Addition is figure-out-able! In this episode Pam and Kim describe some of the fundamental patterns students should learn as they get familiar and automatize with their single -addition facts. We want students at all levels to have access to their facts so they don't bog down every time they encounter them.
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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 00:09
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. Y'all, we take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being mathematicians they can be. We answer the question? If you're not teaching algorithms and step by step procedures, then what are you?
Kim Montague 00:39
Yeah, so last week, we started the conversation around addition facts. And we mentioned some really important ideas about speed, conceptual understanding and fluency. And we also know that although we're talking about something that we wish younger grade students have a great handle on, we are all very aware, right, that we have some older students who still need experience with addition facts. So we want to invite all grade level teachers and leaders to join in today, as we share about patterns with addition facts. These patterns are relationships that we want students to identify, explore, play with, individually own.
Pam Harris 01:18
Yeah, it's going to be important the way we learn the addition facts, the way that we automatize them, because we absolutely want students to know their facts. Not rote memorize their facts, but to know them, to own them down deep inside. So before we get into tons of patterns, the first thing that I want to talk about today is the fact that we need students to understand what addition actually is. And it needs to be connected to experience, to some sort of concrete things. Now this is a little different than just CRA. So listen carefully. You might be right now going, "Oh, yeah, CRA. She's just talking about concrete, representational abstract or concrete pictorial abstract." So I'm actually not. What I mean here is, and I think many, if not most, teachers today are aware of the fact that we don't start with students by saying, "Okay, lovely young students here is: three plus five equals eight, now you do it." Because I personally know of people who looked at that equation as a young student, and said, "Three what? Plus five what? Like what is this?" And literally had no sort of context or meaning behind those symbols. And so we need to make sure that we do things with students where we just give them problems to solve that aren't necessarily represented symbolically. We literally just say, "Hey, I got five - now pick something - markers, and seven markers, how many total markers do I have?" Or, "Whoa, I had eight markers, and you just took three of my markers. How many markers do I have left?" Like, we just throw out these problems. And we let students solve them with concrete materials. And we let them sort of fuss with them. And we then represent what they're doing, using more symbolic kinds of things. We're like, "Oh, when you did that, that could look like this on paper. Like when I just gave you that problem, it could look like..." so for example, when I said I had five... now I already forgot what I had... five markers. And you gave me seven more markers. How many markers do we have? I could represent that with Ooh, this is a five. And then I "gave you" and I put the addition. So you can't see me. I'm just realizing I'm on a podcast. So I wrote the number five in the air. And then I gave you seven more markers. And so then I wrote the addition sign, the addition symbol. And then there's a seven and I wrote the seven. So the 'gave' is sort of the addition symbol. And now I've got seven. I write the seven. I'm like, "How many total is that?" And then I might write the equal sign at some point when we've got sort of those 12 total. And then I might write, "Oh, and you told me there were 12." So I'm just sort of represent the problem itself using symbols. Now, you might be like, "Pam, nobody looks at five plus seven and says, what are you talking about?" Well, I married him. And my husband can remember the days of sitting in first and second grade and looking at the symbols on a paper and having no idea what they meant. And realizing later in life, not too much later, like third grade. "Oh, like you meant five paperclips plus seven paperclips. That's what you... well, I could have done that."
Kim Montague 04:32
Pam Harris 04:32
Now don't look down on my husband. Plenty of people were here. But he literally thought that he was supposed to - it was like a puzzle that he was supposed to memorize. So this symbol plus that symbol. And sometimes, he said, "Pam, I even thought that like five plus seven should be five and then a seven. Like you're putting them together, like 57. Like you're putting five plus seven together." I mean, he was trying to make sense of what was happening and because there was so little experience attached to the symbols that he was having a hard time with that. So just right off the bat, I won't talk too much more about it. But we need to make sure that we give kids experience. Kim, you and I have had experience doing this with young learners where we might do something like, "Hey, let's say we've got red and green apples." Now, we could choose any kind of total, but I'll choose 10. Because 10 is kind of a cool total. We could say, "If I have 10 apples, how many green ones do I have? And how many red apples do I have?" And then let kids go. This is a super nice, open rich task for young students to tackle. Because we can have students that just start guessing. They can start pulling out some red apples and green apples and kind of like, "Oh, here's some green, here's some red. Is that 10? Nope, try again." We've even seen, like, they'll put them all back and then grab, "Okay, how about these number of red?" And they're not even really like judiciously, like taking away a green or red or adding more in or you're like, "Oh, I only had nine, maybe I better get some more." But we can sort of then step in as the teacher and go. "Well, I noticed when you had those three red and six green that you only had nine. But how many are we looking for? 10? Do you need more apples? Do you need less apples?" We start asking those questions to help kids kind of get more clear, you know, if they're kind of haphazardly doing stuff. But we can have all the way where we might have some students in that same exact class where we can ask, "Hey, do you have them all? Do you have all the combinations of all the number of red apples and green apples? For example, I see that you have six and three, six green and three red. Do you have 6 red and three green? Well, I see that you have five and five? Do you also have" - did I say six and three?
Kim Montague 06:43
Pam Harris 06:45
Sorry, six and four. How many apples are we dealing with Pam? I'm alive. I'm awake. I slept last night. Sorry, six and four, and then five and five. And we might have a kid go, "I'm done. I did it all." And you're like, "I don't know, your friend over here had eight green apples, is that a possibility? Could you have eight green apples? Could you have some red apples to make? Do you think you have them all now? How do you know you have all the combinations?" Bam. I mean, that is a great extension question that could keep many kids thinking for quite a while. While I might be helping students that are still making sense of you know, any kind of combinations that are reaching 10. So we really like that. In fact, I'm gonna give Cathy Fosnot credit for first introducing that task to me. We've used it lots with students. We've kind of made it our own in some ways. But that could be a way, that's an example of helping students really understand what we mean, when they're later going to deal with these single digit addition facts. Oh, it's like all the ways that we made 10 apples, not nine apples, Pam, ten. Oh, my goodness,
Kim Montague 07:47
One of the things that I love about that, you mentioned that there's access for so many different experience levels that kids can also do that task without having a lot of notation work, right? They can fill out a chart of two-eight, seven-three. And then it gives you an opportunity to introduce the plus sign or an equal sign right? After they've already done some work. Messing and figuring about the apples.
Pam Harris 08:15
Really nice. In fact, I'm gonna back up from that. You could even just have them draw red apples and green apples. And you can underneath that write the numerals. And then later introduce the plus symbol and the equals. That's very nice, very nice. Yeah, I like it. Let's now talk about some of these patterns that can help kids automatize the facts. If we can help kids identify and explore and play with some specific patterns, then the facts start to make sense. And it's not just what I think some of us think. If I go to the store, and I buy a deck of cards, addition flash cards, here's my deck. It's a pretty thick, old deck, right?
Kim Montague 08:56
Pam Harris 08:56
And it could be quite daunting. And you might have parents that go, "Oh my gosh, my kid has to learn all of these." And 'learn' might feel like rote memorize. And that can feel very daunting. So if we can look at some patterns and go, "Oh, actually, there's a whole sets of cards that we can be like, I understand these. I don't need to even think about these. I could just like do them. I know these." We can kinda set those aside and work on maybe the more difficult facts and how we could think about relationships for those.
Kim Montague 09:26
So one of the things that I have a lot of experience with is that as a third grade teacher, we did this with multiplication, but it's absolutely true for addition as well, that when you connect the commutative property to the facts, right? You help kids make sense of the commutative property, then all of a sudden, half that card deck or half of the facts on the addition table instantly goes away. Right. So if you're looking at an addition table, there's this diagonal line that you can cut the facts in half or if you have a deck of cards that have, you know 100 facts, and once kids make sense of the fact that it is seven plus two, and that's going to give you the same total as two plus seven, then instantly, half the facts are gone.
Pam Harris 10:12
Kim Montague 10:13
And you can connect that so early on even with the rich apple task, because you're going to have three red apples and seven green apples, and seven red apples and three green apples. And if you focus on the fact that you still have the total of 10, then they can start to see that three plus seven and seven plus three still give you 10.
Pam Harris 10:35
Still have the same total, which is the commutative property. That A + B is equivalent to B + A. We get the same total. Nicely done and nice connection back to the apple task. Yeah, yeah. So we like to do this with parents sometimes when we talk to parents. And we're like, "How do you want help your kid? Well, one way is to help them with their addition facts. And we're going to send a flashcard deck home with you." Now we'll talk a little bit more about what that looks like in another episode. Coming up soon. So keep listening. But when we do that we help parents understand, but check it out. Like let's work on this idea that two plus five is equivalent to five plus two. And bam, like we really only have to think about half of the deck. If we can sort of use the commutative property to think about facts. So very nice. It's a great place to start. Let's also talk about a pattern in the addition facts with anything plus zero. So that might seem like the easiest one. You might be like, "Well, Pam, three plus zero, I mean, that's just three and eight plus or zero plus eight, that's just eight." Right, but we do need to take into account that zero was actually created or invented or established fairly late in our history. We had other mathematical concepts and numerals and things happening before we kind of came up with this idea of that we needed to represent zero, an amount of zero. So that can actually be a little tricky for kids. That's why we do problems like the apple problem. Because in that list, in fact, we should have probably asked you guys before I just said that, y'all, if you're looking at that apple problem. And I said to you, "Do you have them all?" Did you include zero red apples and 10 green apples? Or 10 red apples and zero green apples? Like did you include sort of this idea of having 10 plus zero, still being 10 apples? And if you have, then right here and now when a student sees an addition fact like five plus zero, they can be like, "Oh, yeah, like meanie, you didn't give me any more apples. All right, looks like I just have the five apples I started with." And they can sort of make sense of that plus zero. But now that we've made sense of plus zero, bam! Those are out of the deck, like we can sort of take them out of the deck and we can move on and deal with other patterns. Let's talk about another pattern in addition facts. What about plus one, or even plus two. So for example, if I look at nine plus one, or 13 plus one, I mean, even 56 plus one, don't memorize anything, plus one. Don't rote memorize that. That's not a fact you have to memorize. Just think about it, like just do it. Just like 56, 57. 56 plus one is 57. Just think about the next number in the sequence, or plus two just just add on two. And I'm totally okay if kids count at that point. So instead of saying, "Alright, five plus two, you got to memorize it. Five plus two is like your shoe in the door. And so is 4." Wait not four. I mean, I was trying to come up with a rhyme off the bat. Like instead of trying to come up with something to help the kid memorize it, just think about what it means. Just do five plus two, okay, 5, 6, 7, bam, I've got seven. And that takes the pressure off all of those plus ones and plus twos. And we can just sort of think about those as kind of the 'just do it' facts, and we can take them out of the deck. Bam.
Kim Montague 13:53
So I'm gonna interrupt you here for a second because I can hear some of your listeners saying, "Wait a second, wait a second. Isn't that going to get kids stuck in the counting domain of the Development of Mathematical Reasoning?" Right.
Pam Harris 14:07
Oh, nicely done. Yeah. So if anybody's been listening, and you were thinking about the Development of Mathematical Reasoning, and we're talking about not leaving kids in counting, but helping them gain additive thinking. This is one of those times where it's okay to count. Yeah, just take the pressure off of memorizing those facts. Let them count the one and the two. Now after that, now, I'm again - I just said again, like you knew it was in my head. So let me say it for the first time, so I can say again. If a student just knows five plus two, which often they do, right? We're okay with that. We're not saying, "Thou shalt go count 5, 6, 7." No, no, no, if they know five and two, and they just say seven. Yay, good. Good. That's a good thing. This is only if they don't know it. If they don't just, you know, like if I say four plus three, and they don't just pop out with seven. Then in that case - oh I just said three- four plus two, in that case, then I'm okay with him going for five, six. They're going to do that quick enough, it's not going to bog them down. After that four plus three, now I'm going to want to work on some other relationships to help them sort of automatize those facts that I don't want, really want kids counting plus more than one or two.
Kim Montague 15:19
So it's interesting that we, you know, we talk about these patterns in these relationships. But there's also some things that are happening simultaneously. That when you work on this count-on fact, and you've also had this conversation and these experiences with the commutative property, then you also are working on counting on from the larger number. Right? That's not something that we kind of parsed out, but I definitely want to. When you have the problem, five plus two or two plus five, if you have a kid that needs to count 5, 6, 7, you're okay with that. But we're not okay with somebody starting with two and saying there's a two in the problem. So it must be 2, 3, 4, 5, 6, 7. Right?
Pam Harris 16:01
Kim Montague 16:02
At that point what we want them to do is say, "Okay, I see that it's two plus five, I'm going to flip that and use the commutative property in my head. And I'm going to say that's the same as five plus two. So 5, 6, 7." There's a lot going on there. This is a lot of thinking. I had a student once, a group of teachers that were puzzled by the student who, for whatever reason, would not start from a larger number, would not. If the problem was given to him like five plus two, then he would say "5, 6, 7." But if the problem was two plus five, then he would not. And I worked with him for a little bit -
Pam Harris 16:41
He would started at two and count up.
Kim Montague 16:42
He would start at 2. Whatever the problem was, how it was written, would be the way that he -
Pam Harris 16:48
Thou shalt do it in order.
Kim Montague 16:49
He would do it in order.
Pam Harris 16:50
Kim Montague 16:51
And so I was intrigued, and I wanted to know why. And so what I found, I did some work with him. And what I found was that he didn't trust the commutative property.
Pam Harris 17:00
Kim Montague 17:01
So he thought, like, he was surprised to know that they were going to give him the same answer. He was okay with that. But he didn't trust it. And so when he was given a problem, like two plus five, and teachers just said, "Oh, it's going to be the same. Just flip it around, or what else would that be?" Then he wanted to start in the order that the problem was given. And so we did some activities. We pulled out a rekenrek, we built five plus two, we flipped it upside down, saw that it was the same as two plus five. And he-
Pam Harris 17:35
Wait, wait, tell me what you mean by flipped upside down?
Kim Montague 17:39
Flip the rekenrek, the actual rekenrek, flipped it upside down.
Pam Harris 17:43
So okay - So tell me what it looked like first.
Kim Montague 17:46
So if he had the problem, two plus five, then he would move two beads over on the top, and five beads over on the bottom. So that represented two on top five on bottom.
Pam Harris 17:57
Kim Montague 17:58
And we wrote that down: 2 + 5. And then he literally took the rekenrek and flipped it over. So that now it had five on top and two on bottom. With no beads moving in either direction. They just -
Pam Harris 18:11
You just literally turned it. Yeah, upside down.
Kim Montague 18:15
And so we spent some time working with is it going to always be true, that when you flip the rekenrek over, no beads were exchanged in any way. And with some experience doing that, then he started to trust that he was going to get the same total.
Pam Harris 18:32
Isn't that fascinating? I love it.
Kim Montague 18:36
Pam Harris 18:36
And how brilliant that y'all were willing to hang with him and deal with him and not just go, "Dude, do it. It's this, like, make it happen." Like you worked with him and talked to him. I love the flipping the rekenrek and that's pretty cool, rekenrek upside down. Nice. Y'all, these are important patterns that we can identify, explore, play with and eventually own. So important.
Kim Montague 18:57
Yeah. And when you say eventually own, it's not magic, right? It's not going to just magically happen. Sometimes people will say, "Well, hey, I've heard you talk about this. And so I played that game that you mentioned," or "We kind of walked through some patterns on the addition table and they could kind of see while we're talking about it." We're suggesting that kids need repetition, right? They need lots of experience. It's just not the flashcards that we did when we were kids.
Pam Harris 19:25
Yeah, what we're not suggesting is that you just rote, it's just like we're gonna rote drill you over and over and over. Not that. But lots of experience with apple-type problems. Lots of experience with having the kids tell you their reasoning for the facts and how they know it and what it looks like on a number rack. And then playing games where we expect them to spit back facts, but it's in an atmosphere where it's not about rote memory and it's not about speed.
Kim Montague 19:52
Pam Harris 19:52
And to be clear, earlier when I said, "Okay, so if you know plus one and plus two then you can just take those cards out of the deck. If you know plus zero, you can take it out of the deck." When I say that, I kind of mean, take it out of the "Okay, now we need to drill a kid on these". But I don't mean that we never experience them again, like, I want kids to see five plus zero and to see two plus five and think about five plus two in that case. I want them to see all - when I say take them out of the deck, I don't mean don't have experience with them again, I just mean don't fuss with them again. Now let's go on and explore other relationships. But I do want to have those problems come up all the time. So students deal with them over and over and over again. And that will help them as they use relationships to figure them out. That will help them automatize them so that they are less bogged down every time they see them after that.
Kim Montague 20:42
Yeah. Join us next week as we continue to parse out patterns with the addition facts that can help students automatize the facts. So right now Pamela, we have the You Can Change Math Class happening. That challenge is going on right now. It is so fantastic. We have the most amazing teachers and leaders joining us.
Pam Harris 21:02
It's totally one of the funnest things I do all year. Woo! We love it. And it's not too late to join us. So join us for the challenge each evening. You can register at mathisFigureOutAble.com/change. We're also super excited because this Friday is open registration for some fabulous workshops written by somebody we love. They are happening right now this Friday, you can register. Yeah, and y'all we have four super workshops. I'm really, really excited about and proud of Building Addition for Young Learners, Building Powerful Multiplication, Building Powerful Division, and Building Powerful Proportional Reasoning. They are my pride and joy, I am thrilled with what we were able to create for y'all to learn more real math, to take a deep dive into content, whatever content really fits for you. You can register for those at mathisFigureOutAble.com/workshops. Y'all if you're listening to this podcast at a time where registration isn't open, you can still go to mathisFigureOutAble.com/workshops and get on the waitlist and we will notify you the next time that we open registration. Also we'll be doing challenges. So get on the waitlist for the workshops, get on the email list and then you'll know the next time that we do a challenge if you happen to be listening to the podcast when we're not doing a challenge or when registration for workshops isn't open. 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!