Season 2 | Episode 7 – Making Sense of Fractions - Guest: Susan Empson, PhD

Rounding Up

Rounding Up

Season 2 | Episode 7 – Making Sense of Fractions

Guest: Dr. Susan Empson

Mike Wallus: For quite a few adults, fractions were a stumbling block in their education that caused many to lose their footing and begin to doubt their ability to make sense of math. But this doesn't have to be the case for our students. Today on the podcast, we're talking with Dr. Susan Empson about big ideas and fractions and how we can make them more meaningful for our students. Welcome to the podcast. Susan. Thanks for joining us.

Susan Empson: Oh, it's so great to be here. Thank you for having me.

Mike: So, your book was a real turning point for me as an educator, and one of the things that it did for me at least, it exposed how little that I actually understood about the meaning of fractions. And I say this because I don't think that I'm alone in saying that my own elementary school experience was mostly procedural. So rather than attempting to move kids quickly to procedures, what types of experiences can help children build a more meaningful understanding of fractions?

Susan: Great question. Before I get started, I just want to acknowledge my collaborators because I've had many people that I've worked with. There's Linda Levi, co-author of the book, and then my current research partner, Vicki Jacobs. And of course, we wouldn't know anything without many classroom teachers we've worked with in the current and past graduate students. In terms of the types of experiences that can help children build more meaningful experiences of fractions, the main thing we would say is to offer opportunities that allow children to use what they already understand about fractions to solve and discuss story problems. Children's understandings are often informal and early on, for example, may consist mainly partitioning things in half. What I mean by informal is that understandings emerge in situations out of school. So, for example, many children have siblings and have experienced situations where they have had to share, let's say three cookies or slices of pizza between two children. In these kinds of situations, children appreciate the need for equal shares, and they also develop strategies for creating them. So, as children solve and discuss story problems in school, their understandings grow. The important point is that story problems can provide a bridge between children's existing understandings and new understandings of fractions by allowing children to draw on these informal experiences. Generally, we recommend lots of experiences with story problems before moving on to symbolic work to give children plenty of opportunity to develop meaningful fractions. And we also recommend using story problems throughout fraction instruction. Teachers can use different types of story problems and adjust the numbers in those problems to address a range of fraction content. There are also ideas that we think are foundational to understanding fractions, and they're all ideas that can be elicited and developed as children engage in solving and discussing story problems.

Susan: So, one idea is that the size of a piece is determined by its relationship to the whole. What I mean is that it's not necessarily the number of pieces into which a whole is partitioned that determines the size of a piece. Instead, it's how many times the piece fits into the whole. So, in their problem-solving, children create these amounts and eventually name them and symbolize them as unit fractions. That's any fraction with 1 in the numerator.

Mike: You know, one of the things that stands out for me in that initial description that you offered, is this idea of kids don't just make meaning of fractions at school, that their informal lived experiences are really an asset that we can draw on to help make sense of what a fraction is or how to think about it.

Susan: That's a wonderful way to say it. And absolutely, the more teachers get to know the children in their classrooms and the kinds of experiences those children might have outside of school, the more of that can be incorporated into experiences like solving story problems in school.

Mike: Well, let's dig into this a little bit. Let's talk a little bit about the kinds of story problems or the structure that actually provides an entry point and can build understanding of fractions for students. Can you talk a bit about that, Susan?

Susan: Yes. So, I'll describe a couple types of story problems that we have found especially useful to elicit and develop children's fraction understandings. So first, equal sharing story problems are a powerful type of story problem that can be used at the beginning of and even throughout instruction. These problems involve sharing multiple things among multiple sharers. So, for example, four friends equally sharing 10 oranges. How much orange would each friend get? Problems like this one allow children to create fractional amounts by drawing things, partitioning those things, and then attaching fraction names and symbols. So, let's [talk] a little bit about how a child might solve the oranges problem. A child might begin by drawing four friends and then distributing whole oranges one by one until each friend has two whole oranges. Now, there are two oranges left and not enough to give each friend another whole orange. So, they have to think about how to partition the remaining oranges.

Susan: They might partition each orange in half and give one more piece to each friend, or they might partition each of the remaining oranges into fourths and give two pieces to each friend. Finally, they have to think about how to describe how much each friend gets in terms of the wholes and the pieces. They might simply draw the amount, they might shade it in, or they might attach number names to it. I also want to point out that a problem about four friends equally sharing 10 oranges can be solved by children with no formal understanding of fraction names and symbols because there are no fractions in the story problem. The fractions emerge in children's strategies and are represented by the pieces in the answer. The important thing here is that children are engaged in creating pieces and considering how the pieces are related to the wholes or other pieces. The names and symbols can be attached gradually.

Mike: So, the question that I wanted to ask is how to deal with this idea of how you name those fractional amounts, because the process that you described to me, what's powerful about it is that I can directly model the situation. I can make sense of partitioning. I think one of the things that I've always wondered about is, do you have a recommendation for how to navigate that naming process? I've got one of something, but it's not really one whole orange. So how do I name that?

Susan: That's a great question. Children often know some of the informal names for fractions, and they might understand halves or even fourths. Initially, they may call everything a half or everything a piece or just count everything as one. And so, what teachers can do is have conversations with children about the pieces they've created and how the pieces relate to the whole. A question that we've found to be very helpful is, how many of those pieces fit into the whole?

Mike: Got it.

Susan: Not a question about how many pieces are there in the whole, but how many of the one piece fit into the whole. Because it then focuses children on thinking about the relationship between the piece and the whole rather than simply counting pieces.

Mike: Let's talk about the other problem type that was kind of front and center in your thinking.

Susan: Yes. So, another type of story problem that can be used early in fraction instruction involves what we think of as special multiplication and division story problems that have a whole number of groups and a unit fraction amount in each group. So, what do I mean by that? For example, let's say there are six friends and they each will get one-third of a sub sandwich for lunch. So, there's a whole number of groups—that's the six friends—and there's a unit fraction amount in each group that's the one-third of a sandwich that they each get. And then the question is how many sandwiches will be needed for the friends? So, a problem like this one essentially engages children in reasoning about six groups of one-third. And again, as with the equal sharing problem about oranges, they can solve it by drawing out things. They might draw each one-third of a sandwich, and then they have to consider how to combine those to make whole sandwiches. An important idea that children work on with this problem then is that three groups of one-third of a sandwich can be combined to make one whole sandwich. There are other interesting types of story problems, but teachers have found these two types, in particular, effective in developing children's understandings of some of the big ideas and fractions.

Mike: I wonder if you have educators who hear you talk about the second type of problem and are a little bit surprised because they perceive it to be multiplication.

Susan: Yes, it is surprising. And the key is not that you teach all of multiplying and dividing fractions before adding and subtracting fractions, but that you use these problem types with special number combinations. So, a whole number of groups, for example, the six groups unit fractions in each group—because those are the earliest fractions children understand. And I think maybe one way to think about it is that fractions come out of multiplying and dividing, kind of in the way that whole numbers come out of adding and counting. And the key is to provide situations story problems that have number combinations in them that children are able to work with.

Mike: That totally makes sense. Can you say more about the importance of attending to the number combinations?

Susan: Yes. Well, I think that the number combinations that you might choose would be the ones that are able to connect with the fraction understandings that children already have. So, for example, if you're working with kindergartners, they might have a sense of what one half is. So, you might choose equal sharing problems that are about sharing things among two children. So, for example, three cookies among two children. You could even, once children are able to name the halves, they create in a problem like that, you can even pose problems that are about five children who each get half of a sandwich, how many sandwiches is that? But those are all numbers that are chosen to allow children to use what they understand about fractions. And then as their understandings grow and their repertoire of fractions also grows, you can increase the difficulty of the numbers. So, at the other end, let's think about fifth grade and posing equal sharing problems. If we take that problem about four friends sharing 10 oranges, we could change the number just a little bit to make it a lot harder to, four friends sharing 10 and a half oranges, and then fifth-graders would be solving a problem that's about finding a fraction of a fraction, sharing the half orange among the four children.

Mike: Let me take what you've shared and ask a follow-up question that came to me as you were talking. It strikes me that the design, the number choices that we use in problems matter, but so does the space that the teacher provides for students to develop strategies and also the way that the teacher engages with students around their strategy. Could you talk a little bit about that, Susan?

Susan: Yes. We think it's important for children to have space to solve problems, fraction story problems, in ways that make sense to them and also space to share their thinking. So, just as teachers might do with whole number problem-solving in terms of teacher questioning in these spaces, the important thing is for the teacher to be aware of and to appreciate the details of children's thinking. The idea is not to fix children's thinking with questioning, but to understand it or explore it. So, one space that we have found to be rich for this kind of questioning is circulating. So, that's the time when as children solve problems, the teacher circulates and has conversations with individual children about their strategies. So, follow-up questions that focus on the details of children's strategies help children to both articulate their strategies and to reflect on them and help teachers to understand what children's strategies are. We've also found that obvious questions are sometimes underappreciated. So, for example, questions about what this child understands about what's happening in a story problem, what the child has done so far in a partial strategy, even questions about marks on a child's paper; shapes or tallies that you as a teacher may not be quite sure about, asking what they mean to the child. “What are those? Why did you make those? How did they connect with the problem?” So, in some it benefits children to have the time to articulate the details of what they've done, and it benefits the teacher because they learn about children's understandings.

Mike: You're making me think about something that I don't know that I had words for before, which is I wonder if, as a field, we have made some progress about giving kids the space that you're talking about with whole number operations, especially with addition and subtraction. And you're also making me wonder if we still have a ways to go about not trying to simply funnel kids to, even if it's not algorithms, answer-getting strategies with rational numbers. I'm wondering if that strikes a chord for you or if that feels off base.

Susan: It feels totally on base to me. I think that it is as beneficial, perhaps even more beneficial for children to engage in solving story problems and teachers to have these conversations with them about their strategies. I actually think that fractions provide certain challenges that whole numbers may not, and the kinds of questioning that I'm talking about really depend on the details of what children have done. And so, teachers need to be comfortable with and familiar with children's strategies and how they think about fractions as they solve these problems. And then that understanding, that familiarity, lays the groundwork for teachers to have these conversations. The questions that I'm talking about can't really be planned in advance. Teachers need to be responsive to what the child is doing and saying in the moment. And so that also just adds to the challenge.

Mike: I'm wondering if you think that there are ways that educators can draw on the work that students have done composing and decomposing whole numbers to support their understanding of fractions?

Susan: Yes. We see lots of parallels just as children's understandings of whole numbers develop. They're able to use these understandings to solve multi-digit operations problems by composing and decomposing numbers. So, for example, to take an easy addition, to add 37 plus eight, a child might say, “I don't know what that is, but I do know how to get from 37 to 40 with three.” So, they take three from the eight, add it to the 37 get to 40, and then once at 40 they might say, “I know that 40 plus five more is 45.” So, in other words, they decompose the eight in a way that helps them use what they understand about decade numbers. Operations with fractions work similarly, but children often do not think about the similarities because they don't understand fractions or numbers to, versus two numbers one on top of the other.

Susan: If children understand that fractions can be composed and decomposed just as whole numbers can be composed and decomposed, then they can use these understandings to add, subtract, multiply, and divide fractions. For example, to add one and four-fifths plus three-fifths, a child might say, “I know how to get up to two from one in four-fifths. I need one more fifth, and then I have two more fifths still to add from the three-fifths. So, it's two and two-fifths.” So, in other words, just as they decompose the eight into three and five to add eight to 37, they decompose the three-fifths into one-fifth and two-fifths to add it to one and four-fifths.

Mike: I could imagine a problem like one and a half plus five-eighths. I could say, “Well, I know I need to get a half up. Five-eighths is really four-eighths and one-eighths, and four-eighths is a half.”

Susan: Yep.

Mike: “So, I'm actually going from one and a half plus four-eighths. OK. That gets me to two, and then I've got one more eighth left. So, it's two and an eighth.”

Susan: Nice. Yeah, that's exactly the kind of reasoning this approach can encourage.

Mike: Well, I have a final question for you, Susan. “Extending Children's Mathematics” came out in 2011, and I'm wondering what you've learned since the book came out. So, are there ideas that you feel like have really been affirmed or refined, and what are some of the questions about the ways that students make meaning of fractions that you're exploring right now?

Susan: Well, I think, for one, I have a continued appreciation for the power of equal sharing problems. You can use them to elicit children's informal understandings of fractions early in instruction. You can use them to address a range of fraction understandings, and they can be adapted for a variety of fraction content. So, for example, building meaning for fractions, operating with fractions, concepts of equivalence. Vicki and I are currently writing up results from a big research project focused on teachers' responsiveness to children's fraction thinking during instruction. And right now, we're in the process of analyzing data on third-, fourth-, and fifth-grade children’s strategies for equal sharing problems. We specifically focused on over 1,500 drawing-based strategies used by children in a written assessment at the end of the school year. We've been surprised both by the variety of details in these strategies—so, for example, how children represent items, how they decide to distribute pieces to people—and also by the percentages of children using these drawing-based strategies. For each of grades three, four, and five, over 50 percent of children use the drawing-based strategy. There are also, of course, other kinds of strategies that don't depend on drawings that children use, but by far the majority of children were using these strategies.

Mike: That's interesting because I think it implies that we perhaps need to recognize that children actually benefit from time using those strategies as a starting point for making sense of the problems that they're solving.

Susan: I think it speaks to the length of time and the number of experiences that children need to really build meaning for fractions that they can then use in more symbolic work. I'll mention two other things that we've learned for which we actually have articles in the NCTM publication MTLT, which is “Mathematics Teacher: Learning and Teaching in PK–I2.” So first, we've renewed appreciation for the importance of unit fractions and story problems to elicit and develop big ideas. Another idea is that unit fractions are building blocks of other fractions. So, for example, if children solve the oranges problem by partitioning both of the extra oranges into fourths, then they have to combine the pieces in their answer. One-fourth from each of two oranges makes two-fourths of an orange. Another idea is that one whole can be seen as the same amount as a grouping of same-sized unit fractions. So, those unit fractions can all come from the same hole or different wholes, for example, to solve the problem about six friends who will each get one-third of a sub sandwich. A child has to group the one-third sandwiches to make whole sandwiches. Understanding that the same sandwich can be seen in these two ways, both as three one-third sandwiches or as one whole sandwich, provides a foundation for flexibility and reasoning. For those in the audience who are familiar with CGI, this idea is just like the IDM base ten, that 1 ten is the same amount as ten 1s, or what we describe in shorthand as 10 as a unit. And we also have an article in MTLT. It's about the use of follow-up equations to capture and focus on fraction ideas in children's thinking for their story problems. So basically, teachers listen carefully as children solve problems and explain their thinking to identify ideas that can be represented with the equations.

Susan: So, for example, a child solving the sub-sandwiches problem might draw a sandwich partitioned into thirds and say they know that one sandwich can serve three friends because there are three one-thirds in the sandwich. That idea for the child might be drawn, it might be verbally stated. A follow-up equation to capture this idea might be something like one equals one-third plus one-third plus blank, with the question for the child, “Could you finish this equation or make it a true equation?” So, follow-up equation[s] often make ideas about unit fractions explicit and put them into symbolic form for children. And then at the same time, the fractions in the equations are meaningful to children because they are linked to their own meaning-making for a story problem. And so, while follow-up equations are not exactly a question, they are something that teachers can engage children with in the moment as a way to kind of put some symbols onto what they are saying, help children to reflect on what they're saying or what they've drawn, in ways that point towards the use of symbols.

Mike: That really makes sense.

Susan: So, they could be encouraged to shade in the piece and count the total number of pieces into which an orange is cut. However, we have found that a better question is, how many of this size piece fit into the whole? Because it focuses children on the relationship between the piece and the whole, and not on only counting pieces.

Mike: Oh, that was wonderful. Thank you so much for joining us, Susan. It's really been a pleasure talking with you.

Susan: Thank you. It's been my pleasure. I've really enjoyed this conversation.

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

© 2023 The Math Learning Center | www.mathlearningcenter.org