stems from a fake math atmosphere. It stems from students not having been reasoning before they hit the word problem. So when they hit the word problem, they're still not reasoning. And we might try to help them reason. But because they haven't have not been reasoning as they've been, as they've been computing, they continue to not reason well. Or they try to reason but they don't have the numerical relationships, which isn't just numerical relationships, they also aren't reasoning additively or they're not reasoning multiplicatively. And so if I give them a multiplication problem and their reasoning additively, or for heaven's sakes, if they're using counting strategies, then how are they supposed to pick an operation? They're not, it's not going to 'feel' like multiplication. If they're not developing multiplicative reasoning.
Kim Montague 09:27
Pam Harris 09:27
So our goal would be to help teachers and then therefore their students, understand, create those mental relationships, create, like build Additive Reasoning, so that when you reach a word problem that calls for addition or subtraction, your reasoning additively, you have been computing in ways that use these big chunks of numbers, big jumps of numbers. So when you read a word problem, it 'feels' like what you've been doing. It feels like the reasoning you've been doing to solve all the computation problems. If you have a division problem, division word problem, and you've read it, if you have been in an atmosphere where you've been grappling with division as we've been doing Problems Strings and Rich Tasks and asking you to dive in and really grapple with these relationships, you've been developing Multiplicative Reasoning with multiplication and division. When you read a division problem, it's going to 'feel' like your intuition is going to kick in to say, "Oh, yeah, this is like what we've been doing. Look, I can chunk the numbers this way." Like it's, you have to be reasoning multiplicatively, in order to recognize that that is the operation that you want to use in that problem. You have to be reasoning proportionally, in order to recognize that you've got ratios happening, and you're going to be looking for equivalence, or one of the other ratio or proportion strategies. So the upshot is we've got to develop reasoning, in order for students to then have the intuition and the wherewithal and the kinds of thinking, the sophistication in the way they're thinking in order to choose the operation. And then honestly, it's really less about choosing an operation. And it's more about, "Oh, how am I going to solve this problem?" Like, students that are in a thinking place where they've been developing more sophisticated mathematical reasoning as they go, don't look at a word problem and say, "Shoot, which operation am I supposed to do here?"
Kim Montague 11:30
Pam Harris 11:31
They don't do that. They're figuring out the question, they're reasoning as they read the question. Now, Kim, I want to ask you something. So don't let me forget. Just mention that. I'll remember. One other thing I want to bring up is, this is why we don't, we don't have a big emphasis on word problem strategies, like...
Kim Montague 11:55
I can't believe you said that! I was just about to say something about that.
Pam Harris 11:58
Well say it go. Then I have something else.
Kim Montague 11:59
You know, I was gonna say that if you are in a traditional situation where you feel like you have to like kind of spoon feed, "This is how you do it." You might also, 'might' also be somebody who feels like code words, or a particular do these steps in a word problem is also helpful. And so..
Pam Harris 12:18
Shade these words, underline those words, circle the numbers, cross out...
Kim Montague 12:22
For these particular words. And they mean to do multiplication. This word means to add. This word, right?
Pam Harris 12:29
Those key words.
Kim Montague 12:30
That's a very elementary thing to circle keywords, and then the words, as if the words tell you what to do. And so that tends to be the way that traditional, very traditional teachers would solve the problem of their students struggle with word problems, because they hit the thinking for the very first time maybe. And they're not sure what to do.
Pam Harris 12:53
And it's bloomin hard to mimic how to solve a word problem.
Kim Montague 12:58
Pam Harris 12:59
Because the word problems change.
Kim Montague 13:00
Pam Harris 13:01
Right? Like, if you've been in an atmosphere...
Kim Montague 13:02
Pam Harris 13:04
Say it again.
Kim Montague 13:04
And there's no symbol to tell you plus or minus.
Pam Harris 13:06
Yeah, exactly. There's no thing to mimic. If you're in an atmosphere where everything you've done in math is here, when you see this symbol, you write the numbers this way, line them up, and then move the decimal, butt cheek at the end. And that's what math is, is mimicking the steps. Now you get these stupid word problems. You're like, "This isn't math. Like, what? This isn't math. How am I supposed to think about the, where's this symbol? It's not lined up for me? I don't know. Just tell me, just tell me the steps to do." If students say things like that, that should ping, ping, you have, that they have been in a fake math atmosphere. And they haven't been reasoning the whole time. So that's why we don't advocate all of that, all of those, help me, Kim. What are they called? The BEAMS? No, like I can even think.
Kim Montague 13:56
Acronyms names. Yeah.
Pam Harris 13:57
All those acronyms that suggest all of these ways to help kids, ready? 'Mimic', think about it that way. All of those acronyms, all of those step by step procedures to solve word problems are just that. They are an attempt at teachers who don't understand what mathematizing really is to say, "Oh, well, if math is about mimicking, we've got to come up with a step by step procedure that kids can mimic to solve word problems." That just should be a hint that we're stuck like heavily in fake math land when we do that. Kim, the thing that I wanted to remember, not to forget is let's say that you have a student who reads a word problem that maybe traditionally or typically might be solved by subtraction.
Kim Montague 14:47
Pam Harris 14:49
Talk to me about that.
Kim Montague 14:50
Well, so I mean, it depends on the situation in the problem. But we know that there are a variety of ways to solve subtraction problems, right? You could think of subtraction as removal, but you can also think about subtraction as finding the distance or the difference between two numbers. And so a kid might look at it and say, "Oh, for these numbers, I'm going to choose to find the distance. And I'm going to find it as a missing addend problem."
Pam Harris 15:15
Kim Montague 15:16
So those mimic these steps for subtraction might not even make sense to them in that problem.
Pam Harris 15:23
And the same thing can be true for a quote unquote, division problem. A student might think of that, see that problem and vision it, feel it as a missing factor problem. And if the teacher is up there going, "Okay, so did you choose the right operation? It should be division." And the kids like, "Oh, crud I did it wrong again. I was thinking multiplication. Oh, I'm not a math person. Okay, I guess I'll just..." And then we have that, we continue that sense of if it doesn't follow my intuition, I'm not a math person. And so I'll just try to memorize your stuff. And now we're back in this mimicking land and students trying to mimic that don't even feel like they can. We want to avoid those moments at all costs. Especially that when the student's intuition was brilliant, right? That missing factor problem, missing addend problem could absolutely be the way to solve a nice problem. Then we don't have to worry. Yeah, help me, Kim, finish that sentence I was on.
Kim Montague 16:18
Oh, gosh, well, I mean, we want kids thinking, right? And so we got to give them opportunities where their thinking is valued. And when they can trust their intuition, rather than just continuing to say, "Do these steps rotely." We want to put them in situations where they're thinking is important, maybe in a naked number situation, so that when they come to those word problems, then they just go, "Oh, I'm just thinking now about a situation rather than just the numbers."
Pam Harris 16:43
Absolutely. Absolutely. Nicely said. So, Tad, it's not that I don't ever work with context problems. In fact, we work with contexts a lot. We'd like to have many contexts in Problem Strings. We'd like to have some major contexts in Rich Tasks, or what Cathy Fosnot calls truly problematic situations. We deal with context a lot. In my work with teachers, what you might be seeing me put out a lot in social media, is to help teachers develop mathematical relationships. It's to help them develop their Additive Reasoning, help them develop their Multiplicative Reasoning, help them develop their Proportional Reasoning, their Functional Reasoning with the numbers so that they are thinking and reasoning mathematically, so that then they can help their students think and reason mathematically. And if we all get in that vein, we don't need to spend so much time at all helping students decide quote, unquote, which operation to use.
Kim Montague 17:44
We love questions and comments. Right? We're so excited to be able to answer those. Hey, listeners if you have any questions or comments, you are so welcome to send those to me at Kim@mathisFigureOutAble.com. We'd love to hear from you.
Pam Harris 17:57
Or throw them in the Math is Figure-Out-Able Facebook teacher groups, and we will get to them there as well. Thank you so much for joining us on this podcast. Thanks for tuning in, and teaching more and more Real Math. To find out more about the Math is Figure-Out-Able movement, visit mathisFigureOutAble.com Let's keep spreading the word that Math is Figure-Out-Able