When we teach Real Math, students come up with all sorts of cool strategies! But which ones are the main strategies, and how do we make the distinction? In this episode Pam and Kim take a listener submitted strategy and discuss where and what its uses are.
Talking Points:
Check out Episodes 7 and 164 of this podcast for more about I Have, You Need
Check out our free eBook to learn more about these strategies: https://www.mathisfigureoutable.com/big
When we teach Real Math, students come up with all sorts of cool strategies! But which ones are the main strategies, and how do we make the distinction? In this episode Pam and Kim take a listener submitted strategy and discuss where and what its uses are.
Talking Points:
Check out Episodes 7 and 164 of this podcast for more about I Have, You Need
Check out our free eBook to learn more about these strategies: https://www.mathisfigureoutable.com/big
Hey, fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
Kim:And I'm Kim.
Pam:And you've found a place where math is not about memorizing and 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. Hey Kim?
Kim:Yeah?
Pam:Did you know that as of today, at the time we are recording this, we have over 250K downloads. In fact we have,
Kim:Oh, my gosh! That's great!
Pam:...267,000 downloads of the podcast.
Kim:That is super fun!
Pam:I mean, I actually have no idea if that's really any good, but it sounds good, doesn't it? I mean, it sounds amazing! That's like a quarter million! We're at over a quarter million downloads, since I don't know. Hey, let's go for a million. Whoa! Alright, ya'll. Tell your friends. Tell everybody. Let's get a million downloads. That will be exciting. Anyway, I'm thrilled.
Kim:Yeah.
Pam:Who knew? Who knew we would.
Kim:Who knew people wanted to hear us talk about stuff.
Pam:(unclear).
Kim:Sue did. Sue did actually.
Pam:Sue did.
Kim:We should tell her that.
Pam:We should we should totally give her credit. We give. Sue is the one. In fact, here's a little short story. Sue is the one that. We used to, I don't know, go places, do things. We would put on workshops.
Kim:We would travel a lot, yeah.
Pam:Traveled together, and we'd be in the car and you...me, Kim...would beat stuff out, right?
Kim:Yeah.
Pam:Like something would happen at the workshop. And we like,"What do you think about..." We would go back and forth. And Sue's a little bit quieter. She doesn't, you know, usually jump in as much. She doesn't interrupt us.
Kim:How could you (unclear).
Pam:You have to interrupt, if you're going to get in between us, right? And so, every once awhile, she would just quietly go, "We should have recorded that."
Kim:Yeah.
Pam:And we would say, "What do you mean record this?" She's like, "People would listen to that." We were like, "Nah, that's dumb." Okay, here we are. Alright.
Kim:Who knew?
Pam:Whatever. Hey, so another thing that we're going to just quietly announce is that something new, and cool, and I think you're going to like is coming this week. You're going to love it. Just kind of maybe be aware, podcast listeners, that something new is about to happen. Alright, moving on from that.
Kim:Very cool.
Pam:I'm not going to make too big a deal of it. There you go.
Kim:Okay so. We love hearing from listeners, right?
Pam:Absolutely.
Kim:So much fun to get emails, and questions, and comments that they have (unclear) listening each week. And so, in this episode, we are going to actually answer a fantastic email question that we got from Amanda.
Pam:So, thanks, Amanda! Check out this. I'm actually going to read the email, everybody. Ready? Because this is fun. Here we go. So, Amanda said "Hi. First of all, I want to thank you." Well, Amanda, you're welcome. That's excellent. "I have learned so much from you. I'm a K-5 math coach. I've participated in your recent challenge including..."
Kim:Woohoo! Challenges!
Pam:Whohoo! Hey! In fact, ya'll, we're trying to become the challenge math educators. Like, we want to be that in that space, where if someone says"challenge" and "math class", and you're like, "Oh, yeah." Like. "Join the Math is Figure-Out-Able challenges." She said she joined the backstage. She has a shirt. Whoa!(unclear).
Kim:(unclear). Everybody needs one of those.
Pam:Everybody needs one of those. Absolutely. We should. We sell somewhere, right? On the...
Kim:Yeah.
Pam:On the website?
Kim:On the website.
Pam:And to be clear, listeners, my goal on everything that we sell merch wise is to break even.
Kim:Yeah.
Pam:So, we price those things as absolutely... What's the word I want? Less? As least? As little? As...
Kim:Less expensive as...
Pam:Thank you.
Kim:(unclear) the right word is.
Pam:The cost is as little as we can make it, so...
Kim:There you go.
Pam:So, I don't lose money. But it's literally that. Because you guys are teachers, I want it to be as affordable as possible. And so, we do that on purpose. Anyway, so great job getting your shirt, Amanda. Alright, so she said, "I've watched your free class..." Ah! So, listeners, if you don't know that we have a free workshop. Kim, tell us about it.
Kim:Oh, they got to see this, right? Development of Mathematical Reasoning.
Pam:Bam!
Kim:It's amazing! It is the best free workshop you will ever, EVER get to see.
Pam:Totally cool. I went and did my most asked for professional learning with a group of teachers live. We videoed it, turned it into an asynchronous online worship, and it is available completely for free. mathisfigureoutable.com/freeworkshop.
Kim:Yep.
Pam:Is it free-workshop? It's one of those.
Kim:I think it's freeworkshop.
Pam:If you'll just go to the website, mathisfigureoutable.com, one of the very first things that you'll get to is, like she said, Developing Mathematical Reasoning. Check it out totally free. Anyway, so she says, "I've watched your free class." Excellent. "And listen to the podcast." Whoa! Here we're on the podcast. "I am going to sign up for both your addition and multiplication classes this round." Well, that's fabulous, Amanda. Yay! We need one of those sound things, where we can go [Pam mimicks cheering applause]. We don't have one those.
Kim:We do not.
Pam:No, we do not. Kim's like,"No, we are not adding that to the podcast." I'm kind of agreeing with that. But that means that I'm going to make those noises. Is that a good idea? I don't know. Okay, so.
Kim:She's done all the things Amanda's fabulous!
Pam:Yeah. And you are going to love both the addition and multiplication workshops. Absolutely love it. Let me tell you when I answered Amanda, one of the things that I said was like, "Wonderful to have you in two workshops! However, now that we know that you're a leader, do you know that we have a thing called JourneyLEADER..."
Kim:Yeah.
Pam:"...which is not time sensitive. You can sign up at any time." So, leaders, if you're listening right now, or anyone who might be interested in taking two workshops, for just a little bit more, you get access to all my online workshops plus extra leader support, plus everything wonderful in Journey.
Kim:Yeah.
Pam:So, leaders, make sure that you're aware that we can do you better, so.
Kim:Yeah.
Pam:Anyway, so I told her about that in the reply. Cool. So, then, she said, "I'm looking really closely at your 16 page booklet that you shared at the recent challenge."
Kim:Yeah, that might be one of my favorite freebies.
Pam:Yeah. Like, we give away a lot of stuff, right? We're trying to flood the earth with real math and real math teaching, so we created a free ebooklet that's called the... Mmm, what is it called? We named it a couple different things. Major Numeracy Strategies.
Kim:Yeah, I think so.
Pam:Yeah. Where you can get that at mathisfigureoutable.com/big. Like B-I-G. Big. Because it's big, ya'll.
Kim:Yeah.
Pam:So, if you have not downloaded that yet, check it out. Anyway, so she had downloaded it. We've given it to all of our last challenge participants. She said, "Without writing a very long email(chuckle, chuckle), I'm really thinking about how to support teachers, both in their identities as mathematicians." Absolutely! Like, I can't support that more. That's fabulous. She says, "...which, sadly, most elementary school teachers do not feel they are mathematicians the way they know they are readers, writers."
Kim:Yeah.
Pam:Which is interesting, right? Like, I think that many elementary teachers would relate to that. They consider themselves readers and writers, and they teach that and they. But as they teach math, they don't necessarily consider themselves mathematizing, even at that grade level. She said,"...and how to help them up their math teaching game." Fabulous. We support that goal. Absolutely. Then, she says,"There's a relatively finite group of strategies for early addition and subtraction, I think is..." Oh, let me pause there. We would agree with that, that there is a relatively finite.
Kim:Yes.
Pam:(unclear) of strategies for not only early addition, subtraction, but really kind of everything we do. So, pick an operation, pick writing the equation of a line, writing a function rule for exponential data. Like, pick... I'm trying to think of a middle school example...solving a proportion. Like, pick a thing, and I think there is a finite set of important strategies.
Kim:Mmhmm.
Pam:The strategies that we really want and need kids to own, so that they can sort of be successful enough.
Kim:Yeah.
Pam:So, that's important. We'll sort of grant you that. Then, she said, "I think it's almost freeing for teachers because it can give them a measurable goal, as opposed to the vast openness that many felt from common core." Now, I would suggest that, that was a little bit of an anachronistic. They didn't have to feel that from common core. I think they might have felt that more for materials that came along with common core that. Just like the standards themselves. But I would agree that many people felt this vast openness like that there was this infinite set of relationships and strategies, and so we should just teach the algorithm because "What? What does all the rest of..."(unclear)
Kim:Yeah, we can't teach it. Yeah.
Pam:Yeah. We can't quantify it. We can't, like, list it all out. So, I agree with you. That was a little tricky for many people. So, she says, "I'm really looking forward to working with these strategies this year." Well, Amanda, we think that's a fabulous idea. And we would encourage all leaders to do that very thing. Like download the/big...mathisfigureoutable.com/big...and use it all you want! Use it with your teachers. Use it with anybody that you work with. We are putting out that this is the research, this is the part of my work that I've done to say,"Here are the important strategies, and it's this fairly limited finite set." Okay, so here's the meat, then, of her question. She says, "So..." Thank you, Amanda. "So, I have observed a common subtraction strategies that kids use that I think belongs on your early list." Then, she says, "I'm not trying to be presumptuous here. LOL. But I'm very interested in your reasoning why you didn't put it on?" Well, that's fabulous, Amanda. Like, I really appreciate that. And may I invite everybody to push back. Like, I would really invite your insights about what you think the major strategies are. And I'm going to tell you today what I think about your idea, and why we're going to include it or not on our major strategies list. But if anybody else has other operations or areas in math where you think you've got a major strategy that I haven't considered, absolutely that is a possibility, that is maybe even a probability because there's a lot of math, and there's a lot of different things to consider. I have thought really deeply for quite a while about the four operations, but I still want you're. I still would really welcome your input. Okay, Kim, is there any that you want to talk about before I read the rest of the email?
Kim:Well, I'm interested to hear what the strategy is.
Pam:Right, right. Okay. So she said, Here it is: Take from the 10." Now remember, this is a small subtraction strategy, small number subtraction strategy. So, she said, "For example: For 15 minus 7. If it's a take from 10 strategy for 15 minus 7, think 10 minus 7 is 3. But then, you have to tack that 5 back on." You sort of ignored the 5 in the 15.
Kim:Mmhmm.
Pam:"And you just like, 10 minus 7, that's 3. Then, you're going to bring that 5 back that you ignored. So, 3 plus 5 is 8. So, 15 minus 7 is 8." Does that make sense?
Kim:Mmhmm.
Pam:In fact, as I say that Kim. Am I right? You used that strategy as a small child?
Kim:That's a great question. I'm thinking I have, yes, used that strategy.
Pam:Okay. Maybe I'll do another example before we get into it.
Kim:Okay.
Pam:It just occurred to me people might need a little bit more.
Kim:Yeah.
Pam:So, how about 14 minus 6? So, 14 minus 6. I'm going to ignore the 4 I'm going to take from the 10. So, 10 minus 6 is 4.
Kim:Mmhmm.
Pam:But I ignored that other 4. That's kind of a dumb problem. Sorry. I ignored the other 4, so I have the. 10 minus 6 is 4. But the 4 I ignored I put back on. So, 4 and 4 is 8. Let me do one more. 17 minus 9. I'm going to ignore the 7, so I'm going to think about 10 minus 9, that's 1. But I got to bring that 7 back. 1 plus 7 is 8. So, 17 minus 9 is 8. So, the take from 10 strategy. So, Kim, what are you thinking about that strategy?
Kim:So, I like it. I think it's a really great strategy that you. When you say, "Didn't you use that strategy?" I have used that strategy for slightly larger numbers. I don't know that I ever used it for, what I would consider these are, kind of basic facts I have. Because I know combinations of 100 really well, then I have used it for say like a problem like 159 minus 26. Then, I would set aside the 59. Do the 100 minus 26 part to get 74. And then, just add back that 59 on. But here lies the problem for me. Then I have to think about a problem like 74 plus 59.
Pam:Mmhmm.
Kim:And that's not as... It's a little crunchier for me than just subtracting the 26 in a really nice way. So, I have used this strategy. And also, we were just in a situation a couple of days ago, almost a week ago, where we saw this strategy being used by a participant in a workshop. And so, I think we both recognize this as a strategy and have made sense of this as a strategy. Can I tell you what I think, though?
Pam:You can. Well, before you do. So, it's a strategy. It makes sense.
Kim:Yeah.
Pam:We can understand why students would use it. It works.
Kim:Yeah.
Pam:It's mathematically coherent.
Kim:Mmhmm.
Pam:Okay, when you say, "Can I tell you what you think?" Can you tell us what you think? Finish your sentence?
Kim:I don't think it's a major one.
Pam:Yeah. So, could that. Because that's the question, right?
Kim:Yeah.
Pam:The question is. In fact, I'm going to keep reading what Amanda said. She said, "I realized that in a philosophical sense, there are infinite strategies. But in a real, everyday in the classroom, kind of sense, there's a shortlist of strategies that truly both build and exemplify numeracy." And then, she gives an example."It's just like reading and writing. By reading and writing, we are both building and exemplifying literacy. I think these..." Still, Amanda. "I think these shortlists can be game changers for elementary teachers who feel overwhelmed and who don't strongly identify as mathematicians themselves. But that said, I think the take from 10 strategy belongs on[your] list. I'm very curious to hear..." Well, she said "that list". I'm saying "my list" because I'm.
Kim:Yeah.
Pam:Because it's me. "I'm very curious to hear what you and Kim think." Alright. So, Kim, what you're saying is you're not clear. Or maybe you're saying no, that strategy does not belong. So, respectfully, Amanda, Kim is saying no, doesn't belong in a shortlist.
Kim:Well, and I don't know that I have a lot of reasons why other than I have completely abandoned that strategy. So, there's a ping for me that as I've sharpened other really important strategies, that's never one that I went back to. And I feel like if there were a strong need for it in using other strategies, then I would continue to come back to that as something that I have like there's a need for it. And even though I love combinations of 100, combinations of 1,000, combinations of 10, I've abandoned it. It seems like a pretty... It seems less sophisticated than other strategies that I would want to motivate students to make use of.
Pam:So, I'd like to dive in on that comment that, "it seems less sophisticated," and agree with you that. Sorry, Amanda, I'm also going to suggest that it's not going to go on the major shortlist. But let's be really clear that it probably is on a shortlist of strategies that students will create themselves.
Kim:Yeah.
Pam:So, it is a fairly. Especially if you're working with I Have, You need, which is our wonderful routine that helps develop partners of 10, partners of 100. If you're working to develop partners of 10, students are going to recognize those partners of 10, and they very well might use your take from 10 strategy.
Kim:Yep.
Pam:In fact, I wonder if. Ya'll, if you've been listening to the podcast for very long or any of the workshops that we that we have, I often tell the story about my oldest son when he came home from first grade with subtraction with regrouping, and he used a strategy that he made up on his own to solve problems. And I can't. I always usually say. When I tell this story, I'm like, "I don't really remember what he did that year." But then, I will tell you what he did the second year because I remember that one better. But when I saw this email. Amanda, thanks for the ping. I think this is the one he developed the first year. I think when he had a problem like, I don't know, something like 39 minus 14. No, that's not 34 minus 19 isn't one I usually use. So, if you think about 34, minus 19. Instead of carrying or regrouping to make it 14 minus 9, I think he thought about 10 minus 9, and then added of the 4. So, he still brought the 10 over, but he didn't add it to the 4. He's like, "I'm going to grab this 10 over there from the 30, and I'm going to subtract whatever I need to subtract the 9. But then, I have that 4 over there that I ignored. And so, I'm going to tack that back on." That to me, that feels very connected. Would you agree, Kim? Those are connected?
Kim:Yeah. And I'm thinking. I wonder because I didn't have a lot of instruction in my young years about strategies and numeracy and. You know, it was very traditional upbringing. I wonder if that was my similar story. That I kind of tinkered with numbers and made that up, which is why I recognize, you know, when you first mentioned it.
Pam:Well, and it makes sense to me that as teachers are saying things like, "Bring this 10 over here. Grab that 10 from over there. There's this 10 floating around when you're discussing regrouping or borrowing." That kids that are thinking about relationships might go, "Well, then I'm just going to deal with that 10. I'm not going to tack it on to whatever it is. I'm just going to deal with that. Subtract that. I got that 10 There. Oh, but then I did have that number, and I'm going to deal with it later." So, Amanda, I think I would completely agree that this is on the shortlist, the strategies that students might develop themselves, that we could support and help students reason through as they are sort of developing their numeracy. However, I'm not going to put it on the list of important strategies that I want all students to develop. That's a different list. The list that we want all students to develop is a very specific list, kind of like you said where we're really trying to encourage kids to become numerate. That list needs to be the list that kids should learn. I'm trying not to say"have to learn". But in order for them to be really fluent, there is a shortlist of strategies that we want students to think about, to own. We want them to own the relationship, so the strategy becomes a natural outcome. Let me say that again. It's not about memorizing these strategies. It's about developing mental mathematical relationships, developing these connections between numbers, so that certain strategies become natural outcomes, so they become sort of natural things to do for kids. That shortlist doesn't need this take from 10 strategy. Let me tell you one reason why. Kim kind of suggested that it was a little bit less sophisticated. Let me support that. It is very much like all of the least sophisticated strategies that are in that booklet that are listed first. So, if you go look at the addition, major addition strategies for multi-digit, major subtraction strategies for multi-digit, the very first strategy often that's listed is a partial strategy. Partial sums, partial products, maybe partial differences. We don't really suggest partial, sorry, partial differences with the one we don't usually suggest. Partial quotients might be one. Partials is the first kind of bedrock, kind of begining, less sophisticated strategy. And it's the one that we're going to emphasize the least. We're going to do a little bit with it, and move on. Kids soon as we're working with multiplication of multi-digit numbers, we're thinking smart partial products. We're not taking all the numbers and. Now, we do a little bit of it, where we split it into place value, so kids are really thinking about the place values involved. But this feels like,"Let me split the number up by place value." So, for example. I'm not looking at the email anymore. Let me go back to the email. So, for example, the 17 minus 9. You're splitting the 17 into 10 and 7. That's a very splitting by place value, very partial kind of an idea. Dealing with the 10, and then tacking back on the 7. So A, that substantiates the fact that it's kind of a beginning strategy. But B, it also. If I get kids that are only doing that, kids only ever deal with 10 minus a single digit. They don't ever grapple with 17 minus 9. They kick out the 7, deal with the 10 minus 9, and then kick back the 7. Well, I'm not going to tell students not to do it. I'm not going to actively try to create it in all students because I don't want students to stay there. I want students to think about 17 minus 9, and "What does it look like? How can I think about that? And what are friendly numbers near there? Can I subtract 7 to get to 10? And then, what do I have left to subtract? Can I subtract too much? Subtract 10? And then, how would I need to adjust?" So, we really want to develop the major relationships like subtract to a friendly number or subtract a friendly number. Those will carry students further in both single digit relationships and in the multi digit relationships. We need those to continue on. So, either getting to a friendly number or subtracting a friendly number. Either one of those are more important.
Kim:Yeah.
Pam:So, I hope that kind of clarifies. Kim, what do you want to add to that?
Kim:Yeah, I was just going to say, I appreciate your second point that they deal and grapple with bigger and bigger numbers, which is something that we (unclear).
Pam:Need them to do.
Kim:Experience doing, right? Yeah,
Pam:Yeah, exactly.
Kim:Yeah.
Pam:Exactly, yeah. One other just small thing that I'll mention is, this to me is similar to where CGI. Cognitively Guided Instruction. Wonderful resource. I'm so super glad that they did it. And we've learned a ton from it. But sometimes people can get lost in the weeds a little bit as they dive into CGI research, where they'll really start to think about, "Okay, sometimes kids..." And basically CGI did a great job of identifying things. Like when I'm subtracting. So, say that 17 minus 9. Sometimes kids will go, "17..." and put up a finger." 16..." and put up a finger, and then continue to put fingers until they put up 9. And then, what's the answer? Versus a kid that goes 17, and then starts to count. "16, 15, 14..." And so, they don't count the 17. They count sort of where. And then the answer's different. One way the answer is where you land, the other way is the answer. And, ya'll, I'm not even going to be clear on which of those two. Did you notice how fuzzy I just left all that? Because I don't actually want you to be super clear about that because that's not our goal. Mmm I'm not saying this well. Our goal isn't to help students get better at those two strategies. Once kids can count, and they realize they should be removing, we don't want them removing one by one. We want to help them remove bigger chunks. So, we want to move to these relationships that we were just talking about. So, when I work with teachers and CGI research we just mention that is what they noticed that kids will do, but I don't dive into it and make a big deal of it because I don't want you to then go, "Hey, everybody, you better learn this subtraction strategy. And now learn this..." Not if we're counting by ones No, no. Like, we want to be moving to additive reasoning. So, part of our answer today, Amanda, is that all of the strategies that we're going to recommend as the major, the ones that we need to help students develop, are the ones that help students move forward.
Kim:Yeah.
Pam:Not... Yeah. There you go that's. Maybe I'll just leave it at that.
Kim:Yep. So, you and I both have other strategies that we like, right, other strategies that we find interesting, but. And I'll just say that one of my favorites I find really interesting is the Swapping strategy for addition. I love it, but it's not one that I work with Problem Strings and a lot of intensive work with my students because we have a limited amount of time, right? It's not like we have 300 days every year with our kids. And so, given that we have very specifically outline ones that we know move math forward, we know are super important for kids to know. And that's why, you know, you took the time to outline them in this ebooklet that you have made available to everybody, so that they would know, "Where should I spend my time?" Hey, you just made me think of something. So, Amanda, as you're working with teachers, I think it would be a fine thing to have them see some. In fact, Amanda, sent a couple videos of some students using the strategy. Totally believe that kids do it. I think you could absolutely use some of that to help teachers recognize what students are doing. Just be really clear that then it's not a strategy that we want to write Problems Strings for, that we want to have students really be dealing with a lot. It's not one that we actively are trying to create in all students. If students are doing it, we support a minute, but it's not the one that we're actively creating in everybody. Just like Swapping. Let me just say one other thing about Swapping. Swapping is cool because we can write some really nice Give and Take Problem Strings to develop the Give and Take strategy. And if we write them in a super slick way, we can be kind of extending students who are ready because they begin to play with Swapping, while other students who need more experiences just continue building the Swapping strategy. So, it's good for teachers to know the landscape. We want teachers to know the Swapping strategy, but we also want them to be clear, it's not one of the ones that you're trying to fit in your scope and sequence to make sure you get to by the end of the year, if that's your operation for that year. Does that makes sense, Kim? Yep. Yeah, absolutely.
Pam:Yeah. Cool. Alright, ya'll, don't forget our little tease. Be on the watch for something new that's going to happen this week, or maybe be on the listen for it. Alright, 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. Let's keep spreading the word that Math is Figure-Out-Able!