[00:00:01.330] - Pam
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned math-er.

[00:00:10.190] - Kim
And I'm Kim Montague, a reasoner who now knows how to share her thinking with others.

[00:00:16.550] - Pam
Turn the page, Kim. Scroll.

[00:00:20.820] - Kim
Hey, at Math is Figure-Out-Able...

[00:00:24.270] - Pam
We are on a mission to improve math teaching.

[00:00:26.280] - Kim
I was like, why are you not talking?

[00:00:27.890] - Pam
We know that algorithms are amazing human achievements, but they are terrible teaching because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop. I'm so glad you're human.

[00:00:40.200] - Kim
Oh geez. In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:47.040] - Pam
Y'all, we're so glad you're here to make math more figure-out-able.

[00:00:50.440] - Kim
Golly.

[00:00:51.020] - Pam
And to watch Kim and me scroll. We're both so... We're really focused on the content of these last few episodes.

[00:00:56.920] - Kim
Yeah.

[00:00:57.520] - Pam
And I think our brains are stretched a little bit.

[00:00:59.870] - Kim
Crazy, crazy.

[00:01:00.830] - Pam
Okay, so...

[00:01:01.880] - Kim
Okay, so, so we are in the middle of a series talking about explicit teaching, which I think is really important because the word explicit carries a lot of weight, either good or bad, for a lot of people. And so we wanted...

[00:01:15.680] - Pam
And it's defined differently by many, many people, right? Yeah.

[00:01:18.780] - Kim
For sure, for sure. And so, we want to share ways that we absolutely are explicit in our teaching, both in content goals and in the mathematical behaviors and the behaviors of our community. So, last week we talked through, Pam talked through a Problem String about distance versus removal and named some things that she is explicit about when she delivers that Problem String. So, this week, we are going to tackle another Problem String and give some more examples of explicit in a Problem String.

[00:01:50.070] - Pam
Nice. Okay, so, "Class, today we're going to do this Problem String. Let me remind you that during a Problem String, your job is to think and reason using relationships, so if you find yourself starting to mimic something that your teacher before me taught you, then, you know like, maybe put your pencil down and kind of like look at the problem and see what could I do if I didn't mimic those steps that I've seen before? And then if you want, feel free to pick your pencil back up. It's not if you write, it's what you write that's important." So, I'm going to explicitly give that little introduction that, "You may have seen some steps to do this. But today, today, for right now, I just want to encourage you to think about the relationships. It's not if you can do the algorithm in your head and keep track of all the crossy-outies and whatever. It's not important if you write. What's important is what you're writing. So, I want to really have you... Your job here today, your explicit job today is to see what do you know that you can use to think about the problems?"

[00:02:51.000] - Kim
And listen, you say, put your pencil down because that action alone causes students to take an extra moment to think.

[00:03:01.060] - Pam
Mmhm. And there's plenty of times, especially in workshops or if I'm working with older kids, where I will say, "Pick your pencil up and just appear to be writing." Like, I will directly like request, "Please pick your pencil up." It's kind of a command, right? Because I want everybody in the room, if they want to keep track of their mental thinking, that they will feel socially acceptable to do that. They're not like, "Oh, everybody's doing their head, so I better do it." No, no, if you want to write. But if you find yourself mimicking, then put your pencil down.

[00:03:31.770] - Kim
Yeah.

[00:03:32.500] - Pam
Alright, cool. So, let's start off with a problem. I'm just going to say 6 and 1/5. And I'm writing this down. Plus 4. So, I've written that on the board, kind of on the left-hand side. Then I'm going to very intentionally say, "I'm not going to wait very long on that. What is 6 and 1/5 plus 4?" So I'm also intentionally going to kind of say it that way. And then, you know, if I'm doing this Problem String, the kids that I'm doing it with are going to be like, "That's just 10 and 1/5." And so, I'll just write on the board a number line, and I'll start at a tick mark, and I'll write 6 and 1/5 under that number line, under that tick mark. And then I'll draw a jump of 4. And I'll draw a big enough jump. So, this is not a tiny jump. This is a big enough jump that I'm going to be able to do something with it. Landing on 10 and 1/5. And then I'll go over to the problem, and I'll write equals 10 and 1/5.

[00:04:24.780] - Kim
Mmhm.

[00:04:25.670] - Pam
Yeah. And then I'll say, "Okay, cool. Next problem. What is 6 and 1/5 plus 3 and 9/10?" And I've written that problem on the board. "Go ahead and do it any way you want." So, it's very intentional that I say that. Do it any way you want. But I wonder if you can think. I wonder if there's anything up here that could help you. Hmm. I wonder if there's anything on the board that could help."

[00:04:44.650] - Kim
And I think you would say, "And when you've had a chance to think about it, give me a thumbs-up." That's explicit.

[00:04:50.740] - Pam
That is explicit. Yep. When you are ready, you've gotten to, you've doved, you've dove. I hate that word. You've dived in. Dove in, dived in. And you have some good thoughts about this one, give me a quiet thumbs-up. Then I wait. I might circulate around and see what kids are doing. I'm taking note. Are kids lining them up and they're starting to do a bunch of regrouping stuff? That's noteworthy. Do I see some equivalent fractions? You know, I'm taking all this in very intentionally. And then I'm going to see if I find anybody who used the first problem. I may or may not. But then when I get back to the front of the board, I'm going to say, "I'm kind of curious. Did anybody..." Now, that's an intentional thing, by the way, that I say I'm kind of curious. Okay, let's come back to that. "I'm kind of curious. Did anybody use... You didn't have to. You didn't have to. But did anybody use the first problem to help you with this one?" Now, if no one says they did, like I got nothing, then I'm going to say, "Could you?" If no one sparks to that, it's the wrong Problem String.
We're going to have to do some more work with mixed numbers and just adding whole numbers or something. Some more work there. Chances are high somebody's, you know, at one of those points say yes. And then I'm going to think equity in my head, and I'm going to choose somebody who it's a good day to position that kid as a sense maker. And when that kid says, "Well, you know, so if I know 6 and 1/5 plus 4," as they say, "I used the problem before," in some way, I'm redrawing that number line on purpose. So, I'll redraw, start at 6 and 1/5, add 4, end at 10 and 1/5. And then they're going to say, "But I know that..." In some way, they're going to say, "It was too much," or "I jumped too far," or "I only wanted to add 3 and 9/10. So, I needed to back up or subtract one-tenth." Then I'll pause. I might even interrupt them like, "Ooh, okay. So, here we are at 10 and 1/5, and you've got to back up a tenth. Hmm, sure wish I knew something about fifths and tenths." So, this might be a moment where I just lob out, "Do we know something about fifths and tenths?" Because I might have kids who were still trying to figure out common denominators. Or, "What do you do? I can't remember the rule." Or, you know, in this case, if I've got, especially if I only had a few people that said they used the first problem, I kind of want to pause a little bit and give everybody a chance to think about, 10 and a 1/5 minus one-tenth. Trying to look for somebody that, you know, has some thoughts about that. And hopefully somebody will say. And I might even say something like, "How are tenths and fifths related?" And somebody's going to say something like, "Well, two-tenths is a fifth, so if you subtracted one-tenth, you still have one-tenth left. So, I think that's 10 and 1/10." And then I'll go over to the problem, and I'll write 10 and 1/10.

[00:07:30.540] - Kim
Yeah. I want to jump in here because you asked if anyone used the previous problem to help. And I think you mentioned that maybe you had a few that did.

[00:07:40.100] - Pam
Mmhm.

[00:07:41.720] - Kim
Maybe now you've stopped the class. And a few kids hadn't even occurred to them to use the previous problem. But once you ask that, and a few did, there's a couple of kids maybe now who are like, "Oh, I hadn't thought about it. I have..." And you're pausing intentionally to give them a chance, maybe to catch up a little bit, to rethink this new problem, given that they have a connection that they hadn't. So, you know, I do think this is a very intentional move to say, "I know you didn't think of this previously, but we're going to slow down in this moment to see if anyone else did." And almost always there's a few more that are like, "Oh, okay." It's a purposeful opportunity that you're giving to kids.

[00:08:30.580] - Pam
To give, like you said, the kids who might not have thought about it, they now have a higher dose. They've gotten another dose of like, "Oh. Like you meant..." I said, "Oh, I wonder if there's anything on the board that could help you." But now they're like, "Oh, she actually means it. Like, we could. There is something that could help. Huh. I could use that. Okay." Yeah. One more opportunity. So, then I'm going to continue like that with a problem. Something like 7 and 1/3 plus 6. I'm going to not wait long on that one. I'm going to ask somebody to say it. I'm going to draw the number line on the board. Again, try making sure that my jump of 6 is longer than the jump of 4 for the problem before. I'm going to write down the answer, 13 and 1/3, you know, that some student gave me. And then I may... I've done a lot of this lately. I may say, "I wonder if anybody can guess the next problem." Now, I only do that when I'm pretty sure that I've got a few people. The reason I think I've been doing it lately is, you know, we've been at this for a hot minute, and typically when I do a presentation now, I have some people who've seen me before and I have people in the room who've never seen me before.

[00:09:40.310] - Pam
So, because I have that wide range... I just heard every teacher in the world go, "I have a wide range too." Because we have that wide range, I might want to lob something out to kind of keep the experts sort of thinking. And so, I might just say, "I wonder what the next problem might be," because if they've done a Problem String before, especially like this, they're already thinking, "Ooh, what could be? This could be the helper for another problem." And then just give them a chance to kind of... It gives them a challenge, a little bit more of a challenge while the other kids are still thinking about 7 and 1/3 plus a sixth. Or plus 6. Not a sixth. Ah! 7 and 1/3 plus 6. Okay, cool. Then I might give the next problem, 7 and 1/3 plus 5 and 5/6. And then solve it any way you want. So, very intentionally, solve it any way you want. Trying to free kids up. You know, I don't have to do that very often. Or sorry. I don't have to do that as often after I've been working with adults or kids for a while.
Then they're like clear, I'm supposed to solve. You know, solve it using what I know, solve it any way I want. But boy, when I first start working with people, "Solve it any way you want. Go ahead. Try to think and reason though. Try not to mimic. Think and reason, not mimic." Okay, then I'm going to walk around, see what kids are doing. I'm listening for kids thinking about relationships between thirds and sixths. I might look to see if anybody's modeling their thinking on a number line. If I'm pretty sure kids are using the target strategy, I might say, "Hey, could you represent that?" Because there might be a time where I might say, "Hey, what are you thinking? And you need to... I'm not there. You're going to have to write it down, so I can read your paper later." When you said that, Kim, that was so smart. One of the ways Kim gets kids to record their thinking is that she'll say, "But I'm going to look at your paper later, and you won't be there to tell me, so let me help you know ways that you can represent what your brain's doing so that, you know, if you have to communicate with me when we're not together in the same space, then you can write it down, and I can know what your brain was thinking."

[00:11:36.220] - Kim
Yeah. As you're talking about circulating, I'm thinking of things that I would say to kids. So.

[00:11:41.280] - Pam
Please.

[00:11:41.520] - Kim
I could see myself saying, "Look up at the previous problem on the board." Like, if I'm looking at them, and we're kind of having a conversation, and they're kind of all over the place, I might say, "Look at the problem that we just did on the board." I'm not telling them that they have to do something with it, but I'm refocusing them on the problem that they just had. Especially if they don't have something on their paper. I'm like, "Look back up there."

[00:12:04.460] - Pam
Nice.

[00:12:05.260] - Kim
Or I might say. If they're talking, talking, talking, I might say, "Write that first part down. Let's start with 7 and a 1/3."

[00:12:13.320] - Pam
Mmhm.

[00:12:14.490] - Kim
And then, "What did you want to add? Okay, draw that jump of 6." Like, I'm taking their words and helping them organize it in such a way that is meaningful and helpful to them.

[00:12:25.740] - Pam
And once they have the 7, whether they're looking at the board or they're looking at what they just drew on their paper, you could say, "How does 7 and a 1/3 plus 5 and 5/6 relate? How does that 5 and 5/6? How does that compare to that jump of 6? Is it longer than the jump of 6? Shorter than the jump of 6?" Like, you're nudging them along to use their relationships.

[00:12:43.720] - Kim
Mmhm.

[00:12:43.930] - Pam
So, now I've got... So, then I'll elicit somebody. I'll say, "Did anybody do that kind of thing where you use the 7 and 1/3 plus a 6?" I'll elicit that strategy as they're talking about it. I will redraw the 7 and 1/3 plus 6 equals 13 and 1/3 as they're describing their strategy, so that then I have two 7 and 1/3 plus a 6 number lines. On the second one, we'll back up that one-sixth, so that they kind of have this example on the board now of a helper, and then using that helper on the second number line. And then the third number line is a helper and the fourth number line is using that third problem as a helper to solve the fourth problem.

[00:13:22.720] - Kim
Mmhm.

[00:13:22.720] - Pam
So those are intentional moves. Then I might do another pair. But I might not. I might do two other pairs. But I might then end the Problem String with kind of just a clunker. And I might say, "I wonder. I wonder if you could use what we've been doing. Like, I've given you a..." I'll go up to what we have on the board and I'll say, "So, this problem was kind of helpful. Many of you used it to do the next problem." And then I'll go to the third problem. "This one was kind of helpful for many of you to do the next problem." And I'll do that all the way down the string. And then I'll say, "What if I didn't create the helper? What if you just run into this problem on the street, bam, and you have to solve it. And there's no helper. Nobody gave you a helper. I wonder, could you create the helper? You don't even have to solve it, but just create the helper. What about 9 and 5/8 plus 7 and 3/4? Could you come up with a helper that is kind of in the same..." And that was very intentional right now. "...kind of in the same pattern that we've been using? I wonder if you kind of come up with a helper, if you were to follow the same pattern." I've learned to say that one, Kim, over the years. Because if I just say, "Come up with your own helper," sometimes people will come up with helpers, but they're not really following the pattern of today's.

[00:14:29.400] - Kim
Mmhm.

[00:14:29.560] - Pam
So, that's a place where I have an explicit goal. I'm heading towards this Over strategy. And so, I'm saying things very intentionally to help kids focus in on this idea of something like 9 and 5/8 plus 8. And then I could just back up that quarter.

[00:14:46.595] - Kim
Mmhm. And if all kids didn't come up with 9 and 5/8 plus 8 as the helper...

[00:14:53.750] - Pam
Mmhm.

[00:14:53.990] - Kim
...there's no way you're going to say, "That's the wrong helper. Nobody wrote a helper. Everybody write this down. It should have been 9 and 5/8 plus 8."

[00:15:01.840] - Pam
Correct. Correct. Yeah. Yes. I'm not. Nowhere in here am I saying, "Here's how you add mixed numbers. Write the first mixed number. Then write a plus sign. Write the second mixed number underneath it. Now, they don't have the same denominator in the fraction part, so we've got to find common denominator. Well, first create an improper fraction. And then we're going to find a common denominator." Like, none of that. Like, there's none of this step by step by step. There's a lot of intentionality, and I am directing kids to do things like solve this problem.

[00:15:31.930] - Kim
Yeah.

[00:15:32.890] - Pam
Using what you know.

[00:15:34.020] - Kim
Yeah.

[00:15:35.080] - Pam
Yeah.

[00:15:35.460] - Kim
And you're naming that they are helper problems. You're naming that there are helpers to help you with what we would call a clunker problem. When you are going around working with students, you are explicitly naming things that you know would be helpful for them.

[00:15:51.700] - Pam
Ooh, and you just reminded me of something. When I give that last problem without a helper, and I say, "Write your own helper," that is direct. Write the helper. And some people will solve it, and I'm like, "Where's your helper?" And they're like, "Well, I just solved it using the algorithm," and I'm like, "No, I didn't even ask you to solve it. I asked you to write the helper."

[00:16:09.350] - Kim
Yeah. And I know that on the board you leave a little bit of a gap sometimes before that final problem.

[00:16:14.800] - Pam
Yeah, so...

[00:16:15.210] - Pam and Kim
There is space for the helper.

[00:16:16.820] - Kim
Yeah. Yeah, yeah. Yeah.

[00:16:17.930] - Pam
Yep, yep. Yeah. So, that's a very direct instruction right there to create a helper problem. That is a way to get kids to be nudged into that strategy.

[00:16:27.510] - Kim
Yeah.

[00:16:28.130] - Pam
Y'all, I hope you can kind of see in our brains a little bit here why Kim and I get a little bit... Hmm, what's a good word? Saddened by some of the conversations that's out there about direct teaching versus inquiry teaching versus explicit teaching versus discovery learning. Math is Figure-Out-Able, and we can teach it that way. And when we get more clear on what to be explicit about, and what to lob out and let kids grapple with and use relationships, use what they know to logic their way to new things while we are very intentionally high-dosing them with the important mathematical patterns, hopefully. Y'all, let us know, give us some feedback. Is this... Are these episodes where we're kind of letting you into our brain about what's logical-mathematical? What's social? When are we telling? When are we inviting kids to grapple? If this has been helpful, we'd really like to hear from you. Or if you hear a part of it, you're like, "Yeah, but what about..." That would also be super helpful. Give us some feedback in the Math is Figure-Out-Able Teacher Facebook group or anywhere on social media. Just tag me and let us know.

[00:17:32.350] - Kim
Yeah. And I'd like to also encourage listeners again that if you hear somebody say "explicit teaching", that rather than have an image in your mind that you know what they mean, ask a question. Like, can you say more about that? What does it look like to you? And I think more often than we probably even talk about, we've dug into conversations and realized that we're not saying that far off from many other people. It's just that we have different words to use that. Communication's tricky.

[00:18:04.910] - Pam
Yeah. So, let's all communicate better. Thank you. Y'all, 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!