Solving the Problem with Word Problems, Part IV Beyond Whole Numbers
There are two facts that just about every elementary teacher can agree on. First, word problems are essential to how we teach math in elementary school. Second, teachers and students think that word problems are a problem. How can we address these two realities? How can we help students learn how to approach word problems so that they truly are the learning opportunities they are meant to be? In this fourth installment of our series on word problems, I’m going to talk about several strategies for approaching word problems beyond whole numbers.
What I’ve found is that when word problems include fractions, decimals, and integers, it’s almost as if students restart the learning curve. So, I want to revisit some of the big ideas we’ve talked about in our previous posts on word problems and how they apply to problems involving more complex numbers.
Operation Sense and the Math Sandbox
When we talk about making sense of mathematics, no matter the grade level or class of numbers, we use one or more of five representations— words, context, concrete objects, visual images, and symbols (learn more about the five representations). But what I see, and what often presents difficulties for students, is that we ask them to go directly from reading the word problem to looking for an answer. When working with word problems, you are most likely already covering unknown words or filling in unfamiliar background information. That’s great because students must comprehend what they read in the problem. But we need to look beyond words and context because they tell only a part of the story. We need to provide students with time to explore all five representations so they can make sense of the whole story, what’s going on mathematically.
This is where the math sandbox comes in. Students need time to play in the math sandbox. They need to use manipulatives, visual images, and symbols to fully represent, and understand, what is happening in the word problem mathematically. It’s in the sandbox that they gain an understanding of the full range of work performed by each of the operations, or what we call operation sense. [See previous posts for information about the jobs performed by addition and subtraction and multiplication and division.]
Once students know how to represent a particular operation in strategic ways, they can apply this understanding to any quantity, regardless of the class of number. They recognize that fractions and decimals and later integers and algebraic expressions are merely extensions of what they have learned about addition, subtraction, multiplication, and division using whole numbers.
I talked about the strategy of three reads in the first post on word problems, but I want to revisit it with a problem involving fractions.
Priya was baking naan. There were 4 ½ cups of flour in the container and she used 1 ¾ cups to make the bread.
First Read: This first pass is for reading comprehension and context. Depending upon where you live, your students may not be familiar with the name Priya. They may not know what naan is. Some students may have never baked and are unfamiliar with the idea of scooping flour out of a container. On this first read, your students need to understand that naan is a type of bread and that baking is a process that consumes ingredients. The name of the baker, although interesting, isn’t mathematically significant. As students start to mathematize—see the mathematics in a situation—they lean to identify which details are part of the math and which aren’t.
Second Read: The second pass is when we look at the quantities involved, in this case 4 ½ cups of flour and 1 ¾ cups of flour. Here’s where we contemplate what 4 ½ cups of flour look like. How much is that? What does it look like? Can we show it with our hands? Priya used 1 ¾ cups of flour. Is that a little bit of the flour in the container? Most of it? Will I have less flour or more?
Priya was baking naan. There were 4 ½ cups of flour in the container and she used 1 ¾ cups to make the bread. How much flour was left in the container?
Third Read: You’ll notice that it’s only with the third read that I add the question. And before I do that, I may ask students to think about what questions we could ask. Some of the questions students pose may not be able to be answered with the information in the word problem. Some may require additional research. At this point, students can focus on the question this word problem is posing, “How much flour was left in the container?” There’s an action here, something is being consumed—a subtraction type of action. This means finding the difference between the two quantities.
It’s after the third read that students need to spend time in the math sandbox making mathematical sense out of what is going on. Maybe students use measuring cups. Perhaps fraction bars or fraction tiles.
Now let’s think about three areas where students might struggle with the problem and how you might best support their learning.
- Reading Comprehension: If students don’t know what naan is or have never baked, they are struggling with context. You can show video clips of someone baking naan. You might bring in examples of naan and similarly shaped breads, such as a tortilla or English muffin. You might role play baking bread.
- Number Sense: Students may struggle to make sense of fractions. They don’t know how much flour 4 ½ cups represents. Or perhaps they can set up the equation, but they can’t do the subtraction. These students are struggling with number sense. You might have them work with manipulatives, such as measuring cups. You could remind them of the strategies they know for subtracting fractions. You may help them remember what they know about common denominators.
- Operation Sense: Students may not understand why this problem calls for subtraction. Or maybe they are intimidated by fractions. These students are struggling with operation sense. Here, you could make the numbers friendly and let them work through the problem. Or you might remove the numbers all together, so they can better understand the underlying work being performed. Once they understand that it’s a subtraction problem, you can go back to the original quantities.
As we can see there are lots of reasons why word problems can be hard. But over the course of our four-part blog series (and accompanying webinars on EdWeb.net) we have covered many strategies to help you feel confident teaching word problems and students feel confident solving them.