Better Teaching: Only Stuff That Works

[00:00:00] Welcome to Better Teaching, Only Stuff That Works, a podcast for teachers, instructional coaches, administrators, and anyone else who supports teachers in the classroom. This show is a proud member of the BE Podcast Network shows that help you go beyond education. Find all our shows@bepodcastnetwork.com. I Am Jean Taver Netti, the host for this podcast. And my goal for this episode, like all episodes, is that you laugh at least once and that you leave with an actionable idea for better teaching. A quick reminder, no cliches, no buzzwords. Only stuff that works

Today we're going to be talking about a problem that has persisted at least as long as I have been in education, which is 30 plus years. And that is the lament of elementary teachers, saying that the kids just can't do [00:01:00] story problems or word problems. They'll say that, they can do the math, but they just don't understand the problems.

The fact that it's been going on for so long means that maybe we need a different way to solve it. to look at this, a different way to attack and teach how we teach students to solve word problems. And that's what my guest will be talking about today. My guest today is Dr. Sarah Powell. She's a professor in the College of Education at the University of Texas at Austin.

She's a former kindergarten teacher who graduated with her Ph. D. from Vanderbilt University in 2009. Her research and teaching focus on mathematics, particularly for students who experience difficulty with mathematics.

Good morning, Sarah. It's great to have you on Better Teaching.

Hi, Jean. So nice to be here. Thanks for the invitation.

Yeah, well, as I said in my introduction, this is something that is it's a perennial problem. I've been involved in education for [00:02:00] over 30 years, and for 30 years, teachers have been telling me, oh, the kids can't do word problems, which means it's time,

very tricky. Yes.

take, maybe we should another look at them.

And so, I found you. I found you on the internet, and, um, you were talking about solving you were talking about problems, the common problems there are in the way we teach now, but the thing that really struck me was you talked about using schemas. to help kids. Can you talk about schemas and how you do that?

Yeah. So, I work a lot in the area of word problem solving, as you said, and same reason for you. While I'm a former teacher and while I taught kindergarten, word problem solving wasn't exactly top of mind as a teacher. I worked on several different research projects across the country.

Grades two and three, where word problem solving really becomes important because it's how students show off what they know about [00:03:00] math. Now, whether we like that or not, that is just the name of the game in all 50 states at this point in time. And so, I've been really interested in problem solving and what are some of the ways that we can support students in terms of problem solving.

And there's decades of literature that point to this idea that students really need to have an understanding of the concepts that are embedded within these word problems. And another way to say concepts is through schemas, or sometimes people call these problem types or structures. But the idea, and it's actually very cool, is that there are only a few schemas that students see in word problem solving across grades K through 8.

And so if students learn these schemas and if they're aware that these schemas exist in these word problems, then it takes word problem solving from being a really daunting task for many students, to being something that a lot of students could approach in a pretty easy way. Easy and formulaic way. And so I, and there's a lot of other people that do [00:04:00] research in this area.

But my research team and I do a lot of work right now. They're actually out in schools as we speak, doing some tutoring on word problem solving, where we are teaching students about these common schemas. And then students are taught to recognize the schema or the concept of the word problem, and then once they recognize that, once they know what Oh, this is a total problem, or this is an equal groups problem, then if they have an approach to solving a total problem, or an equal groups problem, then it just makes word problem solving much easier for students, and it's cool because it emphasizes conceptual knowledge, which is always a big thing that we want to talk about in mathematics.

And it also helps students problem solve across grade levels. So while they might be doing this in third grade, they're going to see the same schemas in grade four, grade five, grade six, grade seven, and so on. You'll actually see the same schemas as an adult. And so we are helping students develop a deep knowledge of mathematics across the grade levels.

You know, when you talk about schemas it makes me think [00:05:00] of language arts. And so when I have been on a mission for the last couple of years to change the term word problems to story problems.

Yeah. Yeah. Why do you want to make that change?

well, I want to make that change because the word problems really are a story. I mean, when we think about what we do with math, or let's say statistics, there's a story being told.

And what's the story? And if you're a statistician, you take a look at that story and you recognize the schema. Oh, we need to do a covariance test or it's been so long since I've done statistics. I may have said the wrong thing there but I think it's the same idea. So the reason that I like to call them story problems is that I think it's students.

are taught, there's a lot of time spent in reading and reading comprehension. And if we can understand what's going on in the [00:06:00] problem even before we talk about schema, then I think we have a better chance to understand schema. And one more thing that I want before I get to a question, Sarah is that you know, when People identify why kids can't do story problems is that they have a hard time with the reading comprehension.

but as you say that, I think that, like, the first thing that students have to do when they're approaching a word problem or a story problem is they have to read it and they have to understand that. And so that involves an understanding of word reading, sentence reading, reading comprehension.

And it also involves a understanding of vocabulary, both math vocabulary and general vocabulary, general English vocabulary. Maybe that's what we should call it. But that is like the big first barrier for many students. And so if students struggle with reading the problem, maybe they are students that have a learning disability in reading or dyslexia.

Maybe they're [00:07:00] students who are emergent bilinguals and, you know, reading is difficult for them. Then that really prohibits a lot of students from moving forward. So we can't even think about that. Mathematics, if we have a hard time reading the problem and understanding that vocabulary. And I also agree with you that reading and that understanding, that is the first thing.

And then we can start to talk about the schema, like, well, what's going on in this story? Like, are we putting things together? Are we comparing things? Is there a change, right? So then we can start to get into those schemas. But the first and initial thing that is so, so important is that students are able to read these problems, comprehend, These problems and then be able to start to think about, well, what's going on in this story and then what do I need to solve?

Like, what's the question asking me to answer?

Right. So, I work, the bulk of my work is explicit instruction and coaching teachers in explicit instruction. And so one of the things that, that I always focus on with teachers is to identify [00:08:00] what's the problem, what's the hiccup here in this process. And you just identified. You know, it's reading comprehension.

And if you take a look at so many of the word problems that are generated because, and I know they're generated to have different vocabulary and different terms because somebody's paying them to generate them and they're not worried about, they're not worried about the schema.

Yeah.

So, so what do you think about when we're starting to work with students in in trying to identify the schema and understand what's going on is that we do as much as we can to control that variable of vocabulary.

Like, why do we need, you know, six different words for put together or whatever, so that the, so that we could teach the schema, and then we know we have to worry about the vocabulary and all of

Yeah. Oh, this is a good question. Something we've talked about in our [00:09:00] research team. I don't know how to exactly answer this. I think a few ideas come to mind. One is that when we initially teach a schema to students, often we're doing that through a story that's orally presented, and it usually involves the use of manipulatives or gestures or acting something out.

And often we will bring in graphic organizers pretty early, so there's a real nice connection from the schemas to what I would think about are some of the representations we can use in math to help students understand what the mathematics means. So we could be using manipulatives to show students, Oh, look, we're putting these amounts together.

We could be acting that out, you know, like half the classes here, half the classes there. We all walk together. We're all putting ourselves together. We could also, there's a graphic organizer that students could use to see. And then we can get into the point where, Oh, okay, look, this is a story problem in which there are parts that are put together [00:10:00] for a total.

There's also ways that you can work toward working on a total. I'm just talking about total problems. There's several other schemas out there, which I'm sure we'll get to, but at first the first few total problems that students see in our intervention efforts are solved. So I might say, you know, Gene has 5, Sarah has 2, they have 7 altogether.

So there, I'm not asking a question, but I'm just putting in this situation where I can talk about, Oh, look at Gene's part, and then Sarah's part, and then we have this 7, that's the total. Amount of money that Jean and Sarah have, and then you can ask the same question. Oh, Jean has $5, Sarah has $2. How much money do they have altogether?

Right. And this would be a perfect one where I could get money and actually use that money to act out that putting together of those parts. And then, or I could have a graphic organizer that could show. five, two, that is, you know, the same as seven. And then eventually we get [00:11:00] to the point where we are asking students to read these questions and solve them, maybe without those representations and, you know, without a solved problem so that you could, there's lots of different ways that you can phase students into that.

One other way, which I know is very popular in schools, but we haven't done a lot of this in our intervention, is to also talk about numberless word problems. So initially saying, well, Gene has some money. And Sarah has some money. And so how much money do they have altogether, right? So there we can focus on the action of putting our two amounts together for that total.

Sometimes it might be more than two parts, by the way, but there we can, with the numberless problems, we're not getting bogged down in the mathematics as much as we are able to focus on that schema or that concept that's embedded within that word problem.

Hey, people can't see me. But I was smiling really big when you were talking about that because without knowing what you know that there's research behind this and things that you were training teachers to do. I had the teachers that I was working with talked them into trying a couple [00:12:00] things, and one of them was that acting it out and you know, they, instead of the entire class acting it out, they had two students act out.

A total, uh, part Is that was the

Part, total.

Part total. And then after they acted it out, then the kids did did a turn and talk and they talked about what happened and they did it. And the And then all of a sudden, the kids were able to do the problem, and the teacher was blown away, in that, geez, it's just so evident that they just didn't understand what was going on, you overcomplicate it, and this is another reason why I like to call it story problems, is it's a story that has some numbers in it, you know, you know, and what are the numbers, it doesn't matter what the numbers are,

Yeah, it

story it's the same story.

it doesn't often matter what the names are in the story, what the things are, whether we're talking about strawberries or turtles or money, you know, the schema stays the same. And I [00:13:00] think that's the power of the schema is that Yeah, the students learn the total schema probably when they're three, four, five years old, and then it continues with them for the rest of their life.

And eventually they might be doing that with fractions or decimals or even positive and negative integers, but still the schema, the concept of the problem, stays the same. So, so we typically see that there's six schema across grades K through eight that students are going to see regularly. We see that total schema, like we've just talked about that one, that's where parts are put together for a total.

We have a difference schema where students are comparing amounts for a difference. We have a change schema where one amount increases or decreases. And those are what we call the additive schema because students will then typically add or subtract to solve the problems that are represented by those schemas.

And then in third grade, students usually learn the equal groups schema, where we have groups with this equal number in each group. They also learn the comparison or multiplicative comparison schema. We call that SET, by the way, just to make [00:14:00] it a little simpler for kids. But that's where a SET is multiplied a number of times.

And then in middle school, students start to learn a schema that where they often apply a ratio or proportion. So that would be the sixth schema. And then many times in late elementary grades and middle school grades, there are multi step problems, which end up being multi schema problems. So maybe first you have to figure out the total, and then you have to figure out a difference but the schemas really stay the same.

And so there's a lot of power in teaching these schemas early, and then students are going to see them again and again, while the math gets more complex, the stories get more complex, the vocabulary gets more complex, or as you alluded to earlier, the vocabulary changes, the schema does not. The schema always remains very consistent.

Well, a couple things. You talked about a some sort of representation, you know, so, you know, thinking about the CRA representation, concrete representational abstract, and that's one of the things. that I [00:15:00] don't understand why we don't do more of,

Oh, yeah. C R

it, you know, it's like, we do it when we're in kindergarten and we're, you know, first grade, we're trying to teach the kids, you know, to teach the kids these concepts, but we have STEM schools and STEAM schools and, you know, all this engineering stuff.

And the first thing engineers do when they have a problem, As they draw a

Draw it out. Yeah. Yeah.

as they draw it out but in, in most of the text series that I see, that's only a strategy if you can't do it.

Yeah, and it should be something that all students have access to. So, with the concrete, typically we're talking about hands on tools to help students understand math. The pictorial or the representational are those drawings of math. And I would say that the vir all the virtual manipulatives that are out there, kind of fall into that category.

And then the abstract are those numbers and symbols. And I do see they're exactly right. And the elementary, early elementary grades, we see a lot of particularly manipulative use. There's a [00:16:00] lot of very fun and colorful and cute math manipulatives. But there are amazing manipulatives for grammar.

fractions. There's really great manipulatives for algebra that can help you understand the property of equality and what does a zero pair mean, right? And we see less of those used and many times, I don't ever want to fault any teachers. I think sometimes they just don't. I don't know that those are out there, like I've many times I'll say, Oh, there's these algebra tiles or these algebra blocks or these Cuisiner rods for fractions and people are like, Oh, that's cool.

You know, tell me more about that. And sometimes the visuals are in textbooks, but sometimes they are not, but it's also to have the schools or districts invested in purchasing some of these materials or having tablets for every student. So they have access to all these virtual manipulatives. And so I think it's sometimes an access thing, why people don't use them, but I would suggest that as the math gets more [00:17:00] complex, we likely need to be using more and more of these representations, whereas right now we see that, When the math is the, I'll say, like, the earliest and the easiest for students, we use the most representations and then we fade them from use, when in actuality, we should bring in more representations as the math gets more and more complex, so that students can have an opportunity to develop better skills.

This knowledge of math that isn't just, Oh, I do this with my pencil or with my stylus, but I actually understand what the mathematics means. And word problem solving is just one of those avenues where we could be using representations much more than we are so that students have that understanding of what the story is asking them to do.

Well, I'm going to add to your list of reasons why teachers may not be using it. And I think part of it is and when I was researching your work, that, So much of the, [00:18:00] this schema, talking about schemas, is talking about students with mathematical difficulties, special ed students, as if it's, you know, that it's an intervention.

And it seems to me Like, if it's an intervention why don't we start with that? You know, if it's so good why are we waiting?

Well, when you look at a few textbook series, they really hit the schemas really hard, and there's a very popular approach to the teaching of math and for core instruction called cognitively guided instruction, or CGI. Probably a lot of your listeners are familiar with that.

And there, they're talking about the meanings of the operations. And they talk about. The same schema that I'm talking about just in a slightly different way, but they're talking about, oh, when students add, they're putting things together or they're adding on. Those are the total and the change schemas, or when students subtract, they are taking away or they are comparing.

That is the change and the different schemas. And so there's a lot. out there. [00:19:00] But often this is all on top of many of the curriculum materials that teachers are using. And to date, I really haven't run into a curriculum series that's doing an excellent job teaching word problem solving. I think that, you know, most of the curricula focus more on the math content knowledge and making sure that's developing for students.

But problem solving should be a part of every student's mathematics experience every day, but there is a lot out there for core instruction. They may not be calling it schemas. They might be calling about like common problem types, or you might see these graphic organizers that map directly onto the schemas, but that doesn't mean that everyone's doing it or that everyone's doing it as much as they should.

Well, I think it's one of the things that I always think about is that novice expert difference. And I think we didn't talk about this, but I sent you I, I sent you something that I had my thoughts about intuition in math, that it's, there's no such thing as intuition. [00:20:00] It's just automaticity.

Well, it's all learned.

It's learned, and then poof, it's immediate. And I think that's what happens with the experts in the story problems. And we could have a six year old expert in

Yeah.

this sense, where they don't need to have that representational phase or it fades out more quickly. That they're not going to need it as, as long, but I think that's part of the issue as well is understanding the teacher, understanding what's going on in your head that you're able to do this, that your student can't do it.

And I think that's always a great question for a teacher to ask themselves. What do you know that they don't,

Yeah. Agreed.

So having So we talked about, you talked about the different, that there's a limited number of schemas.

Yeah.

And being in special ed sure you have, you know, [00:21:00] broken these things down very discreetly.

Yeah, we've tried. Yes.

So, so, and you talked about the beginning of where this work might be with a problem that is the kids have a problem and it's already solved.

Yes.

And so then that first step might be just the kids identifying. The part, total.

Okay. And then so the kids get lots of repetitions in that.

Maybe we've controlled for the vocabulary. Teachers

and controlled for the numbers, right? You know,

Yeah. Yeah.

simple numbers at the beginning so that students can really focus on the schema and not have to get so bogged down on calculations.

And so the next step after you've done that, would you be doing a matching of the. of the the labeling of the problem to a graphic organizer. Would that be a next

Yeah, that's a great. Yeah, that would be the next step. Yeah. [00:22:00] So as you're talking through it, and you're exactly right. So, you know, the reading and understanding of these problems is first and foremost. And then as you've read the problem with students, and as you start to talk about, well, what's going on in the story, what's the focus of the story?

Then the second thing we'll typically ask is about, well, what's going on in this problem? What kind of problem is it? So we will ask students questions like, is this a total problem? Are parts put together for a total? Is this a difference problem? Are amounts compared for a difference? Is this a change problem?

Does one amount increase or decrease? Is this an equal group story? Are there groups with an equal number in each group? Or is this a set story? Is there a set? compared a number of times. And so then students say, Oh, it's a, let's say it's an equal group story. And I'm like, yeah, it is. You're exactly right.

And I always ask, why is it an equal group

Yes. Yes. Yes.

want them to be like, Oh, I see that I have these houses, and there's the same number of people in each house, right? And I'd say, yeah, like, so, so what are your groups? And they'd be like, the groups are the houses. I would say, what are the equal number in [00:23:00] each group?

Oh, those are the people. Yeah, it does seem like we have an equal group story. So then once they determine the schema, then it's really cool for students to have something to help then organize the information from that schema problem, whether that is a graphic organizer, like you talked about whether it's an equation and it then, you know, usually it's in a graphic organizer equation, then they could also use, you know, Drawli other Drawlians or other representations there.

And so if they decide it's an equal group story, as soon as they make that decision then they should write maybe the equal groups equation. And so then that equation, which we use, like we use GR times N equals P, so groups, times the number in each group equals the product, then they write GR times N equals P, and then it's like, Okay, what talks about the groups?

Do you know that? If so, put that number into your equation. If not, maybe that's the unknown. Do you know the number in each group? Do you know the product? And so that it's what we call a schema equation, or it could be a schemographic organizer, can then help students take all [00:24:00] the information from that story problem and organize it in a way that represents conceptually what's going on in the problem.

So then students, maybe it's eight times seven equals question mark. Then they know they have eight groups with seven things in each group. So then if they multiply eight times seven, that's going to help them figure out how many People they have all together or whatever the story problem might be. And so that's that part, it's once they identify the schema, then if they use that graphic organizer or that equation, that's kind of transitioning into the solving of the problem.

So that's where the mathematics is coming into play. Students might add, subtract, multiply, divide. They might draw, they might use manipulatives, they might use a calculator if they, you know, if you have students using calculators. So like, I think that always the solving of the problem. is kind of secondary to the actual understanding of the problem.

And that understanding is through the reading, the interpretation, and the schema of the problem. And then once you have that [00:25:00] understanding of the word problem, then you move to that solving phase where you use your equation or graphic organizer, do the manipulation with the math, and then importantly, which a lot of students skip, is go back and check your work and make sure your answer makes sense.

Yeah.

Well, and I think the way you described it, it allows discreet, discreet areas for the teacher to practice.

Yeah.

So, so, if I was, now, if you agree with me, this is what I always tell teachers I work with, Sarah, if you agree with me, smile and say, Gene, I like that. And if you don't, just smile.

Okay.

So, so here would be my suggestion to the teacher as they're doing that process. So don't move on to the graphic organizer process until you have enough repetitions of the kids identifying. Those part totals. After you've done that, you know,

yeah, like practicing these things in isolation. So I've talked about these schemas across grade levels. When we [00:26:00] teach, right now we're doing fourth grade, so we're introducing five schemas to the students. And each schema gets anywhere from about six to nine days of practice.

pretty much isolated practice. So we, and we're very formulaic with the introduction of these schemas. And there's also this whole thing that maybe we're about to get into a conversation about in terms of the unknown. So let's just talk about the total schema. In the total schema, there's parts put together for a total.

So there are total problems like the one I talked about before. Gene has 5. Sarah has 2. How much money do they have altogether? That's a problem where the unknown is the total. Like, that's what we have to figure out. But the story could also be, Gene and Sarah have 7. If Gene has 5, how much does Sarah have, right?

That is still a total problem, because there are still parts that are put together for a total. In that situation, we're given the total and one of the parts, and our job is to figure out the other part. So, let's just go back to like, introducing total problems. So on the first day, I would introduce total problems, maybe with nothing missing, [00:27:00] so a totally filled in problem, or maybe with an unknown total.

And then we're probably going to practice that for three or four days, and maybe practice that with an equation, or a graphic organizer, probably do some stuff with manipulatives, so that really students can understand, oh, I've got parts put together for a total. Then we would say, okay, today we're going to still work on total problems, but today's total problem is just going to be a little bit different.

And then that's where you would introduce this idea that sometimes you're given the total and one of the parts, and our job is to figure out the other part. And then you practice those heavily, But still bringing in total problems with an unknown total. So like once students learn a variation of a schema, we want to continually bring those back so that they're practicing those as well.

So then after probably. You know, six to nine days, and that might be over a two week span. Okay, now our students have a pretty good idea of total problems where parts are put together for total. There's also three or four part total problems, so you might want to introduce those at the end if those are going to [00:28:00] be something that students would see regularly.

Then, We would move to the next schema. And on day one, usually it's difference problems with an unknown difference. And so we're going to practice that for a few days with a few total problems as review, and then you get to difference problems with the unknown greater amount, or the difference problems with the unknown lesser amount.

And so There's like a lot of, there's a lot to it. And so it does take a fair amount of time. I would say several months in the school year to introduce these to students, but then once they understand the schemas, and if they're practicing this regularly, then they're moving to the next grade level.

And the next grade level teacher doesn't have to do a whole lot of instruction there because they already understand those schemas. So that was a very long winded answer for how to go about teaching it. But I, you know, When I talk about the schemas, like here, we'll talk about it for 30 or 45 minutes, or sometimes when I do a teacher training, we talk about it for three hours or six hours, but there's a lot to it as you're teaching.

I mean, the first part is getting teachers to have the knowledge around the schemas and what's [00:29:00] important for word problem solving, but then the second part is how do we actually you know, use this information across a school year to help students have a deep knowledge around the word problem schemas.

It's okay, you can use the term interleaving on this podcast.

I figured

With that, but that's what we're talking about,

right, yeah.

continually coming back to have a review of those problems that students have already learned. The other thing that we do within our interventions, and we've done this at whole class, small group, and individual, is that Within the session, we do different types of practice with students.

So a lot of it is modeling and practice on let's work through this word problem together. Let's all do this together. But we also have typically a review where students just identify the schema of the word problem and no solving of the word problem. And we have found that's really helpful for students too.

So the tutor or the teacher would [00:30:00] read a problem. So I would say, you know, Jean has 5, Sarah has 2. How much money do they have altogether? What kind of problem is this? And students have to say, Oh, it's total problem. And you're like, yep. Okay, great. And we go to the next one. And we just do that for usually one or two minutes, but having students just be able to very quickly, and I would say fluently start to identify the schemas of these problems is helpful.

And it's a little counterintuitive. And the first time we usually ask students to do that activity, they think it's very weird that we're not solving the problems because so many things that we've done in math, we always require an answer, but I'm like, I don't want you to like solve this problem.

I just want you to tell me what's going on in the story. And then we're going to do, you know, five more of these. We put them on cards, like five more of these cards before we go to the end of our tutoring session, just to give them that practice with pretty quickly recognizing those schemas.

Yeah. Two things as you, you were talking one of the things again, before I met you and your work you know, virtually [00:31:00] was I was struggling with how do you represent you know, we're working with young kids. How do we represent this graphic organizer? You know, like, like, if it's a, um, you know, part, whole, what do we call those, you know, because, oh, are we going to introduce, are we going to, but you just introduced, no, it's a P, it's a P, and it's a T, and it's a

Part, total PPT. Yep.

Part, total, and The fact that you are working intervention and with special ed and that's not a problem.

No. And you know, a lot of, we, in a lot of our research studies and a lot of the work we do in schools, we do work with some students that have an identified disability, but most of the research that I do is for students who are experiencing difficulty with word problem solving. So their teacher has identified them or through an assessment, they've been identified as a student.

a student who's struggling with story problems and so then we'll provide intervention to those students. And I'm sure you've [00:32:00] talked about this on other episodes of your podcast, but that's the majority of kids in the United States. When we look at data coming from like the NAEP, the National Assessment of Educational Progress, which is all story problems, at grade four, we're seeing that I think it's like 64 percent of students are not meeting a minimum level of proficiency.

I would say that this word problem stuff that we're talking about is important in special education for sure, but also really important for students who struggle with math, but also probably likely important for students who don't struggle with math as well. So really for all students.

No, absolutely. So you you talked me out of being afraid to help teachers, you know, you know, label this graphic organizer and just teach what it means. It's a P, you know, it's, there's a one to one correlation. You labeled it a P up here. Now it just goes down here.

Well, and one of the things that your listeners can't see is that as I talk about the word problems, I'm also using hand gestures to describe them. And so when we ask the question, like, are total problems, or is this a total problem? Are [00:33:00] parts put together for a total? We hold our hands apart when we say parts, and then we move our hands together for total.

And we have found that those gestures, when they're paired with that orally presented question, are very helpful for students because they'll use them. They'll be like, Oh, okay. Jean's got 5. Sarah has 2. Putting together. Oh, yes, that is a total problem. So that type of I mean, there's a lot of really great research in math around the use of gestures and how important they are for student understanding.

And so we have a separate gesture that goes along with each schema to help students. Think about, Oh, what type of problem is this? And so I've done a lot of the schema stuff as early as second grade. So second, third, fourth, and fifth grade and into middle school. My colleague Lynn Fuchs at Vanderbilt university, they've done schemas with students as early as first grade and they teach students to P you know, P plus P equals T that's part plus part equals total.

And so I think that when students receive really good high quality instruction on these [00:34:00] things you know, it's easy to. It's easy to take P plus P equals T or G minus L equals D for the greater minus lesser equals difference for difference and students can learn those things and use these equations and use these graphic organizers with a lot of flexibility.

So, instruction is key but it really is possible to take this and do this with students who are as young as 5, years old.

You know, one of the things that you always see in the teaching of story problems, which, if you're using this schema approach, is so easy, is, okay, now go write your own problem.

Yeah, whole problem

yeah, well, yeah, so if you have, if you understand schema, guess what? Sarah and Gene just became, you know, Jose and Maria.

You know, it's the same problem.

have wow, she'll be so excited that I'm talking about this. One of my current doctoral students is really interested. Actually, two of them are. This is like, kind of falls in this whole idea of problem posing. And there's different types of problem posing. But we are right now running an [00:35:00] intervention.

We're actually post testing in schools today on this, but we taught students about the schemas and then we also asked them to problem pose. And so at the beginning, it might be you know, write a total problem that was pretty similar to the one. Yes, you're exactly right. Like this could be now about Jose and Ella.

Jose has 5. Ella has 2. How much money do they have altogether? Or Jose has 9. Ella has And then eventually kind of removing some of those structures of the problem posing. And so toward the end of the intervention, we would ask students like, can you write a total problem with an unknown part? And the students are doing so well.

Like, I get pictures every day of here's some of the, here's what my kids did in terms of problem posing today. And we, I, The data will tell us, but we're getting an idea that it's actually helped through that problem posing, through that generation of their own word problems that is helping them [00:36:00] understand the schema in a better and deeper way.

So stay tuned. Probably our results will, you know, be forthcoming over the next year. But we're really excited about it. And there's a lot of literature out there about problem posing. A lot of it is I would say a little more free form. So we're trying to put some structure into that problem posing and explicitly teaching students how to problem pose.

And so far, just anecdotally, we're really excited about what the students are doing. And then the post dust data will tell us whether that's impactful, not only for their problem posing skill, but also does that translate to improved word problem solving, and that's where I think that would be very cool.

We're also testing a competing intervention against that one, where the added component is a focus on math vocabulary. Understanding that if you know the vocabulary terms in these words, You're more likely to have a deeper understanding of these word problems, and then does that contribute to more successful problem solving.

So, We're kind of in the [00:37:00] last year of a grant where we're able to test out some of these I won't call them side projects, but we're trying to now to figure out what are some of the real important mechanisms for improving students word problem performance. Because we know the schema stuff is, really important and really helpful.

But now if we layer on problem posing, if we layer on a math vocabulary component, does that actually contribute to even better word problem performance than the, you know, original intervention?

Well, I think the schema getting back to comparison between language arts and story problems, understanding that schema is a lot like understanding a text structure.

Oh

you know, so you're going to get a lot of context clues so I don't know how you, I don't know how you separate, separate that.

And it sounds, that does sound exciting. All right. I have to

You know, just to be, so when I know a little bit about reading and just very little about writing, but I've talked with reading colleagues who are in the reading comprehension and I've [00:38:00] worked with some colleagues who focus on Writing and different like how to support students in writing and they've talked about the same thing.

Although it may not always be explicitly called a schema, but especially in writing, like sometimes students are asked to write a narrative. Sometimes they are asked to write a compare contrast, right? Those are different schemas. And. Students and, I mean, teachers teach and students learn how to write a compare contrast essay versus how to write a narrative essay.

And in math, we're doing the same thing, like, here's how to solve a total problem versus here's how to solve a difference problem. And so I do think that and I should probably spend more time on this, but really understanding how we see these problem types or structures or schemas across content areas.

I think then we could. call attention to it and help students see, Oh, there's only a few different types of essays that I'm going to write. There's really only a few different types of passages that I'm going to read. And there's only a few [00:39:00] types of word problems, story math problems that I'm going to have to set up and solve.

Well, when I was in, when I was in high school, I thought my language arts teachers were the smartest people in the world.

Oh, really?

Okay, this was, again, pre internet, but, you know, you go in, and we're reading a novel, and they could talk about all this symbolism, and all of this, you know, all the structures, and, you know, Oh, I remember Moby Dick, you know, when they're below the water, it's the subconscious, and I'm thinking, boy, how are you're so smart to be able to know all that

Yeah. Yeah. Yeah. That's

But guess what? It's just, it's, they weren't that smart at all. They had just been taught, they'd just been taught, they'd just been taught the structures of the, of those things, and all our kids could be that smart.

Yes. Yeah, for

Okay, I have I have to ask you this, and we can make it short, you know, we can make it short.

A couple of the most common ways that you see how to, that we teach [00:40:00] kids how to solve story problems that are not effective. What methods are the least

Well, the first and foremost, which is the most popular teachers or caregivers use keywords tied to operations. This is Often done through a keyword poster, which probably when I say that people know what I'm talking about. But if you don't, I would say that those are words or terms that are organized by operation.

So maybe, you know, in an addition area of a poster, you might see words like put together, more, total, altogether, some. And that, so some students are taught or either. create this knowledge on their own, that every time they see total, that means to add. Or every time they see share, that means to divide.

When in actuality, there's no term that means a specific operation. And in fact, and you kind of talked about this at the beginning of our conversation the people who write these word problems are very cued into this idea that students use these keywords. And so often when you see a keyword on [00:41:00] a standardized test item or within a textbook, it doesn't mean that.

the operation that a keyword poster would tell you it should mean. And in fact, my colleagues and I collected data from a bunch of released items from standardized assessments and kind of explored how often that keyword would help you solve the word problem correctly. And for single step word problems, it works around half of the time, but for multi step word problems, the keyword strategy works less than 10 percent of the time.

So it's really not a good strategy to use for students. And it really does not work for. particularly problems where you end up multiplying or dividing. It starts to fall apart pretty well by the time you get there. So that's one. So do not use those keyword posters tied to operations. And then the second one is more of just like a way word problem solving is practiced in the classroom.

So that is giving students let's just say like a set of problems or a worksheet where at the top it says subtraction word problems. Well, if you do that immediately, no kid. usually in [00:42:00] their right mind is going to read these problems. They'll be like, cool. And so what they end up doing is they become what is kind of commonly referred to as a number plucker, where they just pluck the numbers from the problem, and then they subtract, or they pluck and they multiply, or they pluck and add, or pluck and divide.

And so, that really is not encouraging students to practice their word problem solving at all. And if you need those word problem activities, which I know a lot of teachers are always asking for more and more word problem resources, sure, go ahead and use those word problems. Just remove the operation from the, you know, the worksheet or.

Don't tell the students. Make them decide for themselves whether they are going to add, subtract, multiply, or divide. And you know, in fact, Jean, there's a lot of word problems where you might choose to add, and I might choose to subtract, and both of us are correct. You know, especially if it's like, Jean and Sarah have 7.

If Jean has 5, how much money does Sarah have? Well, you could do 7 5 to solve that problem. You could also start at 5 and count up to 7 to figure out what's the difference between 5 and 7. [00:43:00] So, both of those approaches are great. And in fact, many times we're trying to get students to think about multiple ways to solve this problem.

And so for that situation that I just gave you, I could not describe that word problem as an addition problem, in the same way I could not describe it as a subtraction problem. In actuality, that's a total problem. where parts are put together for a total. And it doesn't matter if I add or subtract. It doesn't matter if I multiply or divide to solve these problems.

What matters is that I under, understand conceptually what's going on in the problem and that goes directly back to the schemas.

Okay. So here's another time where I'm going to ask you to smile and say, Oh, Gene, I liked that. Last time you avoided both. So I'm going to try it or try it this time. What you just described about, you know, solving that missing part total.

Yeah.

Is, and they gave an example of, Oh, there's, you know, there's a couple of ways you could do it.

There's not just one way to do it. What I think, the way you described that, I think [00:44:00] is, was perfect. And I, and it matches what people say, Oh, there's many ways to solve a problem. Where I think teachers get in trouble is where they think there's many ways to do this. Algorithm. No you chose one of those standard algorithms.

You got to decide which one, but then you still use a standard algorithm to solve that problem.

Well, or could kids could use manipulative so they might not be using an algorithm. I think for the example of like, Oh, if I start at five and move to seven, one of the things we often draw for students is an open number line to try to help them think about, Oh, what is the difference between five and seven?

So, I won't get into like the whole thing about, you know, standard versus non standard. Yeah. Yeah. Yeah. Algorithms, because, you know, the research is not, there's actually not a lot of research out there that says one approach is the better, is better, but I will say that every student needs to understand a way to add, subtract, multiply, and divide and be very comfortable [00:45:00] with that and how teachers develop that in the students.

I think we're yet to really investigate the approaches there, but I think, like when we would teach students that, we would typically either say like, Oh, well, what would you do? And probably most students are going to subtract. So, you know, we would be like, yeah, that's a good way to solve that problem.

Some days we would ask, like, what's another way to do this? Some days we got to move on to the next problem, right? But ultimately we just want students to our biggest thing is like, do they understand the schema of this problem? And then I always see that the solving of that problem kind of follows that and that's secondary to the understanding of

right. It's at least back

I didn't really answer your question,

well, well, going back to the, going back to previous discussion is, it's the least important part.

Yes. Yeah. I mean, it's an important part. And if students don't know 7 minus 5, or if they don't know 5 plus what equals 7, then answering that problem is going to be really tricky. And most students would be able to figure out, you know, 7 minus 5, but what if we're now working with three [00:46:00] digit numbers?

What if it's now fractions with decimals? So there is a lot of important computation skill that we need to emphasize here. But We can't, in the classroom, just teach computation and then, like, hope the word problem solving is going to, like, magically follow. I would suggest that we should be really teaching through problem solving, and then you can embed that computation instruction within your problem solving instruction.

Yeah.

Very good. All right, Sarah, you know what I forgot to do? I was so excited that you were going to be a guest. I always have guests. Think of a question that they might have for me. (Question for Gene)

Oh gosh, well, you didn't prep me on this one, but I do have a question for you in terms of like, how would you teach problem solving if you weren't teaching this using the schemas? Like, how have you talked about it with teachers in the past?

Talked about it in a very similar fashion that we talked about it that does a limited, you know, there's a limited number of schema. I didn't call it [00:47:00] schema because I was

But just like problems. Yeah.

you know, you know, some people call it that's the deep structure. It's whatever. Just try to get across to them that it doesn't matter if it's Marbles, Dollars, Apples, or whatever, it's the same problem.

To get their thinking that way, and the, I always think of the progression and we talked about that a little bit and you helped me out by saying, I give them the problem that's worked. So that we are identifying, so that we are isolating the variable in what we're trying to do.

Yes.

Because there's no there's no, they don't have to be perfect to move on to the next thing, but we have to know that they have some competence, and we have to get them, we have to get, you know, their their, I guess it's extraneous load, like, don't worry about solving this, we're not going to solve this today, don't worry about that, we have to get rid of that.

So I think that's what I would do, and like you said, it could take months to do this. [00:48:00] And in the months that it would take to do that, the kids would have a better chance to develop you know, fluency in their math backs. And so again, that is, you know, where. When we're teaching, we're isolating variables to be able to know what do I need to go back and practice?

When can, what can I move on? How can I release them? When I pulled a kid for intervention, what are we going to work on? So I think I think very similar to, you know, you know, but adding, because you've convinced me now that we can add, you know, those, you know, those initials for those, for the

Oh, for sure. Yeah. And I think that makes it easy for students to say like, Oh, this is P plus P equals T. And if they know that they're either thinking it or they can write that down, then they can organize that information from the total problem using that total equation.

so, and with that, I think that it then the [00:49:00] teachers see it too, because it's not that hard to see, but if you haven't been exposed to it, it's like, wow it's like your whole world opens up for you.

Yeah, exactly. And in my experience, you know, for some students and for a lot of teachers, you know, we read these word problems and we're pretty well versed on problem solving. And so our brains just go to the operation, like, well, what am I going to do to solve this? When we really need to step back and be like, well, what's actually going on in this story?

So I know you're going to subtract, like that's what your approach is to this, but why are you going to subtract? Are you comparing amounts? Are you taking away, you know, like what is it that makes you think you want to subtract? And. It has so many conversations in mathematics always are about this balance of this conceptual and procedural knowledge.

Really, the subtraction is very procedural, but conceptually, we want to have students understand what in this story problem wants you to subtract. Well, here I've got this, you know, this amount and we're taking away from this amount and getting students to get that to that conceptual piece is a real important part [00:50:00] of the problem solving process.

And I think using that graphic organizer, if I was working with a teacher, I would say if a kid said, why did I do this? And they use the graphic organizer to explain it. That's good. We're good. We're good.

great. Yeah, I'd

Yeah. Yeah. Yeah. We're good. So gosh, Sarah, it has been a absolute pleasure. I,

This is the highlight of my week talking about word problem solving. So thank you for the invitation.

oh, it was my pleasure and hope to talk to you again.

I look forward to the conversation. Thank you so much.

If you're enjoying these podcasts, tell a friend. Also, please leave a 5 star rating on Apple Podcasts, Spotify, or wherever you listen. You can follow me on BlueSky at gTabernetti, on Twitter, x at gTabernetti, and you can learn more about me and the work I do at my website, BlueSky. Tesscg. com, that's T E S S C [00:51:00] G dot com, where you will also find information about ordering my books, Teach Fast, Focus Adaptable Structure Teaching, and Maximizing the Impact of Coaching Cycles.

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