Hey fellow mathematicians. Welcome to the podcast where math is Figure-Out-Able. I'm Pam. And you found a place where math is not about
And I'm Kim. memorizing or mimicking, waiting to be told or shown what to do. But 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. Okay, so in this episode, we're going to answer a question you got on Facebook
from Diana. And she said, "I just want to say I have thoroughly enjoyed your podcast and free learning challenge." So exciting, right? "You've opened my eyes to how math could and should be taught to our students. I'd love to hear a podcast of how you'd structure your math block now, if you are in a K-12 setting. I know it would look different for each grade. But I'm really thinking of revamping my lessons in August to make more time for kids to talk about math with
each other. Thanks for all you do. I teach fifth and sixth grade math students. And you're really so inspiring." What a great message.
Awesome. So thanks, Diana. I appreciate that. It's a little later than August when we're answering this. But we did get to it as soon as we could. So, fabulous question. What does it look like? And Kim, you know, we've just gotten a message from a couple of people, but specifically, Tad Watanabe asked about why we only do computation, or we sort of talk
about computation a lot. And that's kind of been pinging and then this thing from and Diana were like, maybe we should talk about the fact that we don't do just computation, but we do other things and what it would look like. So one of the things that we've mentioned briefly before in the podcast is that we do take what we call a two thirds - one third approach, or sometimes maybe a four fifths - one fifth, or a three fourths - one fourth approach. So, Kim, tell us a little bit about that.
Yeah. So when I am thinking about my math classroom, I definitely have some things that my school or my principal or my district requires of me. Right? I have a scope and sequence sometimes, sometimes not. And so I tried to think about what is really important to me, as far as splitting up my time to do what was required of me, but also that I could do some of the things that I know is really
important for students. And so I really consider the two thirds - one third was the two thirds is kind of what I have to do, but also can be really good stuff. And then 1/3-
And the way that you do it.
And the way that you do it. But the one third is the play time for me is the time that like, I throw in the things that I know are super great for kids and things that I love to do with kids. And that was kind of a way that I sorted out how in my mind and made sense of that time. And sometimes the one third is separated from the two thirds, like two different times of the day because I was elementary, and sometimes they backup together. But anyway, that's how I split my math time.
So let's kind of separate that a little bit. Because I think you mean a little bit more of content.
Yes.
And a little bit less of the way you do stuff.
Okay.
So you're not saying you do cruddy teaching during the two thirds time.
I hope not.
And you do the fun stuff during the 1/3. That's not
No, no. So the standards that might have been what you mean. the scope and sequence that my district came to me and said, right now, during this part of the year, you're supposed to be teaching fractions. And so if it was a fraction time of the year, then in that two thirds time, then I'm working on fractions.
And then the 1/3 time, I might be doing a whole host of other things, but I really kept pace with the scope and sequence during the two thirds and filled in other things I knew my kids needed during that 1/3.
Yeah, excellent. Okay, so some of the things that can belong in the two thirds time, hmm things, some of the instructional methods that could be long in the two thirds time while you're doing the content that your scope and sequence is kind of demanding. The way that you attack that content could be things like what we've called Rich Tasks, or sometimes we call them Investigations, or sometimes they're open middle tasks, or low floor, high
ceiling tasks. So lots of different names, we actually kind of have particular ways that we like to do Rich Tasks. So it's not just kind of whatever anybody thinks about what a Rich Task is. But we've talked about that in a couple of other episodes, maybe we'll put that, those episode numbers in the show notes. So Rich Tasks are definitely happening in that two thirds time. But also we
Right. think about if we're doing that Rich Task and this time to get
How those things happen. And that that often is a at that main content that serves in the scope and sequence. What do we need to have percolating? What kind of things need to be bubbling up for students? And so then we think about what kind of instructional routine could we do to help bring up those ideas? Problem String. Sometimes it's a different instructional routine. But often it's a Problem String, we think about what needs to be happening percolating or bubbling up for kids to make the
Rich Tasks successful. Not what kids do already need to have cemented and for sure, like it's all in cinch. No, just what needs to be kind of happening, bubbling, percolating, spouting off, like, sort of pinging around the room.
Yeah.
So we might do a Problem String to get that happening, then we do the Rich Task. And as we come out of that Rich Task with a Math Congress, we do sort of solidify some things. But really in a Rich Task, we envision that kids are struggling, they're, productively, they're grappling, they're in disequilibrium, they're clarifying. They're making sense of things. So as we come out of that Rich Task, we're just going to admit that lots of good things were
happening. But we might not have cinched several of those things.
Right.
We may have cinched something in the Congress. But chances are not really. Like we might have brought some things into more clarity. But we still need to like cinch some things more. And so we might do that following that Rich Task with that Congress with a Problem String to sit somewhere or with another instructional routine. Most often it's a Problem String. But there are some other instructional routines out there that can help us kind of cinch
some things. And we sort of see that kind of as a flow that happens in that two thirds time. Would you agree?
Yeah. Yeah, absolutely. Yep.
Yes, I missed a thing?
Oh, no. I thought,
I said both questions in a row.
And I was like, absolutely.
Did I leave anything out? Did I miss anything? Those are two different-
No, no. So the two thirds time, I think what you described is exactly what it looked like, for me. That's the richness of really digging into the content that we're talking about right now. And letting kids grapple. It's kind of messy sometimes. And then having really good conversations.
Yeah, excellent. So then what happens in the 1/3 time? Tell us about some of the things that are kind of in this other track.
So the 1/3 time, you know, I was thinking about, like, what are some of my favorite things to do during that time, and some of it again, is, you know, it's maybe not fun, but necessary, and some of it is still a little bit of play time. So in the 1/3 time would have been when I pulled small
groups to work on things. You know, I've mentioned to you before that I was a firm believer in having kids look back at their work, whether they, you know, did really well on something or whether there were some things that they needed to be clear about in their notation, or something that I looked at it and there wasn't really a lot of evidence of them making sense of something. So almost, you know I wasn't a huge paper pencil
assignment giver. But almost everything that I handed a kid, they came back and you know, I handed back to them, and they analyzed it, and then they came back to me. And that takes some time. So, you know, small groups, kids looking back at their work, I might pose a Would You Rather, we might do some test formatted practice, right? Because-
What test formatted practice, Kim?
Yeah, I taught in testing grades. And so I wasn't going to let the types of questions or the fact that it was worded in a way that maybe I hadn't had them see before. So you know, there's a little bit of that.
Or the fact that they had to bubble, because they didn't bubble for you for homework. So you wanted to make sure they knew that. You were smart, you were wise that the test day was not going to be this big shock. Was not, "Oh, I've never seen this before. I've never done this kind of thing." So you gave kids experience. Hey, it's all about experience, right? You gave kids experience with, "Hey, you, you are going to run into items like
this." Ways of the questions being worded, the format, the ways that they were going to have to answer, all of that made sense.
What do I do when I don't have a lot of space on my paper? Yeah, that's it. Yeah.
You gave experience to students so that they would be able to better jump into a test time.
Yeah, absolutely.
And actually do as well as they could do, not being hampered because they didn't have experience in some of the formatting or something. Yeah.
So that's one of those maybe not so fun things that I mentioned that we still have to do. It's still a reality for-
Well, I teased you a little bit, like really? Test formatted? But I actually completely support the fact that you don't want your students to be to take that stupid test, and have it, and not be able to do as well as they can for reasons other than math. Like let's get those out of the way. Yeah, absolutely. Yep.
So we might have done you know, if a Problem String wasn't on my radar for, within the Rich Task or Investigation that we were doing, I might have done a Problem String during that 1/3 time.
But I think what you mean by that is like some other content, right? Like you're thinking, hey, for this grade level, say you're in fourth grade, I know that students are going to be working on blank all year long. Like we're gonna be getting multiplication to bigger and bigger chunks, getting better at these strategies all
year long. So you might have this in your head, this kind of separate track happening, of where even though right now we're in geometry, I'm going to keep this separate track of multiplying by bigger chunks and strategies happening as you as you go. Yeah, yeah.
Absolutely. So I just listed a couple of my
Which One Doesn't Belong, Open Middle, Guessing My Number, of course, As Close As It Gets, relational thinking. You know, the, the small group pre and post, like getting kids, you know, if we're about to do some fraction work, just kind of getting into kids heads a little bit about what do you already know, so that I'm not just repeating things. I'm not tackling things that I know, my kids already know, and wasting time. So a whole host of things, and that 1/3 time could be
happening. And they change. They change regularly.
Yeah, and I think this could be true at any grade level that we could kind of have this idea of what are these major things that I know need to happen in my subject, my grade level, but aren't maybe in the unit that we're in, but I want to keep that track going. I'm
gonna give an example. A thing that I might do all through high school might be some work towards factoring quadratics, where we just deal with, "Hey, if I give you these two numbers, what could, what pair of factors would multiply to this product, but would also sum to this sum? And add to the sum? And that we might do that work so that when we actually get to factoring quadratics, students, when we bring the variables in, students are already messing with those
number relationships. And so that sort of slides a little bit easier. And I might do that after we factor quadratics to get kids better at it and keep it alive. Because I know, that's gonna be kind of an important thing. It will show up in Calculus where students need to factor quadratics. And so you know, that just might be a relationship that A is kind of fun to play with. And B you might want to keep alive at
other times. So yeah. So that's kind of the maybe the goal, the overarching thing to keep in mind as you consider what would be some things that would happen in that 1/3 time? Well, we have a lot of these instructional routine kinds of, types of
things. So you can choose from, like you said, and maybe I won't repeat them, or maybe I will repeat them, Problem Strings, Problem Talks, games, Which One Doesn't Belong, Open Middle, Would You Rather, As Close As It Gets, relational thinking, Guess My Number, like all of those
kinds of things. But then you say to yourself, "What do I, what's the content to do in those?" These are the major big ideas, big concepts, big trip up things that I know, we need to start at the beginning of the year, and carry through all through the year to keep building and getting students better at.
Which is such a, in my opinion, a better way of spiraling some ideas that we know we have to give kids experience with instead of just buying a premade spiral sheet that they like, have one question on geometry, and one question on fractions, and one question on addition. You know, those kinds of things where the idea is to give kids experience with lots of different things repeatedly. I just think these are much better instructional routines.
Well, let me dive into that just a little bit. Because there's some really good programs out there that I think brought that to the forefront for many of us. I'll give CPM credit for really helping me think about the fact. In fact, also, oh, golly, what's that? Saxon, I mean, which I don't really recommend a whole lot, but they brought to the forefront for many of us this idea of "Whoa, if we kind of spiral this practice that that can be helpful."
Yep. Yeah.
However, the way that Saxon did it, because of the way they taught the original, I'm kind of being a little negative here, but I'm gonna. The way that they sort of taught the initial stuff was so based on rote memory. And then if you could just mimic what the teacher was doing that what ended up happening was if students were in that mimicking compliant mode, that, "I'll just compliantly mimic what you
want." Then that spiral review, spiral practice overtime worked really well for that particular student. But for anybody who doesn't learn well, that way, it was horrible, because every time that spiraled practice came back around, they're like, "Yep, can't do it again." And it didn't, there was nothing to help them have better access to it. Now, I'm actually gonna say, I just said that it was okay for the kid that was compliant and was willing to kind of just, you
know, mimic and memorize. But in reality, it's not okay for that kid because it gives you this false sense that what you're okay. Define okay. You're getting answers and so you're getting grades. But you're not mathematizing. So, yeah, it really isn't good for any learner. However, the idea of keeping, of building small or starting small and building throughout the entire year so that we keep coming back to these major things. We learned that from those folks. And we definitely
want to apply it. We just think there's a better way to do it, which is through those instructional routines that we just mentioned. Hey, Kim, I'm wondering that if, what might be good as we've been talking, especially as you were talking, I was thinking, "Oh, I want to give an example here. Oh, I want to give an example there." But I'm aware that we're sort of up against kind of our timeline that we usually have for podcasts. What would you think if we did that in the next
podcast episode? We'll give some, give some examples. Maybe we'll choose, you know, like, major grades kind of along the span K-12. We'll give us some examples of what it could look like.
That'd be great. Yeah. Let's do it. Fabulous, cool. So y'all at this time of the year, we want to let you know that there are some things happening that don't happen all that often. Y'all if you're interested to learn more in depth, what this could look like on the ground help for teaching Figure-Out-Able Math, well check out our online workshops, because registration has just opened. If I may, I'll tell you just a little bit about it. We've got Building Addition
for Young Learners. That's for pre K through second grade teachers, where we really dive into that young learning. What does it take to help students develop counting into Counting Strategies into Additive Reasoning? So if you teach young students our Building Addition for Young Learners is fabulous. If you teach older students, Building Powerful Multiplication for grades three and up really helps you develop the major models and strategies for
multiplication. And also looking back, how do you develop multiplication with young learners. But then we go clear into what does it look like to really multiplicatively reason in the high school. Building Powerful Division does the same thing for division for grades three and up. Building Powerful Proportional Reasoning middle grades teachers, you really want to be looking at how can we help develop reasoning not just solving proportions, answer
getting. And also in a high school, we've got Building Powerful Linear Functions. That is just coming out. We are just launching our first high school workshop. Great for eighth grade on up. All about linear functions, where we take students from really not knowing very much at all about linear functions, to writing the equation of a line from no rote memory in about 18 tasks. Yeah, and I'm, y'all, I'm so excited for this launch, because it
might be my best work yet. We've been creating it and getting it ready. And it's coming out. I'm just, I cannot wait for the world to see Building Powerful Linear Functions. So if you teach eighth grade and up, head on into that one, checkout registration, so excited. And you know what, there's so many great workshops and registration only opens three times a year and this is one of those times. It's just in time for the school year to start for
many of us. So you'll want to check out mathisFigureOutAble.com/workshops, or we'll have the link in the show notes for information and to register.
So thank you for tuning in and teaching more and more Real Math. To find out more about them Math is Figure-Out-Able movement, visit mathisFigureOutAble.com. Let's keep spreading the word that math is Figure-Out-Able.