“They know the math. They just can’t do the word problems. It’s their reading comprehension.”
Ms. Hartwell was explaining why her students had struggled on the state tests. She knew her students, and had done ample assessment. She was sure their difficulties were a literacy issue.
“I don’t understand why they make the reading so hard. The kids already take a language arts test.”
I had been matched with Ms. Hartwell (and others in her school) as part of a New York City program to boost math achievement. The goal was to bring conceptual math to “Tier 1” schools. Tier 1 schools were those that led the city in three areas. Lowest income, lowest test scores, and highest percentage of students of color.
I had heard the ‘literacy’ explanation from other teachers in the program. And it did make sense. If the students knew the math, why else were they struggling on the test, which had a lot of word problems?
But there was a catch. Over the past three years, literacy proficiency had gone from 30%-40%, citywide. Over the same period, math scores had stayed mostly flat.
If this were a reading issue, increases in reading scores should unlock explosive growth in math. Besides, the language in the word problems just didn’t seem complicated enough to cause trouble.
The next period, I was scheduled to visit Ms. Hartwell’s class. She had planned a lesson on word problems, so I could see the issue first-hand.
Over-Scaffolding Word Problems
The students sat at round tables, and she stood at the front. She and her students all had hand-held whiteboards.
They opened their textbooks to page 47 and she read a word problem out loud. “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 that she will give to each horse. So what operation do we need?”
She pointed to an anchor chart listing the ‘key words’ for each operation. ‘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?”
The class was quiet. 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.
She assigned a similar problem, and pointed to the “CUBES” poster on the wall. “Remember to circle the numbers and box the key words.”
She circulated as students worked. Some asked where to put the decimal. Others had trouble lining up the subtraction portion of the long division. Some students worked in pairs, with one doing the calculating, while the other copied.
By the end of the lesson, everyone had the correct answer on their paper. “See?” Ms. Hartwell remarked, “they all know the math.”
What Went Wrong?
While this is an extreme example, many of the mistakes Ms. Hartwell made are actually pretty common.
Over-scaffolding occurs when we provide students too much support. While it is common for educators to take pride in providing students ‘as much support as they need,’ overscaffolding 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 scaffolding only applies to skills already in a learner’s ZPD. There are a whole range of skills that are currently out of a learner’s reach. When we attempt to scaffold these skills, we do more harm than good.
One problem with over-scaffolding is that students never internalize 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.
She also removed the entire process of ‘formulation’ from the problem-solving process. Many educators think of comprehension and formulation as the same thing, but formulation is a separate mathematical competency.
And it may just be the missing ingredient from the way you currently teach word problems.
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Formulation: The Key to Success with Word Problems
Formulating word problems involves understanding them. But it goes beyond that. Formulation is more like translation than understanding.
When a student reads a word problem, first, they have to turn letters into sounds (decoding). Then, they need to understand what the words mean (vocabulary). They need to understand how the words fit together in a sentence (syntax). Finally, comprehension comes from putting these pieces together to make meaning.
Most 6th grade students could read Ms. Hartwell’s problem and easily understand that Sarah is handing out carrots to her horses. But far fewer will connect the “sharing” of carrots to division.
This is the part that trips up a lot of educators. It seems so obvious to us. “She’s literally dividing up the carrots. How could they not see that it’s division.”
They just don’t. If they did, you wouldn’t see key word charts hanging in math classrooms. And it all goes back to when we first teach operations.
Why Students Struggle with Formulation
Many teachers in many schools teach students that math is a subject of sense-making. But in others, math is taught as a subject of remembering.
An example of ‘math as remembering’ is when students memorize math facts. But algorithms are also about remembering. Put 120 on top of 47. Follow these steps to add, these to subtract subtract, and these to multiply.
Students can repeat these steps without understanding the size of the numbers or even what the operations mean.
When we start with concepts (by using visual models or number sentences), students learn what the operations mean. When we go straight to an algorithm, students think of operations as a series of random steps.
Year after year, we have students remember and repeat. They complete the same algorithm 30 times for homework. Change the numbers, and the process remains the same. But give them a word problem, and now they need to make sense.
What the Tests Tell Us About Reasoning
We could give our students 10 problems with the title “Division Word Problems,” to make it easier. 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 sit for a state test, the goal isn’t to make it easier. The goal is to assess what they can do on their own.
Test makers don’t want to tell them what operation to use. They’ll even put key words for division and addition in the same problem. “There are 20 students in the class. If each child has three dollars, how much do they have in total?”
So when students struggle with word problems, it’s not just about the word problems. It usually means they are doing math by memory. We need to teach them that math is about sense-making.
Students Need Strategies and Process to Solve Word Problems
Part of the problem with a system like CUBES is that it calls itself a strategy, but it’s really a process.
A process is a series of steps. Once you identify an efficient process, you can repeat it again and again without thinking too much about. Algorithms are a process. As is lining up ‘tallest to shortest’ for the walk to lunch.
A strategy is different. Strategies require deep understanding. And the ability to apply understanding in new situations.
We can’t prepare students for every scenario they’ll encounter in a word problem. Some require they combine different math skills they’ve never combined before. Or they may require that students change the order of steps to come up with the correct solution.
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.
The Polya Process for Solving Word Problems
George Pólya was an influential Hungarian mathematician and Stanford professor who found a process that could be used to solve any problem. His work has touched countless mathematicians and educators, most of whom don’t know his name.
In fact, CUBES and other so-called problem solving strategies are loosely based on his work. The problem is that they try to simplify it. In the process, they remove anything that resembles deep thinking.
Polya believed that every problem could be solved in four steps: Understand, Plan, Solve, and Reflect. I think Polya’s original four-step process is just fine for use in schools.
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, 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.
Next is the tricky part. The ‘plan’ is where we formulate the problem. We’re taking a real world scenario and turning it into another mathematical representation.
Finally, students look back and check their work. They should also reflect on their process. If I made a mistake, why? Could this strategy help me to solve similar problems in the future?
Strategies for Solving Word Problems
Having a process to approach word problems is definitely helpful. But we can’t pretend that meaningful math is learned just by following steps. The whole point of word problems is to expand math beyond “math facts” and algorithms.
There are some helpful strategies to help students with formulation. At first, I explicitly teach a number of different strategies.
As I teach the strategies, I have students solve problems that go well with that strategy. Some problem types go really well with ‘Guess and Check.’ Others are more suited to ‘Draw a Picture.’
Eventually, I give them challenging word problems that can be solved in several ways. Once they have the tools in their toolbelts, they can pick which is best for the situation.
Combining strategies can also be very effective. For example, drawing a picture can often be helpful to help with formulation. If a farmer has 18 rows of carrots and 10 carrots in each row, a student may start by drawing the field. At some point, they may think “Wow, this is an array,” and realize they can multiply using mental math, an expression, or an algorithm.
Other times, students can use visual models to formulate and solve the problem.
Help Your Students Tackle Challenging Word Problems
If you’re like many teachers, you might be worried about how your students will handle all this. Processes, strategies, critical thinking…there’s a lot involved.
You may even be thinking that word problems seemed easier before this article. If so, I apologize.
The key is to not get overwhelmed trying to do everything at once. Start by solving a simple word problem using Polya’s process. Then, introduce a new strategy each week. By the end of the year, you’ll be amazed at how far they’ve come.
If you’re ready to bring some Polya to your classroom tomorrow, this graphic organizer is a great way to start. It will scaffold the four-step process and help 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