Solving the Problem with Word Problems, Part I
No matter where I am in the country, when I talk to teachers, they all tell me that word problems are a problem for a whole variety of reasons. That got me to thinking, why is that every time we talk about word problems it’s with a wince and a frown? And how can we turn that around, because in the same breath that teachers share their challenges with word problems, they also tell me that word problems are an important instructional tool. So, over four posts (and an accompanying edWeb.net webinar series) we’re going to explore the world of word problems.
The Purpose of Word Problems
As counterintuitive as it may sound, I don’t think that word problems are an ideal way to teach problem solving as they don’t look like the problems we encounter in real life. In my opinion, word problems are examples of mathematical applications and connections. They provide students with a chance to learn the work of the underlying operation, in the case of elementary math—addition, subtraction, multiplication, and division. And in strengthening student knowledge of the underlying mathematical operations, we provide them with the building blocks needed to solve real-world problems.
Three Skills Word Problems Develop
Although word problems look different across the grade levels, there are three skill sets we want to develop—language comprehension, mathematical comprehension, and solution paths.
I talk about language rather than strictly reading because word problems involve oral as well as written language. Think about how kids learn a language, they start talking, they learn vocabulary, they speak in progressively more complex sentences, and only then do they start reading.
Understanding the language of math, just like learning any language, is a progression. We start with language students know and only then do we connect that language to a particular mathematical representation. For instance, initially we talk about a bird flying away, something has left, or eating a cookie, something disappears, or sharing a pizza, something is taken away, and only then do we move to mathematical comprehension, the concept of minus and the operation of subtraction.
This is what I call the missing link, that connection between everyday terms and more formal mathematical language. I’m not talking about keywords, because those only work in limited, prescribed situations, but rather what we might call the verbs of mathematics, what action am I taking (adding, subtracting, multiplying, or dividing) and what language in the word problem connects to a particular operation. This connection, a step we all to often skip, helps students figure out what operation to use, and that is the most critical function of word problems.
Only after we make the connection between language and mathematical comprehension, can we move on to finding a solution to the word problem. Finding the solution path is different than finding the answer. This is where we encourage students to explore the tools and strategies they might use to help them solve a particular problem.
Two Strategies That Support Word Problems
In this post, I’m going to talk about two general strategies that support all word problems regardless of operation. In future posts, we’ll discuss particular strategies for each of the four operations—addition, subtraction, multiplication, and division.
Let’s look at the following problem about a shepherd, 125 sheep, and 5 dogs.
The shepherd has 125 sheep and 5 sheep dogs. How old is the shepherd?
The first thing, you’ll notice about this word problem is that it can’t be solved with the information we have. When researcher Robert Kaplinsky gave this task to a class of eighth-graders, four students calculated the answer by adding or subtracting, sixteen students used division or multiplication, and four students guessed. Most of the students didn’t actually read the problem, they went directly to the numbers, a phenomenon John SanGiovanni calls number plucking. So, how can we help our students be thoughtful when approaching word problems and not simply go directly to using the numbers to find an answer?
Encourage kids to slow down and really understand the problem—read it three times. For the first two reads, we remove the question. On the first read, we read for story and context. And I’d encourage you to read it aloud and read it more than once, as this helps kids make sense of the story. Now you may live in an area where your students have never seen sheep and don’t have any idea what a shepherd does or why dogs would be involved. You will have to explain the context as that’s going to help students understand what’s happening in this very brief story.
On the second read, we concentrate on quantities. When you think about the quantities, you also need to think about units, as that helps with meaning. It’s not just 125, it’s 125 sheep. It’s not just 5, it’s 5 dogs. What does that mean? There are a lot of sheep and not very many dogs. Are the dogs keeping the sheep in one place? Are they moving them from point A to point B? Are the dogs guarding the sheep from something?
On the third read, we add the question. And now, students have thought about the problem and learned about the situation. They know there are 125 sheep and 5 dogs, but they don’t really know anything about the shepherd. They are ready to think if the quantities they have can help them answer the question about the shepherd’s age.
Alternately on the third read, you can still eliminate the question and ask students what kinds of questions they could answer with the information they have. Then look at the question the word problem posed and have students decide if they want to answer that question (can they answer that question) or one of the questions they developed. Whether they answer the original question or a question they came up with, students are really thinking about the math behind the story and not simply plucking numbers.
The Math Sandbox
The sandbox, or sand at the beach, is a magical place where kids can play and build things. And if they don’t like what they’ve created, they can simply smash it and start over. Sand is open-ended, it’s endlessly malleable, and it encourages kids to use lots of fun tools. And solving word problems needs to feel as exploratory as the sandbox.
Once students understand the language behind a word problem, they are ready to head into the math sandbox. Instead of using buckets and shovels, they may be using fraction pieces, base-10 blocks, counters, or drawings. This is the mathematical comprehension step. How do they represent 125 sheep and 5 dogs? The tools they use will depend on the specific question being asked. The important thing is that when students are in the sandbox they have lots of different mathematical representations that allow them to make sense of what the word problem is asking. They may start with counters and realize that’s not practical and move to a number line or writing an equation or number sentence.
Both of these strategies encourage students to take their time and really understand what is happening in the word problem. It provides them with a process and a protocol to solve word problems. And by the time they get to those end-of-year high stakes tests, students have assimilated these strategies so they can rapidly assess a word problem, break it down into steps, and arrive at a solution.
Join us for the following one-hour webinars on edWeb.net:
Making Sense of Multiplication and Division Word Problems: Solving the Problem of Problem Solving on Tuesday, November 1 @ 4:00–5:00pm EDT
Word Problems Beyond Whole Numbers: Solving the Problem of Problem Solving on Tuesday, December 6 @ 4:00–5:00pm EST