Drs. Aina Appova and Julia Hagge, Making Sense of Word Problems

Mike Wallus, Vice President for Educator Support


Contextualized story problems are an important tool that educators use to bring mathematics to life for their students. That said, navigating the meaning and language found in story problems is a challenge for many students. Today we’re talking with Aina Appova and Julia Hagge from The Ohio State University about strategies to help students engage with and make sense of story problems.

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Schema-Mediated Vocabulary in Math Word Problems


Mike Wallus: Story problems are an important tool that educators use to bring mathematics to life for their students. That said, navigating the meaning and language found in story problems is a challenge for many students. Today we're talking with Drs. Aina Appova and Julia Hagge from [The] Ohio State University about strategies to help students engage with and make sense of story problems. 

A note to our listeners: This podcast was recorded outside of our normal recording studio, so you may notice some sound quality differences from our regular podcast.
Welcome to the podcast, Aina and Julia. We're excited to be talking to both of you.

Aina Appova: Thank you so much for having us. We are very excited as well.

Julia Hagge: Yes, thank you. We're looking forward to talking with you today.

Mike: So this is a conversation that I've been looking forward to for quite a while, partly because the nature of your collaboration is a little bit unique in ways that I think we'll get into. But I think it's fair to describe your work as multidisciplinary, given your fields of study.

Aina: Yes, I would say so. It's kind of a wonderful opportunity to work with a colleague who is in literacy research and helping teachers teach mathematics through reading story problems.

Mike: Well, I wonder if you can start by telling us the story of how you all came to work together and describe the work you're doing around helping students make sense of word problems.

Aina: I think the work started with me working with fifth grade teachers, for two years now, and the conversations have been around story problems. There's a lot of issues from teaching story problems that teachers are noticing. And so this was a very interesting experience. One of the professional development sessions that we had, teachers were saying, “Can we talk about story problems? It's very difficult.” And so we just looked at a story problem. And the story problem—it was actually a coordinate plane story problem—it included a balance beam. And, you know, you're supposed to read the story problem and locate where this balance beam would be. And I had no idea [where] the balance beam would be. So when I read the story, I thought, “Oh, it must be from the remodeling that I did in my kitchen, and I had to put in a beam, which was structural.” So I'm assuming it's balancing the load. And even that didn't help me. I kept rereading the problem and thinking, “I'm not sure this is on the ceiling, but the teachers told me it's gymnastics.” And so even telling me that it was gymnastics didn't really help me because I couldn't think, in the moment, while I was already in a different context of having the beam, a load-bearing beam. 

It was very interesting that—and I know I'm an ELL, so English is not my first language—in thinking about a context that you're familiar with by reading a word or this term, “balance beam.” And even if people tell you, “Oh, it's related to gymnastics”—and I've never done gymnastics; I never had gymnastics in my class or in my school where I was. It didn't help. And that's where we started talking about underlying keywords that didn't really help either because it was a coordinate plane problem. So I had to reach out to Julia and say, “I think there's something going on here that is related to reading comprehension. Can you help me?” And that's how this all started. [chuckles]

Julia: Well, so Aina came to me regarding her experience. In fact, she sent me the math problem. She says, “Look at this.” And we talked about that. And then she shared [the] frustration of the educators that she had been working with, that despite teaching strategies that are promoted as part of instructional practice, like identifying mathematical keywords, and then also reading strategies have been emphasized, like summarizing or asking questions while you're reading story problems. So her teachers had been using strategies—mathematical and also reading—and their students were still struggling to make sense of and solve mathematical problems. Aina’s experience with this word problem really opened up this thought about the words that are in mathematical story problems. And we came to realize that when we think about making sense of story problems, there are a lot of words that require schema. And schema is the background knowledge that we bring to the text that we interact with.

For example, I taught for years in Florida. And we would have students that had never experienced snow. So, as an educator, I would need to do read-alouds and provide that schema for my students so that they had some understanding of snow. So when we think about math story problems, all words matter—not just the mathematical terms, but also the words that require schema. And then when we think about English learners, the implications are especially profound because we know that that vocabulary is one of the biggest challenges for English learners. So when we consider schema-mediated vocabulary and story problems, this really becomes problematic. And so Aina and I analyzed the story problems in the curriculum that Aina’s teachers were using, and we had an amazing discovery.

Aina: Just the range of contexts that we came across from construction materials or nuts and bolts and MP3 players that children don't really have anymore, a lot of them have a phone, to making smoothies and blenders, which some households may not have. 

In addition to that, we started looking at the words that are in the story problems. And like Julia said, there are actually mathematics teachers who are being trained on these strategies that come from literacy research. One of them was rereading the problem. And it didn't matter how many times I reread the problem or somebody reread it to me about the balance beam. I had no kind of understanding of what's going on in the problem. The second one is summarizing. And again, just because you summarized something that I don't understand or read it louder to me, it doesn't help, right? And I think the fundamental difference when we solve problems or the story problems [is that]in … literacy, the purpose of reading a story is very different. In mathematics, the purpose of reading a story is to solve it, making sense of problems for the purpose of solving them. The three different categories of vocabulary we found from reading story problems and analyzing them is there's “technical,” there's “subtechnical,” and there’s “nontechnical.” I was very good at recognizing technical words because that's the strategy that for mathematics teachers, we underline the parallelogram, we underline the integer, we underline the 8 or the square root, even some of the keywords we teach, right? “Total” means some or “more” means addition.

Mike: So technical, they're the language that we would kind of normally associate with the mathematics that are being addressed in the problem. Let's talk about subtechnical because I remember from our pre-podcast conversation, this is where some light bulbs really started to go off, and you all started to really think about the impact of subtechnical language.

Julia: Subtechnical includes words that have multiple meanings that intersect mathematically and other contexts. So, for example, “yard.” Yard can be a unit of measurement. However, I have a patio in my backyard. So it's those words that have that duality. And then when we put that in the context of making sense of a story problem, it's understanding, “What is the context for that word, and which meaning applies to that?” Other examples of subtechnical would be “table” or “volume.” And so it's important when making sense of a story problem to understand which meaning is being applied here. 

And then we have nontechnical, which is words that are used in everyday language that are necessary for making sense of or solving problems. So, for example, “more.” More is more. So more has that mathematical implication. However, it would be considered nontechnical because it doesn't have dual meanings.

So … categorizing vocabulary into these three different types helped us to be able to analyze the word problems. So we worked together to categorize. And then Aina was really helpful in understanding which words were integral to solving those math problems. And what we discovered is that often words that made the difference in the mathematical process were falling within the subtechnical and nontechnical. And that was really eye-opening for us.

Mike: So, Aina, this is fascinating to me. And what I'm thinking about right now is the story that you told at the very beginning of this podcast, where you described your own experience with the word problem that contained the language “balance.” And I'm wondering if you applied the analysis that you all just described with technical and subtechnical and the nontechnical. When you view your own experience with that story problem through that lens, what jumps out? What was happening for you that aligns or doesn't align with your analysis?

Aina: I think one of the things that was eye-opening to me is, we have been doing it wrong. That's how I felt. And the teachers felt the same way. They're saying, “Well, we always underline the math words because we assume those are the words that are confusing to them. And then we underline the words that would help them solve the problem.” So it was a very good conversation with teachers to really, completely think about story problems differently. It's all about the context; it's all about the schema. And my teachers realize that I, as an adult who engages in mathematics regularly, have this issue with schema. I don't understand the context of the problem, so therefore I cannot move forward in solving it. And we started looking at math problems very differently from the language perspective, from the schema perspective, from the context perspective, rather than from underlining the technical and mathematical words first. That was very eye-opening to me.

Mike: How do you think their process or their perspective on the problems changed, either when they were preparing to teach them or in the process of working with children?

Aina: I know the teachers reread a problem out loud and then typically ask for a volunteer to read the problem. And it was very interesting; some of the conversations were [about] how different the reading is. When the teacher reads the problem, there is where you put the emotion, where the certain specific things in the problem are. Prosody?

Julia: Yes, prosody is reading with appropriate expression, intonation, phrasing.

Aina: So when the teacher reads the problem, the prosody is present in that reading. When the child is reading the problems, it's very interesting how it sounds. It just sounds [like the] word and the next word and the next word and the next word, right? So that was kind of a discussion too. 

The next strategy the math teachers are being taught is summarizing. I guess discussing the problem and then summarizing the problem. So we kind of went through that. And once they helped me to understand in gymnastics what it is, looking up the picture, what it looks like, how long it is, and where it typically is located and there's a mat next to it, that was very helpful. And then I could then summarize, or they could summarize, the problem. But even [the] summarizing piece is now me interpreting it and telling you how I understand the context and the mathematics in the problem by doing the summary. So even that process is very different. And the teacher said that's very different. We never really experience that.

Mike: Julia, do you want to jump in?

Julia: And another area where math and reading intersect is the use of visualization. So visualization is a reading strategy, and I've noticed that visualization has become a really strong strategy to teach for mathematics as well. We encourage students to draw pictures as part of that solving process. However, if we go back to the gymnastics example, visualizing and drawing is not going to be helpful for that problem because you are needing a schema to be able to understand how a balance beam would situate within that context and whether that's relevant to solving that word problem. 

So even though we are encouraging educators to use these strategies, when we think about schema-mediated vocabulary, we need to take that a step further to consider how schema comes into play and who has access to the schema needed, and who needs that additional support to be able to negotiate that schema-mediated vocabulary.

Mike: I was thinking the same thing, how we often take for granted that everyone has the same schema. The picture I see in my head when we talk about balance is the same as the picture you see in your head around balance. And that's the part where, when I think about some of those subtechnical words, we really have to kind of take a step back and say, “Is there the opportunity here for someone to be profoundly confused because their schema is different than mine?” And I keep thinking about that lived experience that you had where, in my head I can see a balance beam, but in your head you're seeing the structural beam that sits on the top of your ceiling or runs across the top of your ceiling.

Aina: Oh, yeah. And at first, I thought, the word “beam” typically, in my mind for some reason, is vertical.

Mike: Yeah.

Aina: It's not horizontal. And then when I looked at the word “balance,” I thought, “Well, it could balance vertically.” And immediately what I think about is, you have a porch, then you see a lot of porches that balance the roof, and so they have the two beams …

Mike: Yes!

Aina: … or sometimes more than that. So at no point did I think about gymnastics. But that's because of my lack of experience in gymnastics, and my school didn't have the program. As a math person, you start thinking about it and you think, “If it's vertically, this doesn't make any sense because we're on a coordinate plane.” So I started thinking about [it] mathematically and then I thought, “Oh, maybe they did renovations to the gymnasium, and they needed a balance beam. So I guess that's the beam that carries the load.” So that's how I flipped, in my mind, the image of the beam to be horizontal. Then the teachers, when they told me it's gymnastics, that really threw me off, and it didn't help. And I totally agree with Julia. 

You know when we do mathematics with children, we tell them, “Can you draw me a picture?”

Mike: Mm-hmm.

Aina: And what we mean is, “Can you draw me a mathematical picture to support your problem solving or the strategies you used?” But the piece that was missing for me is an actual picture of what the balance beam is in gymnastics and how it's located, how long it is. So yeah, yeah, that was eye-opening to me. 

Mike: It's almost like you put on a different pair of glasses that allow you to see the language of story problems differently and how that was starting to play out with teachers.

I wonder, could you talk about some of the things that they started to do when they were actually with kids in the moment that you looked at and you were like, “Gosh, this is actually accounting for some of the understanding we have about schema and the different types of words.”?

Aina: So the teacher would read a problem, which I think is a good strategy. But then it was very open-ended. “How do you understand what I just read to you?” “What's going on in the story problem?” “Turn to your partner.” “Can you envision? Can you think of it? Do you have a picture in your mind?” 

So we don't jump into mathematics anymore. We kind of talk about the context, the schema. “Can you position yourself in it?” “Do you understand what's going on?” “Can you retell the story to your partner the way you understand it?” And then we talk about, “So how can we solve this problem?” “What do you think is happening?” based on their understanding. That really helped, I think, a lot of teachers also to see that sometimes interpretations lead to different solutions, and children pay attention to certain words that may take them to a different mathematical solution. It became really about how language affects our thinking, our schema, our image in the head, and then based on all of that, where do we go mathematically in terms of solving the problem?

Mike: So there are two pieces that really stuck out for me in what you said. I want to come back to both of them. The first one was, you were describing that set of choices that teachers made about being really open-ended about asking kids, “How do you understand this? Talk to your neighbor about your understanding about this.” And it strikes me that the point you made earlier when you said context has really become an important part of some of the mathematics tasks and the problems we create. This is a strategy that has value not solely for multilingual learners, but really for all learners because context and schema matter a lot.

Aina: Yes.

Mike: Yeah. And I think the other thing that really hits me, Aina, is when you said, “We don't immediately go to the mathematics, we actually try to help kids situate and make sense of the problem.”

There's something about that that seems really obvious. When I think back to my own practice as a teacher, I often wonder how I was trying to kind of quickly get kids into the mathematics without giving kids enough time to really make meaning of the situation or the context that we were going to delve into.

Aina: Exactly. Mike, to go back to your question, what teachers can do, because it was such an eye-opening experience that, it's really about the language; don't jump into mathematics. The mathematics and the problem actually is situated around the schema, around the context. And so children have to understand that first before they get into math. 

I have a couple of examples if you don't mind, just to kind of help the teachers who are listening to this podcast to have an idea of what we're talking about. One of the things that Julia and I were thinking about is, when you start with a story problem, you have three different categories of vocabulary. You have technical, subtechnical, nontechnical. If you have a story problem, how do you parse it apart? OK, in the math story problems we teach to children, it's typically a number and operations.

Let's say we have a story problem like this: “Mrs. Tatum needs to share 3 grams of glitter equally among 8 art students. How many grams of glitter will each student get?” So if the teacher is looking at this, technical would definitely be grams: 3, 8, and that is it. Subtechnical, we said “equally,” because equally has that kind of meaning here. It's very precise; it has to be [an] exact amount. But a lot of children sometimes say, “Well that's equally interesting.” That means it's similarly, or kind of, or like, but not exact. So subtechnical might qualify as “equally.” Everything else in the story problem is nontechnical: sharing and glitter, art students, each student, how much they would get. 

I want the teachers to go through and ask a few questions here that we have. So, for example, the teacher can think about starting with subtechnical and nontechnical, right? Do students understand the meaning of each of these words? Which of these could be confusing to them? And get them to think about the story, the context and the problem. And then see if they understand what the grams are, and 8 and 3, and what's happening, and what do those words mean in this context? 

Once you have done all this work with children, children are now in this context. They have situated themselves in this. “Oh, there's glitter, there's an art class. There's a teacher; they're going to do a project.” And so they've discussed this context. Stay with it as a teacher, and give them another problem that is the same context. Use as many words from the first problem as you can, and change it up a little bit in terms of mathematical implication or mathematical solution. For example, I can change the same problem to be, “Mrs. Tatum needs to buy 3 grams of glitter for each of her 8 art students. How many grams of glitter does she need to buy?” So the first problem was [a] division problem; now [it] becomes a multiplication problem. The context is the same. Children understand the context, especially children like myself, who are ELL, who took the time to process, to learn new words, to understand new context, and now they're in this context. Let's use it. Let's now use it for the second piece. 

So, Mike, you've been talking about the two things going on. There's a context and then there's problem solving, or mathematical problem solving. So I believe posing the same question or kind of the same story problem with different mathematical implications gets at the second piece. So first we make sense of the problem of the context, schema. The second is, we make sense of that problem for the purpose of solving it. And the purpose of solving it is where these two problems that sound so familiar and situate in the same context but have different mathematical implications for problem solving. This is where the powerful piece, I think, is missing. If I give them a division problem, they can create a multiplication problem with the same Mrs. Tatum, the art students, the glitter. But what I'd like for them to do and what we've been discussing is, “How are these two problems similar?”

Mike: Mm-hmm.

Aina: This kind of gets at children identifying some of the technical. So the 3 is still there, the 8 is still there, you know, grams are still there. But then, “How are these two story problems different?” This is really schema-mediated vocabulary in the context where they now have to get into subtechnical and nontechnical. “Oh, well there, there's 3, but it's 3 per student. And this, there were 8 students, and they have to all share the 3 grams of glitter.”

So children now get into this context and difference in context and how this is impacting the problem-solving strategies. I'd love for the teachers to then build on that and say, “How would you solve the first problem?” “What specifically is in the story problem [to] help you solve it, help you decide how to solve it? What strategies? What operations?” And do exactly the same thing for the second problem as well. “Would you solve it the same way?” “Are the two problems the same?” “Will they have the same solutions or different?” “How would you know?” “What tells you in the story?” “What helps you decide?” So that really helps children to now become problem solvers. 

The fun is the mathematical variations. So, for example, we can give them a third problem and say, “I have a challenge for you.” For example, “Mrs. Tatum needs to buy 3 grams of glitter for each of her 8 art students for a project, but she only has money today to buy 8 grams of glitter. How much more glitter does she need to still buy for her students to be able to complete their art project?” Again, it's art, it's glitter, it's 8 students, there's 3, the 8. I didn't change the numbers, I didn't change the context, but I did change the mathematical implication for their story problem. I think this is where Julia and I got very excited with how we can use schema-mediated vocabulary and schema in context to help children understand the story, but then really have mathematical discussions about solutions.

Mike: What's interesting about what you're saying is the practices that you all are advocating and describing in the podcast, to me, they strike me as good practice: helping kids make meaning and understand, and not jumping into the mathematics, and recognizing how important that is. That feels like good practice, and it feels particularly important in light of what you're saying.

Julia: I agree. It's good practice. However, what we found when we reviewed literature—because one of the first steps that we took was [asking] “What does the literature say?” We found that focusing instructional practice on teaching children to look for key mathematical terms tends to lead to frequent errors.

Mike: Yep.

Julia: The mathematical vocabulary tends to be privileged when teaching children how to make sense of and solve word problems. We want to draw attention to the subtechnical and nontechnical vocabulary, which we found to be influential in making sense of. And as, in the examples Aina shared, it was the nontechnical words that were the key players, if you will, in solving that problem.

Mike: I'm really glad you brought up that particular point about the challenges that come out of attempting to help kids mark certain keywords and their meanings. Because certainly, as a person who's worked in kindergarten, first grade, second grade, I have absolutely seen that happen. There was a point where I was doing that, and I thought I was doing something that was supporting kids, and I was consistently surprised that it was often, like, that doesn't seem to be helping.

Julia: I also used that practice when I was teaching second grade. The first step was “Circle the keywords.” And I would get frustrated because students would still be confused in the research that we found. When you focus on the keywords, which tend to be mathematical terms, then those other words that are integral to making sense of and solving the story problem get left behind.

Mike: The question I wanted to ask both of you before we close is: “Are there practices that you would say, like, ‘Here's a way that you can take this up in your classroom tomorrow and start to take steps that are supportive of children making sense of word problems.’?”

Julia: I think the first step is adding in that additional lens. So when previewing story problems, consider what schema or background knowledge is required to understand this word, these words, and then what students would find additional schema helpful. So thinking about your specific students, what students would benefit from additional schema and how can I support that schema construction?

Mike: Aina, how about for you?

Aina: Yeah, I have to say I agree with Julia. Schema seemed to be everything. If children don't understand the context and don't make sense of the problem, it's very hard to actually think about solving it. To build on that first step, I don't want teachers to stop there. I want teachers to then go one step further. Present a similar problem or [a] problem that includes [the] same language, same words, as many as you can, maybe even [the] same numbers, definitely [the] same schema and context, but has a different mathematical implication for solving it. So maybe now it's a multiplication problem or addition problem. And really have children talk about how different or similar the problems are. “What are the similarities?” “What are the differences?” “How their solutions are the same or different? Why that is.” So really unpack that mathematical problem-solving piece now, after you have made sense of the context and the schema. 

As an ELL student myself, the more I talked as a child and was able to speak to others and explain my thinking and describe how I understand certain things and be able to ask questions, that was really, really helpful in learning English and then being successful with solving mathematical problems. I think it really opens up so many avenues and to just go beyond helping teachers teach mathematics.

Mike: I know you all have created a resource to help educators make sense of this. Can you talk about it, Julia?

Julia: Absolutely. Aina and I have created a PDF to explain and provide some background knowledge regarding the three types of vocabulary. And Aina has created some story problem examples that help to demonstrate the ways in which subtechnical and nontechnical words can influence the mathematical process that's needed. So this resource will be available for educators wanting to learn more about schema-mediated vocabulary in mathematical story problems.

Mike: That's fantastic. And for listeners, we're going to add this directly to our show notes. I think that's a great place for us to stop. 

Aina and Julia, I want to thank both of you so much for joining us. It has absolutely been a pleasure talking to both of you.

Aina: Thank you.

Julia: Thank you.

Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling all individuals to discover and develop their mathematical confidence and ability.