Ep 265: Show Your Work

Should you require student's to show their work in mathematics? Whose work? When? How much work? For what purpose? In this episode Pam and Kim discuss classroom culture and the importance of communication in mathematics, diving into these questions and more.

Talking Points:

• Whose work are you asking student to show?
• Motivations for showing work
• Don't fix the math, mentor the mathematician - Cathy Fosnot
• Communication is a mathematical behavior
• Determining 'just enough' is a mentoring opportunity

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Transcript

Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  0:10  
And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  0:17  
Ya'll, we know that algorithms are amazing human achievements, but they are not good teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:32  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical

Kim  0:38  
relationships. 

Pam  0:38  
Thanks for joining us to make math more figure-out-able.

Kim  0:43  
Hi there. 

Pam  0:43  
Kim! 

Kim  0:44  
Hi.

Pam  0:45  
Sup! How's it going in the hot, hot Texas summer?

Kim  0:48  
It's good. It's good. Running outside is pretty

Kim  0:52  
brutal. 

Pam  0:52  
Yeah, I was going to say, you've been running a lot.

Pam and Kim  0:56  
Yeah. 

Pam  0:57  
Because your kid is running, right? 

Kim  1:03  
Yeah, cross country. Love it.

Pam  1:03  
Yeah! Super cool. Yay to all those cross country people out there. Who I was never one of. No. 

Kim  1:03  
Yeah.

Kim  1:03  
Well, I mean, I was not a basketball player, so props to you for that. I'm not coordinated enough.

Pam  1:09  
Oh, I don't know about that. 

Kim  1:10  
Oh, for sure, not. 

Pam  1:12  
We all have our we all have

Pam  1:13  
our things, right? 

Kim  1:14  
Yes. Okay, so today we're going to have a conversation about something that popped up in our Math is Figure-Out-Able teacher Facebook group. And it's been a while. But, you know, we always get the best questions. If you are not a part of that teacher Facebook group, you really should join us there. There's a lot of really thoughtful people and asking questions. And we get to pipe in every once in a while and leave some comments. So, I wanted to read one, Pam, that somebody left a bit ago. 

Pam  1:40  
Okay.

Kim  1:41  
Katrina said, "So, I've been following for a while now and implemented a lot this year." (unclear). 

Pam  1:46  
Way to go, Katriana! 

Pam  1:48  
Yeah, Whoo! Ow-ow! 

Kim  1:48  
"I'm trying to catch up on all the podcast episodes, so I can understand how to help my students think for everything taught in grade six."

Pam  1:56  
Nice.

Kim  1:56  
Yeah, it's great. "But the question I haven't really heard an answer for yet is do we still need to tell students to show their work?" And then she says, "Are they showing their thinking with a model? For instance, I've been teaching area, and to start, we do simple area of rectangles. And, of course, some can just do in their heads. Traditionally, they have to show all their steps. So, what do they show on paper? I'm worried about testing and future teachers not realizing the thinking that is happening if they don't see traditional steps or the algorithm." So, Katrina, let's chat about it! 

Pam  2:20  
Nice, yeah. 

Kim  2:21  
(unclear). So, the really cool thing about this group is that a lot of thoughtful people pipe in their answers, so I'm going to just give you a few of them, okay?

Pam  2:36  
Yeah, go. 

Kim  2:36  
Jim said, "If it's thinking..."

Pam  2:38  
Is that Jim from Alaska?

Kim  2:39  
It's Jim from Alaska. Jim said, "If it's thinking, everyone's work will look different, and just enough is okay." So, I'm kind of paraphrasing. Robin said something about teachers may want to give points for some steps, and if you don't do that young, then it won't go well older. I think she's a high school teacher. 

Pam  2:58  
Okay. 

Kim  2:58  
Dana said, "We want to give kids opportunities to use strategies." Another high school teacher talked about proofs in geometry.

Pam  3:05  
Mmm, mmhm. 

Kim  3:06  
And somebody older grade said how do they do it younger. "How they do it younger is not really relevant." I typed in something about how maybe the answer lies in why they're showing their work. Is it for compliance? Is it for grading? You know, and I kind of talked about a little bit about Mikayla, who I worked with, and I knew that she really needed to get stuff out of her head on paper. And so... 

Pam  3:28  
I'm going to slow you

Pam  3:28  
down just a little bit. 

Pam  3:29  
Mikayla was a seventh grade student. 

Kim  3:31  
Yeah, yeah. 

Pam  3:32  
You'd had her in fifth grade. She came back. You were tutoring her a little bit.

Kim  3:35  
Yeah. 

Pam  3:35  
So, you knew her well in fifth grade. Okay, carry on.

Kim  3:37  
Yeah. And so, you know, we have some students that can hold a little bit more, and it doesn't get mixed up. And so, Mikayla was a sweet, sweet student who needed to just think of it, write it down. Think of it, write it down.

Pam  3:48  
And could absolutely math when she did that. 

Kim  3:52  
Oh, for sure, for sure. But

Kim  3:52  
if she's held stuff for too long, you know, got a little switched up. Anyway, so I commented a little bit about that. And Laurel said, "Let's write enough down, so that I can tell what they're thinking about, and they can also look back at their work, so that they can correct it when they get an opportunity to make corrections. So, lots of really great ideas and different different perspectives about answering the question that Katrina put in. So, I want to know, and I'm sure these guys want to know, what do you think? What? Talk about "show your work".

Pam  4:21  
Yeah, so it's interesting. As I kind of made this wonderful eye opening transition from what I was doing, which was rote memorizing and mimicking algorithms, to realizing I could actually use what I know to reason, to think through problems, to do the mental actions that mathy people actually do.

Kim  4:42  
Mmhm. 

Pam  4:42  
As I was doing that, I had a similar question, but I began to realize that when my teachers had said, "Show me your work," I think they actually meant, "Show me my work." And you might have heard me say that before, but I think it's so true. And I'll often say that if I give a first kind of presentation in front of a group or something, I'll often talk about what does it mean to show your work versus showing the teacher's work? You know, the teacher has given you the rule and shown you three examples, and all the steps are on the board. And so when they say, "Show me your work," I think they've actually, at least mine did, meant, "Show, mimic the work that I've done. Show me that you can mimic the steps that I've given you. And then if you goof a little bit, I'll give you partial credit for almost mimicking me well." What my teachers didn't mean, at least my teachers didn't mean, was "Show me what you're thinking about. How are you using relationships and connections to solve this problem? How are you reasoning through things?" Now, what's the word I want? I will give credit to a couple of professors that I had who said like, "Prove this thing." So, I had a geometry professor at BYU and I had an Advanced Algebra Professor at BYU. The geometry guy actually wasn't too bad. The Advanced Algebra guy was terrible. But both of them actually did want to see my work. Which, at that point in my career, was super tricky because I didn't have any work. Like, I didn't have any thinking going on. Now, any thinking? Not about my own thinking. I didn't have  my own reasoning. And so, when this Advanced Geometry class, the professor would say, you know, "Prove this non-euclidean geometry thing," I would be like, "I'm not even sure what that like..." And he's like, "Well, does it make sense to you?" And I'm like, "Well, yeah. I think I can follow it." And he kind of helped me make sense of the thing that we were proving and how I could reason that it made sense. And then he's like, "So, write that down. Like, put that logical structure that you just did, and make sure you have reasons for everything that you're saying." And I kind of, for the very first time, learned, "Oh, this isn't like geometry class in high school where it was like this two column silliness." I'm going to say "silliness" because no mathematician uses a two column proof. Put that out there. And so, that that really has become something that we... It's an artifact of math teachers trying to make geometry proofs easier and gradable. I would not use two column proof if I was teaching geometry today. That's a longer story. We could sometimes do more with. But I learned, "Okay, it's literally can you reason through something using relationships?" Then, when I get into abstract algebra, and the expectation in the entire course was that you were proving things. Kim, we proved long division. Like, it was so interesting and bizarre, and... 

Kim  7:24  
Yeah.

Pam  7:24  
And I kept saying, "I don't know!" And this guy was not any good. And he just was like. I literally said to him, "If you will give me the proofs ahead of time, I will memorize them, and I will spit them back out." And he's like, "Yeah, that's not what we do here." I'm like, "Well, what do we do here?! Help me!" Like, I... Yeah, it was terrible. Worst grade I got at BYU. It was horrible! I'm glad I kind of pulled it out at the end. I figured some things out. But I realized that we can say to students, "Show me your work," and actually mean, "Show me my work." So, Kim, in all the things that you just mentioned, great comments from people. Ya'll, join the Math is Figure-Out-Able teacher Facebook group if you're not in there yet. It's great place to ping ideas off of other really well thoughtful educators. I think there's kind of, maybe if I had to parse this out, sort of two things happening here.

Kim  8:11  
Mmhm. 

Pam  8:12  
So, maybe one is what kind of what I just said. There's my work versus your work. So, what does it mean if you're in kind of a repeating, rote memorize steps classroom. If that's the perception that you've had of what math teaching is. You kind of this distorted thing that that's  what we do. We give kids the one and only one way. It's clear. It's concise. It will work every time. They won't mess it up because it's just like, "Just memorize these steps." If that's the classroom that you're running, then, yeah. You want kids to show that because if they do it in their head, and they goof it up, you don't know what part to fix, right? You don't know how to go in and fix the math. Gal, Kim, this reminds me of a quote Kathy Fosnot had. Which I have a harder time finding. I tried to find it the other day. Yeah, in one of her books, she says, "Don't fix the math. Mentor the mathematician." 

Kim  9:03  
Yeah. 

Pam  9:03  
Kim, I had to think about that for a long time because from my fake math perspective? Yeah, you have kids show my work, mimic the algorithms, so that I can fix the math, so I can help them know, "Ooh, it's this step right here. Fix that."

Kim  9:19  
Mmhm, mmhm. 

Pam  9:18  
It also reminds me of an experience I had. We videoed a lot of Problem Strings in a lot of classrooms. I was in a particular high school. We used those in our Journey, in our teacher coaching program. I was in a high school classroom one day. Now, I videoed in lots of high school classrooms, so you're not going to know where this was and who this was. But great teacher. Teacher walked in. You could tell had a super rapport with the kids. Joking around with them. The kids liked him. We started to teach and had a good context. I was like, "Okay, this is nice. The kids are thinking." Well, I say the kids are thinking. It was a nice context. And then he kind of said, "Alright, start working." And I literally watched this delightful person walk up and down the aisle and point on a student's paper and say. "Oh, add, not subtract." And then on another student's paper, "45 not 39." And then on another student's paper, "Multiply here not..." And I was like, "Oh, oh." Like, I just sort of watched it happen before my eyes, where what this teacher was doing was fixing the math not mentoring students to reason, and think, and use what they know like a mathematician would.

Kim  9:19  
Yeah.

Pam  9:22  
To actually do the mental actions of a mathematician. So, there's kind of two. So, in this my work versus your work thing. If you're in kind of this traditional mimic algorithms thing, then I get it why you're asking kids to show your work. We're inviting you to consider there's actually a different purpose to teaching math, and that's to mentor students to actually use the mental actions of a mathematician. Like, mentor them to be mathy people. So, that's one kind of thing. You might also be a teacher that it's just about compliance.

Kim  10:51  
Mmhm. 

Pam  10:51  
Kim, I feel like you might have...

Kim  10:51  
We've done an episode about this. 

Pam  10:51  
Well, I think I feel like you might have something to say here if I just say "compliance". 

Kim  10:59  
Oh,

Kim  10:59  
yeah. I mean, we have maybe even two episodes about compliance and where it does or does not belong in a math class. And I think there are a lot of classrooms that have a lot of compliance in them. And...yeah.

Pam  10:59  
When you say that, you mean like, "Do the thing just because I said to, not

Pam  11:07  
(unclear).

Kim  11:08  
Right, "This is the way it looks. This is what I expect from you. There's a lot of points off for not having it a perfect way or a specific way." And, you know, at my core, I try very hard to assess and grade based on academic understanding. And I think, you know, I run into a lot of classrooms where it's the way things look are graded. Or if you don't do a certain thing in a certain way, you're less than. 

Pam  11:44  
And

Pam  11:45  
it's not that the math is incorrect. 

Kim  11:45  
I struggle with that a lot. 

Pam  11:46  
It's not just a certain way, right?

Kim  11:48  
Yeah.

Pam  11:48  
It's not like... I think you said incorrect a minute ago. if it's incorrect, you're going to...

Kim  11:53  
Oh, yeah. Sorry. The way that the teacher wants it to be because of their preference. If it's not adhering to the preference of the teacher, and it has nothing to do with the actual mathematics.

Pam  12:04  
That rubs you a little bit wrong. You'd prefer that we don't sort of just like demand this compliance for compliance sake. 

Kim  12:08  
Right. 

Pam  12:08  
So, I'm going to push on that just a little bit. If a kid doesn't write their name on the paper, do you take points off? I do not. What do you do?

Kim  12:18  
I help them. First of all, you know, I have a stack of papers. And, you know, frankly, when you're an elementary teacher, you get to know handwriting for a couple of classes. I understand that in high school you can't do that. But I've got kids who are in high school, and sometimes they go find their paper. And, you know, they're always  grateful for the grace that they just forgot a name. You know, sometimes I think a lot of people say, "Well, in the real world..." And, you know, I struggle with how much we don't give grace to students in compliance type things for the sake of preparing them for the future. Yeah, it's a big conversation for me. So, no. I do not take points off for no name on the paper. And for a whole variety of other things that people take points off for.

Pam  13:07  
And you're not looking for a recitation or a repetition of steps that you showed them.

Kim  13:13  
Unh-unh. 

Pam  13:13  
So, to me then I think we get into maybe the second. I said I think there was kind of two things happening here. So, one was kind of my work versus your work, and this sort of fake math versus compliance kind of thing happening.

Kim  13:26  
Mmhm. 

Pam  13:26  
But I think another thing that we can bring up in this "show your work" conversation is all about communication. 

Kim  13:33  
Yeah. 

Pam  13:34  
And if I could talk about just mathematical behavior for a minute, I think there is mathematical behavior of communicating your reasoning, your thoughts, your results to other people. I think that's part of mathematics. If a mathematician cannot communicate what they've proved, or discovered, or answered or solved to other people, then they're not it's they're not worth as much like we can't we. If we can't communicate to the world, then we can't use it. We can't build on it. We can't go from there. So, part of mathematical behavior needs to be that we can communicate our thoughts and our reasoning, and that we can do that in a way that other people can gain from Yeah?

Kim  14:18  
Mmhm.

Pam  14:18  
So, if that's part of mathematical behavior, then I do think there's a world where we ask students to hone, to develop, and hone, and get better at the mathematical behavior of communicating their thoughts, and their reasoning, and their problem solving, and then their answers. So, if I'm mentoring a kid to be a mathematician, part of what I'm going to do is try to encourage them to communicate well. We can talk about a few ways to do that. One way that we strongly promote here at Math is Figure-Out-Able is as students as we pull out their reasoning, we're going to attempt to represent it, make it visible, make it point-at-able and discussable. Who did I just chat with the other day that was like, "You make everything ian 'able'." Point-at-able

Kim  15:05  
That's true. 

Pam  15:05  
Discussable. I was like, "Oh, yeah. I guess I do." So, when we do that, then students have a chance to go, "Okay, when my brain does that, it could look like that. Therefore, I could use those representations. I could use what you just put on the board to make my thinking visible. Alright, now I'm getting a feel for I could use those representations to represent my thinking to communicate with other people." So, we do think it's important that students develop this part of mathematical behavior, which is communicating their path of reasoning or their path of proving that someone else can follow. And then maybe there's sort of two reasons for that. One is it's a mathematical behavior, and I think that's the most important reason. I think there's also maybe the older the student gets. Maybe not. I'm still grappling with this. But I do think, at some point, we grade stuff.

Pam  16:01  
Mmhm. 

Pam  16:01  
At some point we give... 

Kim  16:02  
Sure. 

Pam  16:03  
...you know, points for things. And I think that's a really good conversation that I would love to have with a professional thought partner as I give homework, quizzes, and tests.

Pam  16:14  
Like, how can I ask this question in such a way that I can actually grade academic competence coming out of this, the mathematical understanding and skill coming out? How can I grade that and not just this compliance of, you know like, "Do it the way I expect it to show up"? Does that make sense? Like, what do

Pam  16:35  
you think about that?

Kim  16:36  
It does. Yeah, let me give you an example. If I'm grading a student's paper, and I am kind of very traditional, then I'm focused on the answer, right? It's like got my red pen. I'm like, X, check, X, check, X, check. And what I really appreciated when I was able to model students thinking and they would be able to get to the point where they were able to model their own thinking is I would just say to them, "Hey, ya'll, I don't grade papers at school. Like ever. I'm with you at school, and we're doing things." You know, I just... No judgment on people that do grade at school, but I was a take it home grader. And so, I...

Pam  17:10  
Well, at least... Can

Pam  17:11  
I say at least outside of class grader. In other words, you didn't grade it right next to the kids. So, you couldn't... I think... Sorry I'm interrupting you. But I think your point is they weren't sitting there... 

Kim  17:23  
They weren't there.

Pam  17:23  
...(unclear) to can tell you what was happening.

Kim  17:23  
Right.

Kim  17:23  
So, I would take it home. And I would say to my students, once we got to a point where they could model their thinking, I would say, "Hey, I'm not with you to ask you the questions that I would normally ask if I was sitting next to you." Because when I would circulate and I would talk to kids, I would say like, "Tell me more about this." And, "How are you thinking about it?"

Pam  17:38  
You pull it out. 

Kim  17:39  
And if I wasn't... If they weren't there, I couldn't ask them the questions. And so, I would ask them to put just enough. And I think that's part of the conversation that was happening in the teacher Facebook group is just enough felt good to me, so that I could make sense of what they were thinking about. Because let me tell you, there's enough times where kids would write a problem on a piece of paper, and they would get a digit wrong. They just copied the problem wrong. Or they, you know, whatever. And if I were just checking out the answer, it might be a wrong answer because they wrote the problem wrong. But if I have something. Like, I'm making sense of their work. I'm like, "Oh, their strategy was fantastic. They were thinking through this. They got a wrong answer because it was the wrong question. So, correct. There were plenty of times where what was incorrect was not the answer to the problem. Does that make sense? Where like if they wrote down the question wrong, they couldn't get the answer that I was supposedly looking for.

Pam  18:38  
Yeah. I'll even give you another example. They might have had the question correct, but then in their work written something a little bit messy. 

Kim  18:46  
Yeah.

Pam  18:46  
I can almost tell from one line to the other.

Kim  18:48  
Mmhm. 

Pam  18:48  
Or the model to the other side of the array or something. That what I know is a 2 looked like an 8 or something or vice versa, and so then they wrote the 8 down over there because they, you know, as they were going, they just a little messy. And so, yeah, are we grading?

Kim  19:03  
Yeah. Well, and if your goal is to mentor mathematicians, then you're using that as an opportunity to have a conversation about like, "Hey, we got to be able to make sure we understand your handwriting." Or, like, you're using a fantastic strategy, and there's just like this little tweak that needs to happen. There are plenty of opportunities where analyzing a wrong answer can reveal so much. And I just really appreciated the point where I could look at students work and like it was fun for me to analyze, and think about what they were doing, and did I see the same strategies appearing over and over again? So, there is something for me about having just enough. But if they didn't write something, it's not a big fat X wrong. You didn't put something. It's just a start to a conversation about how that's helpful.

Pam  19:51  
And part of what could be happening here is that if you are teaching kids to rote, memorize and mimic, and a kid just writes an answer down, then you're like, "No, no, no. You got to show your work because that's the definition of math." 

Kim  20:03  
Right. 

Pam  20:03  
But in our classrooms, if we've got really students reasoning, then very rarely are they just going to write an answer down because they're clear that it's not all about the answer. It's about their reasoning. And so, they're going to give it, hopefully, and then we're going to mentor them. What is just enough? Maybe if they write something down, you're like, "Yeah, that wasn't enough." Like...

Kim  20:23  
Yeah.

Pam  20:23  
..."Help me out. And then they explain. You're like, "Oh! Hey, if you just put this, then I would have known what you did." That's mentoring, right? 

Kim  20:29  
Yeah, right. 

Pam  20:30  
Like, you're helping them realize what is enough. I could see, Kim, very clearly that you could go to a different student and say, "You don't have to write that much. Like, that's a lot! Like, this is a... Did you use a ruler on that array? Like you you could do a little less perfect work as far..." Perfect. "...as far as like penmanship goes. You could write a little less here, and that would be enough. Like, you don't have to waste your time on that." Does that make sense? 

Kim  20:53  
Yeah, absolutely. 

Pam  20:54  
When you're mentoring, you're actually reacting to what students are doing to help them become better at reasoning, and recording, and communicating, not at rote memorizing and mimic. 

Kim  21:03  
Yeah,

Kim  21:03  
I have one other comment about communication. I think that it really matters who you're communicating for. So, there are plenty of times where it's just for me. Just like I have this thing on my head, I'm kind of in my head, and I'm kind of switching to something else, and I just want to put a number down because I don't want to hold that number while I'm switching to thinking about something else. I might literally just write a number because I need to go back to the paper and look at that number again. In that case, I'm putting just enough for me. I think you know, when we do Problem Strings live here on the podcast, my paper is kind of chicken scratch because I'm not trying to share with somebody else what's happening. But if I'm communicating for you, and you need to say. You know, where you're like, "Hey, Kim, what strategy did you use?" I might write a little bit more. I might communicate for other people in a way that they can look at the paper and make sense of it, where if it's just about only me looking back, and I know what I did, I think that's drastically different.

Pam  22:02  
So, that sounds like we're going to invite everybody to mentor students to actually consider for what purpose am I solving problems right now? Do I need to communicate with other people? Okay, then that's going to mean I'm going to do a little bit more to communicate. I'm going to put more down in order to get my message across. If I'm just doing this for me, and I'm really clear what I did, and I'm going to use it and move on, then just enough. Well, maybe in either case, it's just enough. But when it's just enough for you, it's probably less than if it's for somebody else. Yeah. And I think mentoring, that's a life skill. To realize, you know, when is it just for me. Okay, good. I've got enough. I can move on. I can

Pam  22:42  
keep going. 

Kim  22:42  
It makes me think of a grocery list, honestly. Like, if I'm thinking a grocery list for me, I'm totally writing T-O-M if I want tomatoes. Like, I'm lazy. I don't want to write all the word. But I know what that means. And L-E-T is lettuce. Right? But if I'm sending that grocery list with my kid, they're like, "What? TOM, LET. What do you mean? Same, same. 

Pam  23:00  
It's even

Pam  23:01  
worse with my husband, because then I have to like put the amount next to it. You know like, I need a whole bunch of that or I need two of those. Or, like. Yeah, if it was just for me, I clearly know how many I already have in mind to buy. Unless I have some random, weird recipe. My daughter and I love to cook together, and so we will try new things. Well, then I'm writing the amount down because it's a recipe that. Yeah, sure. So, what's the purpose? And so, we invite you to think about, like why are you asking kids to write stuff down? Show whose work? So, if you're in an answer getting classroom, you might kind of be thinking about mimicking you, showing your work. And we would invite you to consider what would it be like to actually get kids to use what they know more like in a Math is Figure-Out-Able classroom. If you're in a Math is Figure-Out-Able classroom, then you're thinking, you're helping mentor that mathematician to communicate and who they're communicating with. Is it just for them? Well, they could probably write less down, If they're communicating with their classmates, they probably need to be even more clear than they need to if they're just communicating with me but write enough down, so that I can help you grow as a mathematician. But like you said, just enough. Is

Pam  24:09  
there anything else you want to add? 

Kim  24:11  
I think that's it. 

Pam  24:12  
Bam.

Pam  24:13  
ya'll, let's show some work. But just enough. Just enough. Thank you 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. Thanks for spreading the word that Math is Figure-Out-Able!