“They know the math. They just can’t do the word problems. It’s their reading comprehension.”
It was my first coaching session with “Ms. Hartwell,” a 5th grade math teacher. She was explaining why her students had been struggling on standardized tests, which were heavy on word problems.
I had been matched with Ms. Hartwell and a few of her colleagues at this East Harlem public school, as part of a New York City program to boost math achievement in underperforming schools.
And while many of the teachers in the program echoed the ‘literacy’ explanation for low math achievement, it didn’t quite add up. For one, the language used in the word problems was pretty basic. Here’s an example of a released question from the 5th grade state test:
Compare that to one from the grade 5 language arts test:
It’s true that increases in literacy are correlated with improved math performance. But the reasons are complicated. For one, students from wealthy families tend to be stronger in both math and reading, because of their additional resources.
In addition, strong readers can develop their math skills by reading their textbook. They’re not solely reliant on explanations given orally by their teachers.
I know of no studies that say students who know the math fail state tests because they can’t read the questions. Standardized tests are specifically designed to prevent such issues. (Not that the tests are perfect, but judge for yourself based on the samples above).
I knew there had to be another explanation. Fortunately, I was about to watch Ms. Hartwell teach a lesson involving word problems. Hopefully, a first-hand look would uncover the issue.
An Eye-Opening Word Problems Lesson
Ms. Hartwell stood at the front of the room. “Open to page 47,” she announced.
Her students sat at round tables with their textbooks and hand-held whiteboards. As instructed, they opened to page 47: Division Word Problems with Decimal Quotients.
Ms. Hartwell read aloud, “Sarah has 40 lbs. of carrots for her horses. If she has 100 horses, how many pounds does each horse get?”
“OK class, she has 40 pounds for 100 horses. How much does each get? What’s our operation?”
She pointed to an anchor chart of ‘key words.’ ‘Each’ was at the top of the division list.
“Division!” the class responded.
“Right. So how do we divide a small number by a big number?”
Silence. Ms. Hartwell drew a vinculum (long-division house) on her whiteboard. The students copied as she put 40 under the house and 100 outside.
“Now 100 doesn’t fit into 40, so we need to add a decimal.” She changed 40 to 40.0. “How many times does 100 go into 400?”
A few students called out “Four!”
“Exactly. Now, we need to put a decimal in our quotient as well. Our final answer is?”
The class read the answer, “0.4,” off of her whiteboard.
The students then worked in pairs on a similar problem about 20 students running a 5-mile relay race.
Ms. Hartwell pointed to the CUBES poster on the wall. “Remember to circle the numbers and box the key words.” She circulated, showing students where to put the decimal or how to line up the subtraction for the the long division. In most pairs, one student calculated, while the other copied.
Once everyone had the correct answer, Ms. Hartwell turned to me. “See, they know the math.”
Classroom Resources for Teaching Word Problems
Adding and Subtracting Integers Activities | Digital Word Problems$5.00 Add to cart
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Dependent and Independent Variables Word Problem Activities | Digital and Print Options$5.00 Add to cart
Lesson Plans and Classroom Activities
Fractions as Division Word Problems – Complete Digital and Print Lesson$5.00 Add to cart
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Math Word Problem Graphic Organizer | Digital and Print$2.00 Add to cart
3 Common Mistakes When Teaching Word Problems
The mistakes in this lesson may seem extreme, but I know I’ve made most of them myself. And most are relatively common in math classrooms.
We all want our students to succeed. So once we set a learning objective, we do everything we can to ensure all of our students meet it.
But we can only provide so much help before we are no longer teaching what we meant to teach.
Once we cross a certain line, instead of learning to problem-solve, our students are just learning to mimic us. And when they just follow along without real understanding, they can only do the work when we’re standing next to them.
So if your students’ performance on state tests doesn’t match what they seem able to do in class, look out for these 3 common teaching mistakes.
1. Teaching Key Words Instead of Strategies
Have you taught your students to use key words to solve word problems? The idea behind this ‘trick,’ is that instead of making sense of what’s happening in a word problem, students can simply memorize a list of key words that will tell them which operation to use.
Not only does this defeat the purpose of word problems, key words simply don’t work.
The reason we teach students word problems is to help students think about math in real-world contexts. If they just skip the sense-making by memorizing key words, we may as well skip the word problems altogether.
Some cynics might say that none of that matters…as long as students pass the test. But consider the following:
Mr. Smith is teaching his class the difference between pints and quarts. He asks 10 students to each bring in 3 pints of lemonade. He asks another 10 to bring in 2 quarts of iced tea. How much iced tea will the students bring in altogether?
Students who rely on key word strategies will come up with everything from -5 to 600.
If you make sense of this problem, it’s pretty straightforward (10 students x 2 quarts per student = 20 quarts). But it contains key words for subtraction, division, addition, and multiplication. (Some key word resources also list ‘each’ as a multiplication word). Not to mention extra numbers that won’t be calculated.
The test makers are hip to the whole key word thing. So while key words may have worked 20 years ago, today’s tests are specifically written to outsmart that approach.
2. Pre-Formulating Word Problems
For students to be effective in solving word problems, they need to master the art of formulation. Formulation is the process of ‘translating’ a problem from word form to a mathematical form, such as an equation or a visual model.
Once we’ve translated the problem, we can use the math strategies we’ve learned to solve it.
We could make things easier by giving our students a set of 10 word problems with the title “Division Word Problems.” They won’t need to understand the meaning of division. They won’t even need to read the problems. They just need to figure out (or guess) which number is the divisor and which is the dividend.
But when students encounter math in the real world (or on a state test), the problems won’t be pre-formulated like this. They will need to figure out which numbers and operations to use, what order to complete the steps, and so on.
So just like key words, teaching a lesson where every problem involves the same operation and types of number (two-digit, rational, etc) takes the challenge and thus the learning, out of the word problems.
3. Over-Scaffolding (Fear of Productive Struggle)
Over-scaffolding occurs when we provide students too much support. While it is important to adjust instruction to meet each students’ needs, over-scaffolding is different.
The term ‘scaffolding’ comes from the work of Lev Vygotsky. He coined the term “Zone of Proximal Development.” The ZPD includes things that are just out of a learner’s reach. They can’t quite do it on their own, but they can do it with support. By ‘scaffolding,’ we can move these skills into a learner’s core skill set.
What is often overlooked is that we can only scaffold skills in a learner’s ZPD. There are a whole range of skills that are currently out of a learner’s reach. Attempting to teach students those skills leads to over-scaffolding.
One problem with over-scaffolding is that it prevents students from internalizing the targeted skills. When we help a student with something outside of their ZPD, they will always need our help. At least until we identify the intermediate skills that are in their ZPD.
Another issue is that over-scaffolding gives us a false sense of success. In this case, Ms. Hartwell believed her students could ‘do the math’ in the word problem. In reality, they were just imitating her actions. There was no reason to believe they could find a similar solution on their own.
A Better Way to Teach Word Problems
Meaningful learning requires productive struggle. That means giving students an appropriately challenging problem, and then stepping back.
Some traditional educators have critiqued inquiry-based learning as “not teaching anything,” and just leaving students to spin their wheels when they don’t understand something.
While some have certainly tried such a laissez faire approach, that is not how well designed inquiry-based classrooms function. Effective IBL requires that we teach students how to learn through inquiry. First, we provide students with the tools and strategies that support collaborative problem-solving. Then, we facilitate the learning process while they learn through discovery. And finally, we help them cement their learning through sharing and reflection.
For more on planning and teaching Inquiry-Based Lessons, read Lesson Plans that Promote Student Engagement.
Word Problems: Process vs Strategies
When it comes to helping students struggle productively with word problems, it’s important to provide them with two tools they will need to be successful: a problem-solving process, and a set of strategies.
A process is a series of steps that can be repeated again and again to achieve consistent results. Algorithms are a process. As is lining up ‘tallest to shortest’ for the walk to lunch.
CUBES is a process for solving word problems. It’s just not a very good one, as it relies on key word, in an effort to avoid the strategic learning needed for formulation.
The Polya Process, which we’ll use for problem-solving, still provides students’ with a list of steps to follow. But it also leaves room for strategic thinking within that process.
Consider the following:
You have 5 gallons of juice for a school event with 100 students. If each cup holds 3 oz, how many cups can each student drink? How much will be left over.
A problem-solving process can help students begin to tackle this problem. “Identify what is being asked.” “Consider what information is being given.”
But there is no set process we can teach them to actually solve it. That’s where strategy comes in. They need to convert units. Recognize that we are dividing with remainders, rather than dividing completely. They also should probably reorder the operations, dividing the juice into cups before dividing by the number of students.
There’s no “trick” that will get students there. They need to understand the meaning of operations. They need to understand math as a subject of sense-making. And they need repeated practice applying strategic thinking to word problems.
Why the Polya Process is Better than CUBES
George Pólya was an influential Hungarian mathematician and Stanford professor who found a process that could be used to solve any problem: Understand, Plan, Solve, and Reflect. His work has touched countless mathematicians and educators, most of whom will never know his name.
In fact, CUBES and other so-called ‘problem solving strategies’ are loosely based on his work. The problem is that they oversimplify it, removing anything that requires deep thinking.
Understand: To understand the problem, we identify what information is given and what we want to find. This is where the ‘C’ and ‘U’ in cubes come from. If students want to circle the numbers and underline the question, fine. I prefer they write them (with units) in a graphic organizer [link], at least at first. This helps them process what they are reading. It also helps me, as the teacher, to assess their understanding as I walk around the room.
Plan: This is the tricky part. The ‘plan’ is where we formulate the problem. Formulating word problems involves understanding them. But it goes beyond that. Formulation is more like translation than understanding. This is where our strategies come into play, so we’ll come back to the plan phase when we look at strategies.
Solve: The third step is to Solve. If we’ve formulated correctly, we simply calculate with an algorithm, equation, or visual model.
Reflect: Finally, students look back and check their work. They should also reflect on their process. If I made a mistake, why? Could I use this strategy to solve similar problems in the future?
The second step, ‘Plan,’ is where the strategic thinking occurs.
This process of formulation is where Polya’s Process differs from keyword-based approaches. And since many of us don’t consciously think about formulation when solving a problem, this stage is often overlooked, or misunderstood as a ‘reading comprehension’ problem.
Most 6th grade students could read Ms. Hartwell’s problem and easily understand that Sarah is handing out carrots to her horses. But fewer will connect the ” equal sharing” of carrots to division.
The problem goes back to procedural approaches to teaching math. If students learn how to multiply without understanding the meanings of multiplication; or how to operate on fractions without learning the foundational concepts, they won’t be able to recognize these numbers and operations when they encounter them in context.
When we use conceptual strategies, like visual models or number sentences, students learn what the operations mean. Then, a simple 4-step process, like Polya’s, is enough. They will already be exposed to strategies like ‘draw a picture’ and ‘make an equation.’
Then, you can model strategies, like ‘work backwards’ and ‘guess & check,’ to give them additional tools to draw upon. Allowing students to ‘choose their strategy,’ is important for engagement and making the learning student-centered. But they can only make educated decisions about which strategy to use once they have learned the options.
Students may also combine strategies, using one to formulate and another to solve. For example, if asked to find the total carrots in a field with 18 rows of 10 carrots each, they may formulate by drawing a picture. Once they realize it’s an array, they can multiply using an equation, algorithm, or mental math.
Bringing it Back to Your Classroom
So if your students struggle with word problems, it’s probably not about their reading comprehension. In fact, it’s probably not even about the word problems. It’s about conceptual understanding of math topics.
So before you even introduce word problems, make sure your students are learning the concepts behind the math. Then, instead of resorting to tricks like CUBES and key words, your students will be making sense of problems and formulating like pros.
If your school or district is ready for a more engaging, conceptual approach to teaching word problems, we’re here to help. We offer online and on-site workshops that allow teachers to learn through exploration and collaboration. Each session includes all the print and digital resources you’ll need to bring what you learn back to your classroom the very next day.
All math workshops are offered for Elementary School (Grades 1 through 5) or Middle School (Grade 6 through Algebra I). To learn more and schedule your session, click on a topic below or Schedule Your FREE Consultation
If you’d prefer to bring the Polya Process to your classroom tomorrow, this graphic organizer will get you started. It works with literally any word problem, scaffolding the four-step process and helping you identify where your students are going off-track.
About the Author
Jeff Lisciandrello is the founder of Room to Discover and an education consultant specializing in student-centered learning. His 3-Bridges Design for Learning helps schools explore innovative practices within traditional settings. He enjoys helping educators embrace inquiry-based and personalized approaches to instruction. You can connect with him via Twitter @EdTechJeff