Ep 81: Single Digit Addition Facts Pt 1

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 make the case that mathematizing is not about mimicking steps, or rote memorizing facts. But it's about thinking and reasoning; about creating and using mental relationships. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keep students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:42

In today's episode, we are going to start a brand new series all around math facts, specifically, addition facts. What we think about them, whether or not we focus on them, and ways that we think that teachers can help students know that the facts are Figure-Out-Able.

Pam Harris  01:00

Now before you shut off, because you're like, "Oh, I'm a higher grade teacher, you know, like, if this is single digit addition facts, then this is not the podcast episode for me." Well hang on just a second. So if you teach Pre-K, kindergarten, first grade, second grade, third grade, this episode is absolutely for you. However, if you teach higher grades, this episode might also absolutely be for you. Because let's be clear, we've got students at all grade levels that struggle with single digit addition facts, but maybe more importantly, understanding that those facts are Figure-Out-Able. And how that impacts so much of what we do in mathematics. Kim, I remember vividly a day that I went to work at a local middle school. And I was doing some things because I was working with the district. And I went into a particular eighth grade math classroom. And I happened to see a student I knew. I did a little bit with the students. I had some permission from the teacher. And she's like, "Yeah, come on, do a little bit of numeracy." And so I walked in, and I was like, "Hey, how's it going?" And a thing that I will sometimes do when I'm working with students, eighth grade students, I said "Hey, y'all, what is 99 plus anything?" And we started kind of messing around, they threw a number at me. And we started kind of like thinking about whether we could do that. And as I was walking, or somebody said like 47. And I was like, "Okay, what is 99 and 47?" And I walked around for a second because a lot of students immediately dove to their paper and started writing stuff. And I was kind of curious, you know, like what's happening. And I happened to glance at the student who I know, her name is Kayla. And I glanced over her shoulder. And I noticed that she had lined up 99 and 47. And she then proceeded to go with her fingers and add the nine and the seven. And I saw her count by ones to add the nine and the seven. And I thought that's interesting. Like, it's interesting that to figure nine plus seven, this particular eighth grade student who, let me be clear, is a bright, intelligent, fun loving - I mean, I know her mom really well. I know her family well. Like they are a blast to hang out with. She's like, I can't tell you, like she's an amazing young lady. And I watched her count by ones to add nine plus seven. 

Kim Montague  03:27

Yeah, you were suprised. 

Pam Harris  03:28

So, yeah I was surprised. Yes, I was like, this is important. And then, as we had the conversation about 99 plus anything, I could tell that, you know, like students who are kind of stuck in counting strategies have a more difficult time kind of getting out of that. Now, when they do, you know, bing! Bright and beautiful, and we see all the excitement, and you know, like, "Oh, I can think about these numbers!" You know, there's definitely that. But when I sort of am working with them, if they're kind of stuck in counting strategies it's harder to get them to think about bigger quantities and to think simultaneously, if they don't own single digit addition facts. And that's interesting. So teachers of older grades, parents of older students, this might be an episode for you to consider as well, as we want to help really dive in and tackle this idea of what does it mean to own single digit addition facts. What does it mean to own - like Kim, I'm just gonna ask you, flat out. Oh, actually, before I do that, I want to remind listeners, if you are a teacher of older grades, you might be interested to listen to an earlier podcast episode series that we did all about single digit multiplication facts. Yeah, so we have done that series about single digit multiplication facts. Check that out, we think you'll really like that. But in this particular series, we're gonna focus a little bit younger on single digit addition facts. Alright now, Kim. Now I'm going to ask you a question flat out. Do we want kids to know their facts?

Kim Montague  05:07

(pauses) Depends on what you mean. Right?

Pam Harris  05:10

That's interesting that you pause. Right? Like listeners are probably like, why did she pause? 

Kim Montague  05:13

Yeah. 

Pam Harris  05:14

Yeah. So what do we do? What do you mean?

Kim Montague  05:17

Well it depends on what your definition of 'know'.

Pam Harris  05:20

Of 'know'. And that became really apparent to me, the more that I talked with people when they would push back on me, and so I would sort of tell them what I do. And sometimes people, you know, I'm sitting on a plane, "What do you do? What do you do?" And "I teach math teachers," and they were like, oh, you know, they kind of tried to scoot further away from me, sometimes. And sometimes they would get a little combative. And they're like, in my face, like, "Why don't you want kids to know their facts?" I'm like, "Excuse me." And they're like, "You know, you guys, you new math people, you're all making this fuzzy and everything. It's important for students to know their facts and you're kind of dumbing it down." To which I began to get better at saying, "What do you mean by that? What do you mean by 'know your facts'? Do you mean that we want students to own them deep down to have them at their fingertips so that they don't bog you down in the work that you're doing?" Well, that's one definition of 'know'. Or do you mean that you want them to rote memorize the facts, like with a rhyme or a rap or a song or a poster with like, some kind of picture? Like, do you want it to be like you memorize a phone number? Okay, we don't do that so much anymore. Like you memorize a password like, you memorize your address? Is it something that we need to create a pneumonic for because it's not Figure-Out-Able? And that is the crux of the issue. Is a single digit fact or the set of single digit facts, are they something that must be rote memorized? Because they're not Figure-Out-Able? Or by nature are they in fact, Figure-Out-Able? Can students figure them out? Now, if we can agree that the facts are Figure-Out-Able, do we want students to be counting all the time for those single digit facts? Well, I'm going to answer that the same way Kim just did: it depends. But what does it depend on? Well, it depends on where the student is, what does the student own. If the student is a young learner who's just beginning to really understand number and sequence and the counting sequence and more than and less than and comparing. And if we're building hierarchical inclusion, and all those sort of young things, then it's brilliant when a student counts to figure out if I have six watermelons, and you bought three more watermelons, because who needs more watermelons, we all need nine watermelons. Like if I'm asking students to think about those six watermelons, those three watermelons and a student counts all or counts on to get those nine watermelons, that's a brilliant first set of strategies. But eventually, we want students to not only be quicker, because it's not about speed. But we want them to be more sophisticated in thinking about that six plus three, or whatever numbers we're using, more sophisticated, but we want those facts at their fingertips, that it doesn't bog them down, because that's more simultaneity. That's them, their brain dealing with that fact, in a more sophisticated fashion, not just reading off the answer. Not just counting out six, counting out three, pulling them together, recounting, and they're like, "Hey, who knew?" It's gonna be, you know it's that number that you end up with. That's less sophisticated than owning something about six and owning something about three and owning something about how it's going to be more than six, it's going to be more than three. And can we hang on to one of those numbers, all of that is, or hang on to both of those numbers, ideally, and the relationship that they have. And maybe knowing something about six and four and how that relates to six and three. Or knowing something about six and two and how that relates to six and three. Or knowing something about three and six, to relate to six and three. So maybe we're at least, counting on from the larger or adding on from the larger. All of those things means that their brains are more sophisticated, means they're thinking more sophisticatedly. We need kids to continually gain more and more sophisticated understanding of things like single digit facts. So, I have some personal experience with the single digit facts, right? Because I have some kids who are a little bit younger than yours. And so it's still very real to me, right? It's something that we're still dealing with. And I want to at least acknowledge that we have had some really really good experiences with some K-2 teachers who spend some quality time giving kids experiences with building relationships.

Kim Montague  10:08

Can we say, maybe if we look at the gamut of teachers we work with often K-2 teachers are the ones that already have a really good sense of what it means to build relationships?  Yeah, yeah, absolutely. 

Pam Harris  10:20

With number and math and with their students, like they're just natural relationship builders. So we honor that.

Kim Montague  10:26

Right. And so I think what happens sometimes, or at least what I've noticed happening is that at some point, there comes a time where a lot of teachers that I've seen personally say, "Oh, okay, so we built these relationships. But now there comes a time where they just have to memorize them." Right. So we've built it. And we've done some things. But now it's time. And so the experience that I had was with one of my sons, and he was in a situation where they did Formative Loop. And I don't know if you know much about Formative Loop, it's...

Pam Harris  11:01

That's a program. That's a thing. You purchase it. Okay. 

Kim Montague  11:04

Yeah, it's a for purchase program. And the idea is that there's all these different sheets that they get. And so like, you get tested, and kids can go kind of at different paces through it. But like, once you master whatever's on one sheet, then you move on to the next sheet. 

Pam Harris  11:23

And master means that you got the right answer, right? 

Kim Montague  11:25

Correct. And what's tough about it for me is that it's timed. And there's several other things that I don't love, which -

Pam Harris  11:35

Let me just be clear, when you say time - sorry - there's a time limit.  

Kim Montague  11:38

Yeah, it's like a five- ten minute limit. 

Pam Harris  11:40

And so kids are under time pressure as they're completing these. Sorry keep going.

Kim Montague  11:43

Absolutely. So they like get five minutes, or maybe it's three minutes, I'm not really sure. But you do it. And then you, the next day, do the same thing for the three or the five minutes. And then the next day, and when you master it, you move on to the next sheet. Now I've looked at some of these pages. And sometimes what's on a particular page may not be a bad idea. But the time component, you know, gets me. And what was interesting to me is that one of my sons, up until this moment, felt really confident about what he understood about mathematics. Great relationships. His teachers had done a nice job of building things. But they had reached this moment where - He felt like he was a mathematician, right? He was a confident mathematician. Yep. But they'd reached this point where now, apparently, it was time for everything to be memorized. And so this was the way they went about it was, let me give you these speed drills. And it was the very first moment for my son, who got really upset and thought, "I'm not good at this. I'm not good at math." And it was a very real moment for us as a family because I had to say, "Oh, do I step in? And what do I do? And how do I coach him through this?" And he still remembers that; still speaks to that activity. 

Pam Harris  13:08

A really poignant memory. 

Kim Montague  13:09

Uh-hum. Where the speed and the no instruction in between. So I'm not a real big fan. Not a fan, not a fan of doing things that make kids feel like they're not mathematicians. And that was the first one for him.

Pam Harris  13:26

And we're not only gonna dog Formative Loop, we're gonna dog any kind of Mad Minute speed drill that at its core has the philosophy of more and timed speed, speed pressure is the thing that's gonna make you good at this. I even hesitate say 'good at it', because good at what? Like, good at spitting out something rote memory. And what we know from research and from our own personal experiences is, it doesn't work. Now, let's define what 'work' means. When we say it doesn't work. We've got students like your son, who sort of crumble under that and don't appreciate it. My daughter was the same way. It was not a pleasant, positive experience. We know so many people who have trauma over it because it was such a traumatic experience. Kids who were more like I was, that kind of found it to be like a game. I was really into competitions so it was okay and whatever. And I did quote unquote, well on them. But what didn't work for me is that I didn't develop any relationships. It didn't help me mathematize. In fact, it hid the fact that I wasn't mathematizing. 

Kim Montague  14:41

Right. 

Pam Harris  14:41

And if anything, it sort of pigeonholed me. It kind of stuck me in a place where I was like, "Oh, look, I'm doing okay." And had this false sense that I was doing well in math, which then bit me later on. When in reality, I wasn't doing real math at all and couldn't hang with people who were doing real math. So when you say, "Yeah, but it works." Hmmm, I'm gonna, to push back on that definition that you're using that it works there.

Kim Montague  15:10

So one of the things I want to mention is that we're going to actually talk quite a bit in the series about addition facts. But a thing to note is that if our goal is the algorithm, then it's understandable why you might think that the single digit facts are the be it end all, right? 

Pam Harris  15:28

Uh-hum.

Kim Montague  15:28

But the algorithm clearly is not our goal. So we have some different ideas and different things that we want to talk about throughout the series that addresses that.

Pam Harris  15:38

Yeah, we'll get more into that, I think as we go. But I think we will grant that if your goal was the algorithm. Okay, we understand why that might be something that you are thinking you need to push in your classroom. But we're also going to argue against that. We're also going to say we get why you think that's true. Would you consider that maybe it's possible that we can get to automaticity of facts, without rote memory. Without all the rote memory kinds of things that we've been told work, but actually not so well. Because in a huge way, we want to decrease the emphasis on speed and on that recall from rote memory. And we want to instead emphasize building additive reasoning. How can we really help students reason more additively. Y'all, we have a download for you that can help you understand the difference between students using counting strategies and what it looks like for students to use additive reasoning. We created an infographic all about additive reasoning. It's a one page cool download that has videos attached to it. You're going to love it. So rather than print it, you're going to want to watch it online. So you can click on those videos and check it out. You can grab that free download. It's a bitly link, so it's bit.ly/pharrisinfo. Now when you go there to bit.ly/pharrisinfo, or we'll put the link in the show notes. You'll notice that we have several infographics. For today's episode you're going to want to look at the additive reasoning infographic. But by all means you're welcome to look at the other ones. We like them. We think they're pretty cool as far as helping really understand sort of what we mean by reasoning. Particularly for today's episode, additive reasoning.

Kim Montague  17:25

Absolutely. So also super cool right now, we're excited because the You Can Change Math Class Challenge registration is currently open. 

Pam Harris  17:35

Oh yeah!

Kim Montague  17:36

This is completely free. It's open to all teachers and leaders - I mean, we think math teachers are gonna love it. It is great for anyone who knows that there's got to be a better way to teach all students. So you can check out that registration and join us for the You Can Change Math Class Challenge.

Pam Harris  17:54

Yeah, we'll put that link in the show notes or head over to mathisfigureoutable.com. We'll have a big banner or something up there for the You Can Change Math Class Challenge. You are going to love it. Like she said, it's completely free. Y'all if you like the podcast, but when you listen to it, you're like, oh, I wish I could see that. This is your opportunity to get it visual. Pam Harris visual in You Can Change Math Class Challenge, totally a blast. See us there. So if you want to learn more mathematics and refine your math teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word. That Math is  Figure-Out-Able!