Ep 114: Moving Students Forward Pt 2

Pam Harris  00:01

Hey fellow mathematicians, you're listening 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 you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But y'all 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 and the rest of us from being the mathematicians they can be.

Kim Montague  00:39

So in the last episode, we jumped into some conversation about how to get started if you or your students are new to numeracy, and focusing on building thinking relationships rather than simply answer getting. In today's episode, we're going to take a look at some of the early strategies that you can and should build with your students as you begin the year together. So listen, if you or your students have only ever really experienced algorithms year after year, they've never really had an opportunity to examine their thinking, you may have to spend just a little bit of time messing around with number in ways that we suggested in the last episode. They're gonna need some experience dissecting problems, and talking about their thinking. And considering that mathematics is not all about you solve one problem for one answer.

Pam Harris  01:27

Yeah. And so as you think a lot of us are starting the school year, this is sort of a time of in a lot of places where you're just meeting your students, and you're thinking about where do I start with them? How do I move them forward from wherever they are. And like Kim said, if you had students who had less experienced reasoning, or even more experience reasoning, this could be the perfect time to get a feel for how your students think about mathematics, what their view of themselves as mathematicians. Sometimes there's things that teachers will do, Kim, where it's kind of these inventories. In fact, in my university classes, I do ask my students to tell me about a highlight in their math background, a low point in their math background, how they feel about themselves as mathematicians. And that's always good information for me. But I gotta tell you, I learn more, I get better information by starting off with a Problem Talk. Like I throw out a problem. And then I see how students react. 

Kim Montague  02:30

Absolutely.

Pam Harris  02:30

I see their, the looks on their faces. And I see if they write if they don't. Now it's less important if they write or not. It's more important what they write. And then how they respond. Are they you know, is it a step by step procedure, someone else has told them? Are they nervous? Are they thinking? Do they have only one strategy? Do they have more than one strategy? Are they flexible? Are they having fun playing around with different ways? You know, so I always give them a really rich problem. And then let's see. So you might, teachers, you might consider starting the year with a really rich problem to gather information. Let that be your pre-assessment, rather than some of those unhelpful assessments where kids start the year with this long boring, heavy, one and one only right answer. And if you want more information on that we have a whole podcast on how not to start your school year. And I don't remember the number of that one, Kim, but we'll put it in the show notes. And I think we call it how not started the school year. So you go check that out. But you could start with a Problem Talk or two, to gather some information. So then where do you go now that you kind of have a feel for if your students are thinking and reasoning, if they only have one strategy, if they've got some numeracy behind them? Then what? So let's talk today about early strategies. We had a podcast not too long ago. Sorry, we had a challenge not too long ago, where we talked about the most sophisticated strategies. Today, we thought we'd back up a little bit. And since it's this time of year, where teachers are often starting with students, what might be some of those groundwork strategies? What might be sort of the places that you would expect students to kind of come in on, and find good success with and, but then build from. Does that sound like a good plan, Kim?

Kim Montague  04:34

Yeah, sounds great.

Pam Harris  04:36

So early strategies, here's an interesting thing to consider. There are programs out there that recognize that students come into operations with some intuition. That they come into, and I'm just going to use multiplications as an example. So Everyday Math was a fine program. Zalman Usiskin you did a great job of helping us all think about transformations in a different way. But in a way the what was written and then what was read by teachers was a little bit different. Here's Pam's take on Everyday Math: They kind of gave us lots of different prior algorithms to mess around with. And the idea was to get students to sort of play with numbers and go, Oh, like and kind of learn relationships. Unfortunately, they used a lot of algorithms that aren't very transparent. And so yeah, students kind of played around and they, but teachers read it as: We used to have students rote memorize this one algorithm, now have them rote memorize all these other algorithms. Teachers kind of get lost in the weeds. Students got lost in all the memorizing. But then, then they made this other sort of interesting decision, which was to end to say, Ooh, like, our preferred strategy that we want all students is kind of, like make sure they have and then we're good enough if they have that. They kind of ended with I would call the partials. So partial sums, partial differences, partial products, partial quotients. Then they're like, "Okay, as long as students understand those, then we're good to go." But unfortunately, I think that's really short sighted, because then when students got to more complicated numbers, like decimals in the middle school, or the higher grades, then all of a sudden those teachers were like, "What? Like, we can't have students just a little (just choose multiplication) we cannot have students just doing partial products with decimal multiplication, because it's way too inefficient." Might that be a starting point for students? Absolutely. Like that could be a helpful starting point to like, oh, what's actually happening with decimal multiplication? But boy, we can't, we can't end there. Because then it left teachers who are teaching decimal multiplication, it left them kind of throwing their hands in the air going, "Well, if you never learned the algorithm for multiplication, I guess I'm gonna have to teach it to you now. So that then you can move the decimal at the end. So we can be successful with decimal multiplication." We would suggest there's a better way. And the better way is do we need partials? Absolutely. But then we need to move them not to the traditional algorithms, we need to help students construct more and more sophisticated relationships. And you might be like, "Pam, what are all these strategies?" Alright, so we have put out, you may have heard, this wonderful free ebook, I'm so excited,  Kim, that we created this ebook. Everywhere I go, I'm telling people about this ebook, it's free, it's free. And it took us a while to create it because we really wanted to make sure that we had the necessary stuff in there to help teachers really think about models versus, or excuse me, strategies versus algorithms. What's the difference? And models versus strategies. What's the difference? And then how do we bring that all together? And what are the major strategies represented on the major models for each of the operations? And so if you have not downloaded that yet, y'all, what is happening? Maybe you just joined the podcast. Well, welcome. Welcome. So download this free ebook, and we'd like to talk about those beginning strategies today. And we'll reference that ebook. So if you haven't yet, download that free ebook at mathisFigureOutAble.com/big. Yep. That was the URL we chose, because we know that this-

Kim Montague  08:11

Huge! 

Pam Harris  08:12

A big ebook. It's gonna be really, really super helpful. So mathisFigureOutAble.com/big. Download our FREE eBook. And I think, in fact, what did we even call that? I'm opening it up so I can, cuz I forget the name. Oh, yeah, we call it Major Strategies, at least that's the name on that on the thing. Okay. So when you check out this big ebook, you'll notice that we have, like I said, a page on strategies versus algorithms and a page on strategies versus models. And then we'll go through the four major operations. And we outline kind of in order. So in case you didn't know, for each operation, there's a page. And on that page, there's sort of an order of strategies from less sophisticated, kind of the beginning the entry strategies, to more sophisticated where we kind of want students to end up. So teachers, if you're working with your students, in addition, and you're looking at that page, you'll notice the very first strategy, we call Splitting by Place Value. And y'all another way to talk about that is that partial sums. We're looking at sums because we're looking at addition. So we're summing numbers up. And as we are summing that beginning thing that often students will do naturally, if we don't superimpose, we don't force a traditional algorithm on them. Students will start naturally splitting by place value. They will sort of think about things like 48 and 29. They'll think about 48 and 29, as 40 and 20. They'll sort of pull those tens together. What was leftover? Oh, yeah, the eight and the nine, they'll pull that eight and the nine together from the 48 and the 29. So now they get the 40 the 20 together to make 60. They've got the eight and the nine together to make seventeen and now they're thinking about, "Alright, 60 and 17." That is brilliant thinking that we want students to mess around with. How do they think about pulling those tens together? How do they think about pulling those ones together? And then pull all those, then 60 and the 17 together to get 77. So that's an example of a starting place, not an ending place, Everyday Math, but a starting place. And research has proven that if we haven't superimpose them for students to start with the ones place, so think about the traditional algorithm, if we line those up, 48 and 29. Students are going to start with those tiny numbers, the eight and the nine first. But that's not student's intuition. Intuition is to start with what the numbers are made of the 40 of the 48, and the 20 of the 29. That we want students to do that. We want them to mess around. That's going to help students reason about place value. They're not just thinking about digits, columns of digits, but they're thinking about the big numbers, 40 and 20. And having to make sense of those and grappling with those big numbers. Brilliant place to start.Though, as you're looking at that free ebook, then you'll notice that after that, that we want to nudge students to keep one addend whole. We give Cathy Fosnot, a lot of credit for helping us really think about this idea that we can start with less sophisticated strategies, and then keep one addend whole and do something with the other addend. And so for example, we keep one addend whole and we can Add a Friendly Number. So take that other number and decompose it into a friendly number. And then whatever's leftover, so add that friendly number part of it. So for example, in 48 and 29, we're gonna have 48 and 20. And then sort of once we've got that 48 and 20 is 68, then we can kind of mess around with the leftover nine. And that's really not so important. But we've kept one addend whole and then we've added a friendly number. But we can also keep one addend whole and Get to a Friendly Number. We can start with that 48 and then get to that friendly 50, just add the two to get to the 50 and then add what's leftover. So this idea of keeping one addend whole is more sophisticated. And it requires a little bit of pre-planning, a little bit of anticipatory thinking. And then to really bring in pre-planning, and anticipatory thinking, that's when we start thinking about, "Ooh, if I can just Give and Take a little bit, then bam, I can do something where I can create an equivalent problem that's easier to solve." And whenever we create an equivalent problem that's easier to solve, we're really dealing with the most sophisticated strategies for those operations. So for addition, we really do want to give students as you're starting off the year, it's a perfectly fine places to start to have students think about place values, and mess around with splitting by place value. That's a great place for an addition start. Now when you get to subtraction, Kim, people might find it interesting that we don't actually suggest that you do a lot of partial differences. 

Kim Montague  13:03

Right.

Pam Harris  13:04

Correct me if I say this wrong. If students are messing with partial differences, we will support them. We will help look at their work when they sort of mess up or whatever. And we'll help question them through, you know, like help them decide kind of how they're handling that. But in class, we're going to actively support that we're just going to think about first removing. And so if I've got a problem like 56 minus 29, I'm going to think about 56. Subtracting some version of 29. Start with that 56. I might subtract 20. Remove a Friendly Number. What's friendly and 29? I might just subtract 20. I also might Subtract to a Friendly Number. Well 56, subtract anything, I might just subtract the six to get to 50 and then subtract the rest of the 29. I might do the Over Strategy, which I don't think I mentioned in addition, Over Strategy, in addition is that a version of Adding an a Friendly Number. We just add a bit too much. So similarly, when we're subtracting, removing over strategies, removing a bit too much, removing a friendly number, that's too much. But all of those sort of come after we kind of just, like, let's just start removing, let's think about minus, subtracting from a number and then kind of like what's your gut instinct? What do you want to remove? What's a friendly thing to remove? Either removing a friendly number or removing to a friendly number. That's kind of the start place for subtraction. Then later we'll get to the more sophisticated strategies like thinking about the relationship between addition and subtraction and finding the difference. That comes fairly quickly. But we want students to understand removal. And then we also want them to understand that they can think about subtraction as difference. And then later the most sophisticated subtraction strategy, finding that equivalent problem that comes later where we use constant difference. So as you begin with students with subtraction really help them think about subtraction, and what would be friendly to remove?

Kim Montague  15:10

Can you jump in for just a second before you move on? 

Pam Harris  15:12

Absolutely.

Kim Montague  15:12

So I find it really important, and I want to emphasize that you're saying that it's necessary that we give kids some time and experience because a lot of times we hear from teachers, like, "Oh, my kids are stuck it" whatever, they're they're working with partials or they're working with just removing small jumps of numbers at earlier ages, younger grades. When we dig into that conversation a little bit, they'll say things like, "Well, I've done a problem string or two." or "I've done a thing or two." And you and I are-

Pam Harris  15:44

Often, sorry, Kim. Often it's, "I've done problem talks," "I've done number talks." 

Kim Montague  15:49

Yes, yes. 

Pam Harris  15:50

We're doing all the work. We're doing, but when we dig in all the work is like some, maybe it's a lot of number talks, not Problem Strings. 

Kim Montague  16:00

Problem Strings. 

Pam Harris  16:01

Yeah.

Kim Montague  16:02

Even if they have done a Problem String it's been a Problem String. 

Pam Harris  16:05

One or two. 

Kim Montague  16:06

Right? 

Pam Harris  16:07

Yes. 

Kim Montague  16:07

Of each type of strategy. And I want to emphasize that we feel really strongly about giving kids some experience, so that then they can get to more sophisticated strategies. And you're not dragging them along. Right? It should be a nudge for sure, you should be able to lob things out, and kids kind of grasp a hold. But if we're just kind of shoving them along and saying we did one Problem String of this type and one Problem String of this type, then the results are not going to be as successful as maybe it feels like it might want to be in your head.

Pam Harris  16:42

Yeah. And when you say give students experience, you and I both have this vision in our head, where it's not a one and done. It's, "Alright. Today we are working on subtraction, we're going to do a Problem String toward-" now pick where you are maybe toward Removing to a Friendly Number. We're not done. It's not "I do, we do, you do. Alright, we've got that thing. that one strategy did on Tuesday, we're handling it. So Wednesday, bam, we're moving to the next strategy. Done. Alright, Thursday, the next day. It's not, that's not how we build mathematical relationships and students heads, therefore, then the strategies become natural outcomes. So it's much more about giving students experience through solving Rich Tasks and Problem Strings, where students grapple with these relationships, they gain more and more mental connections. And then the strategies become natural outcomes, as we continue to give them more and more problems to solve, discuss those, making the thinking visible as we're discussing the strategy so that we have the visual models to discuss. All of that comes into play making those relationships. Yeah, it's really important. So what about multiplication? What's a beginning strategy for multiplication? You might find it interesting that in our free ebook, we don't even list Partial Products as the first strategy. Now part of that is because we kind of didn't have room, you'll notice that we stuck like lots, like even more strategies in the space there, because multiplication is so abundant. There's some really nice things that we want to have happen in multiplication. But but in a huge way, Partial Products should be, it is a necessary starting point, we do want students, so maybe I should define Partial Products. So partial products with multiplication, if we were doing a problem, like 18 times 25, would be thinking about 18 times 25 as 10 and eight, the 18 is 10 and eight times 25 is 20, and five. So I split the numbers, usually by place value 18 into 10 and 8, 20 to 2o and 5. I split them by place value. And then we make sure that we distribute all of those. And if I was using an area model to exemplify or visualize that distribution, then we would have each of the numbers multiplied by each other. So the 10 of the 18 is multiplied by the 20 of the 25. The 10 of the 18 is multiplied by the five of the 25. And then also the 20 of the 25 is multiplied by the 10. So, we'd make sure that, and I probably just repeated them, because I'm not drawing them. So we'd want to make sure that we've just got each number, each part of the number. That's why it's Partial Products. So we're finding all of the partials. And that's the distributive property in a huge way. But we need teachers from the get go to realize that as soon as students make sense of Place Value Partial Products, we want to nudge them to be smart about how they choose Partial Products, to try to be clever about Partial Products. That we don't ever want students to get in a rut where whenever they see two digits times two digits, "Oh, I know what to do. I am going to split this into place." Like we don't want them to be robots where they instantly split both numbers into place value, and then they find all of the parcels. Right? So  because it's not, that's not fun. That's just different steps to memorize. Instead, we want them to be, like clever about how they choose us. So we could call it clever partial products are we in the in the free book, we call them Smart Partial Products, where they can think about something like 18 by 25, as splitting the 18, into 10, and eight, but keeping the 25 whole, because I can think about 10 twenty-fives. That's just 250. And I can think about 8 twenty-fives because I can reason about eight quarters. And I can find that as 200. And then I don't have to split the 25 into 20 and five. I can keep that 25 whole. And whenever we can, we want to have bigger, fewer chunks. That's more sophisticated. And that involves a little bit of anticipatory thinking. That we're sort of looking at the problem, we're asking ourselves, "Ooh, how can I mess with these numbers? What can I do?" And we want to encourage students to do that, from the get go. So that's going to be our beginning multiplication strategies. Smart Partial Product, clever, clever Partial Products. Cool. What about division? So with division, we actually are going to suggest that a beginning place for division is to think about division multiplicatively. Think about division as the missing factor. And how can you find a missing factor, you can kind of multiply up to can take that factor that you know, and multiply up until you get to the product. Or if I was to use the division words, we have the quotient and we, sorry, we don't have the quotient, we have the dividend. And we have the divisor. That divisor is kind of that, we have kind of the factor that we have. We don't have the missing factor. We don't have the quotient, but we have the dividend. We have the product so we could kind of multiply up to find that missing guy. And that could be a really nice begin that we can have for division. And then of course, we want to get more sophisticated with division. But in a huge way, when we start division really considered, teachers, that we want to build the multiplication strategies a lot as we're making sense of division problems, before we ever then go straight to straight division. Kim, would you agree with that?

Kim Montague  22:21

Yeah, absolutely. I was gonna say I'm so glad that you mentioned how much work you'll do with multiplication, before some of the more sophisticated division strategies.

Pam Harris  22:29

And that includes making sense of division problems. So that we're reading both multiplication and division, word problems and scenarios, situations and contexts. And we're making sense of what's happening. And we're really reasoning in a huge way multiplicatively. We're thinking about multiplication to solve the problems. And as we do that, we get better and better. And we do more and more Problem Strings to get all of the multiplication strategies under our belt. And then we turn to get to more and more sophisticated division strategies. So that's a way of kind of thinking about structuring the timing for division strategies. So remember, these are beginning strategies that we are thinking as you start the school year. It's great for students to spend some time here, but don't pigeonhole them there. Like, we want to be promoting the more and more sophisticated relationships. Teachers, it's so important to know your content, know your kids. Know your kids? Sive in, give good questions, start doing Problem Strings, figure out where they are. But know your content, know this landscape of the strategy so that you're always nudging individual students forward. As you're working with them on any particular string, you're noticing what they're doing, and you're lobbing things out, you're asking questions to move them forward. But you're also using this idea of what the more sophisticated strategies are to plan for your instruction, like, what are, we need to spend some time here and develop this strategy, but you're always looking towards, "Ooh, so then after this one, I've got it kind of under most students belts." Now what's the next strategy that we're going to do Problem Strings to help develop those relationships towards? So it's not just about number talks, it's about construction. We need Problems Strings, to really help students construct the more sophisticated relationships, which then make the more sophisticated strategies become natural outcomes.

Kim Montague  24:19

So speaking of knowing your content, knowing your kids. Right? We have a fantastic opportunity coming up tomorrow to have some help with that. So we are about to have the You Can Change Math Class Challenge. Super excited, it starts tomorrow. If you've never participated in one of these challenges you are missing out. We are going to dive in with some examples and assignments, things to do and try. It's a super cool community that's really supportive and it's a really great way to get energized to start your school year. It is not too late to grab a friend and get signed up. You can register now at www dot mathisFigureOutAble.com/change.

Pam Harris  25:00

And if you're listening to this podcast some other time of the year, check out that mathisFigureOutAble.com/change to get some ideas of when our next challenge might be running. Or just see us in social media and MathStratChat on Wednesday nights, and we'll always be announcing things there as well. Y'all 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