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Give Cindy Cliche a math word problem, and she can tell you exactly where most students are going to trip up.
Cliche, the district math coordinator in the Murfreesboro City school district in Tennessee, has spent decades teaching elementary schoolers how to tackle their first word problems and now coaches teachers in how to do the same. Kids鈥� struggles, for the most part, haven鈥檛 changed, she said.
Take this problem, which students might work on in 1st grade: There are some bunnies on the grass. Three bunnies hop over, and then there are five total. How many bunnies were there to begin with?
The problem is asking about a change: What鈥檚 the starting, unknown quantity of bunnies, if adding 3 to that quantity equals 5? In other words, x + 3 = 5. But most 1st graders don鈥檛 make that connection right away, Cliche said. Instead, they see the numbers 3 and 5, and they add them.
鈥淣ine times out of 10 they鈥檙e going to say, 鈥榚ight,鈥欌�� Cliche said. 鈥淭hey鈥檙e number pluckers. They take this number and this number and they add them together or they take them apart.鈥�
This is one of the biggest challenges in word problem-solving, educators and researchers agree鈥攇etting students to understand that the written story on the page represents a math story, and that the math story can be translated into an equation.
Making this connection is a key part of early mathematical sense-making. It helps students begin to understand that math isn鈥檛 just about numbers on a page, but a way of representing relationships in the world. And it鈥檚 one of the ways that kids learn to unite conceptual understanding of problems with the procedures they will need to solve them.
鈥淲hen students struggle [with word problems], it tends to be everything else they have to do to get to the calculation,鈥� said Brian Bushart, a 4th grade teacher in the West Irondequoit schools in Rochester, N.Y.
There are evidence-backed strategies that teachers can use to help students make these connections, researchers say.
These approaches teach students how to understand 鈥渕ath language,鈥� how to devise a plan of attack for a problem, and how to recognize different problem types. And though they provide students tools and explicit strategies, these techniques are designed to support kids鈥� sense-making, not circumvent it, said Lynn Fuchs, a research professor in the department of special education at Vanderbilt University.
The goal, she said, is 鈥渦nderstanding the full narrative of what鈥檚 being presented.鈥�
How word problems are used in early grades
Story problems serve a few different purposes in early grades, said Nicole McNeil, a professor of psychology at the University of Notre Dame who studies students鈥� cognitive development in math.
They can help connect children鈥檚 preexisting knowledge to the math they鈥檙e learning in class鈥�"activating that knowledge kids have in their everyday life, and then showing, how do mathematicians represent that?鈥� McNeil said.
Cliche likes to use word problems in this way to introduce the concept of dividing by fractions.
鈥淲e鈥檒l tell the kids, 鈥業 have three sandwiches here and I need to divide them in half so that everyone will get a piece,鈥欌�� she said. 鈥溾�楬ow many people can I feed?鈥欌��
After students solve the problem, Cliche introduces the operation that students could use to divide by fractions鈥攎arrying this conceptual understanding with the procedure that students would use going forward.
But word problems can also be used in the opposite direction, to see if students can apply their understanding of equations they鈥檝e learned to real-world situations, McNeil said.
And there鈥檚 another, practical reason that teachers practice word problems: They鈥檙e ubiquitous in curriculum and they鈥檙e frequently tested.
There are lots of different kinds of problems that kids could work on in math classes, said Tamisha Thompson, a STEAM (for science, technology, engineering, the arts, and math) instructional coach in the Millbury public schools in Massachusetts, and a doctoral student in learning sciences at Worcester Polytechnic Institute.
Many story problems have one right answer, but there are also problems that could have multiple answers鈥攐r ones that aren鈥檛 solvable. Spending more time with a broader diversity of problems could encourage more creative mathematical thinking, Thompson said. 鈥淏ut we鈥檙e really driven by standardized tests,鈥� she said. 鈥淎nd standardized tests typically have one right answer.鈥�
In general, between 30 percent and 50 percent of standardized-test items in math feature these kinds of story problems, said Sarah Powell, an associate professor in the department of special education at the University of Texas at Austin.
鈥淯ntil things change, and until we write better and different tests, if you want students to show their math knowledge, they have to show that through word problem-solving,鈥� Powell said.
Why students struggle with word problems
Sometimes, students struggle with word problems because they don鈥檛 know where to start.
Just reading the problem can be the first hurdle. If early-elementary schoolers don鈥檛 have the reading skills to decode the words, or if they don鈥檛 know some of the vocabulary, they鈥檒l struggle, said McNeil.
That can result in students scoring low on these portions of standardized tests, even if they understand the underlying math concepts鈥攕omething McNeil considers to be a design flaw. 鈥淵ou鈥檙e trying to assess math, not reading twice,鈥� she said.
Then, there鈥檚 math-specific vocabulary. What do words like 鈥渇ewer than,鈥� or 鈥渢he rest,鈥� mean in math language, and how do they prompt different actions depending on their placement in a problem?
Even if students can read the problem, they may struggle to figure out what it鈥檚 asking them to do, said Powell. They need to identify relevant information and ignore irrelevant information鈥攊ncluding data that may be presented in charts or graphs. Then, they have to choose an operation to use to solve the problem.
Only once students have gone through all these steps do they actually perform a calculation.
Teaching kids how to work through all these setup steps takes time. But it鈥檚 time that a lot of schools don鈥檛 take, said Cliche, who has also worked previously as a state math trainer for Tennessee. Word problems aren鈥檛 often the focus of instruction鈥攔ather, they鈥檙e seen as a final exercise in transfer after a lot of practice with algorithms, she said.
A second problem: Many schools teach shortcut strategies for deciphering word problems that aren鈥檛 effective, Powell said.
Word problem 鈥渒ey words鈥� charts . These graphic organizers are designed to remind students which math words signal different operations. When you see the word 鈥渕ore,鈥� for example, that means add the numbers in the problem.
Talking with students about the meaning of math vocabulary is useful, said Powell. But using specific words as cues to add or subtract is a flawed strategy, Powell said, because 鈥渢here is no single word that means an operation.鈥� The word 鈥渕ore鈥� might mean that the numbers need to be added together鈥攐r it might mean something else in context. Some problems have no key words at all.
In a , Powell and her colleagues analyzed more than 200 word problems from Partnership for Assessment of Readiness for College and Careers (PARCC) and Smarter Balanced math tests in elementary and middle school grades. Those tests are given by states for federal accountability purposes.
They found that using the key words strategy would lead students to choose the right operation to solve the problem less than half the time for single-step problems and less than 10 percent of the time for multistep problems.
Evidence-based strategies for helping struggling students
So if key words aren鈥檛 an effective strategy to support students who struggle, what is?
One evidence-based approach is called . This approach categorizes problems into different types, depending on the math event portrayed, said Fuchs, who has studied schema-based instruction for more than two decades.
But unlike key words, schemas don鈥檛 tell students what operations to use. Instead, they help students form a mental model of a math event. They still need to read the problem, understand how that story maps onto their mental model, and figure out what information is missing, Fuchs said.
One type of schema, for example, is a 鈥渢otal鈥� or 鈥渃ombine鈥� problem, in which two quantities together make a total: 鈥淛ose has five apples. Carlos has two apples. How many apples do they have together?鈥� In this case, students would need to add to get the answer.
But this is also a total problem: 鈥淭ogether, Jose and Carlos have seven apples. If Jose has five apples, how many apples does Carlos have?鈥�
Here, adding the two numbers in the problem would bring students to the wrong answer. They need to understand that seven is the total, five is one part of the total, and there is another, unknown part鈥攁nd then solve from there.
To introduce schemas, Vanderbilt鈥檚 Fuchs said, 鈥渨e start with a child and the teacher representing the mathematical event in a concrete way.鈥�
Take a 鈥渄ifference鈥� problem, which compares a larger quantity and a smaller quantity for a difference. To demonstrate this, an early-elementary teacher might show the difference in height between two students or the difference in length of two posters in the room.
Eventually, the teacher would introduce other ways of representing this 鈥渄ifference鈥� event, like drawing one smaller and one larger rectangle on a piece of paper. Then, Fuchs said, the teacher would explain the 鈥渄ifference鈥� event with a number sentence鈥攖he formula for calculating difference鈥攖o connect the conceptual understanding with the procedure. Students would then learn a solution strategy for the schema.
Children can then use their understanding of these different problem types to solve new problems, Fuchs said.
There are other strategies for word-problem-solving, too.
- Attack strategies. Several studies have found that giving students a consistent they can use to approach every problem has positive effects. These attack strategies are different from schemas because they can be used with any problem type, offering more general guidance like reminders to read the problem and pull out relevant information.
- Embedded vocabulary. A from Fuchs and her colleagues found that math-specific vocabulary instruction helped students get better at word problem-solving. These vocabulary lessons were embedded into schema instruction, and they focused on words that had a specific meaning in a math context鈥攖eaching kids the difference between 鈥渕ore than鈥� and 鈥渢hen there were more,鈥� for example.
- 鈥楴umberless鈥� problems. Some educators have also developed their own strategies. One of these is what鈥檚 called 鈥渘umberless鈥� word problems. A numberless problem has the same structure as a regular story problem but with the quantities strategically removed. An initial statement might say, for example, 鈥淜evin found some bird feathers in the park. On his way home, he lost some of the feathers.鈥�
With numberless problems, instead of jumping to the calculation, 鈥渢he conversation is the goal,鈥� said Bushart, the 4th grade teacher from New York, who has created a .
The teacher talks with students about the change the story shows and what numbers might be reasonable鈥攁nd not reasonable. The process is a form of scaffolding, Bushart said: a way to get students thinking conceptually about problems from the start.
Balancing structure and challenge
These approaches all rely on explicit teaching to give students tools that can help them succeed with problems they鈥檙e likely to see often in class or on tests.
But many math educators also use word problems that move beyond these common structures, in an attempt to engage students in creative problem-solving. Figuring out how much structure to provide鈥攁nd how much challenge鈥攃an be a delicate balance.
These kinds of problems often require that students integrate real-life knowledge, and challenge them to 鈥渢hink beyond straightforward applications of mathematical situations,鈥� said McNeil of Notre Dame.
There may be an extra number in the problem that kids don鈥檛 have to use. Or the problem might pose a question that would lead students to a nonsensical answer if they just used their procedural knowledge. For example: 65 students are going on a field trip. If each bus can hold 10 students, how many buses are needed?
Students might do the calculation and answer this question with 6.5, but that number doesn鈥檛 make sense, said McNeil鈥攜ou can鈥檛 have half a bus.
In a , McNeil and her colleague Patrick Kirkland rewrote some of these challenging questions in a way that encouraged students to think more deeply about the problems. They found that middle school students who worked on these experimental problems were more likely than their peers to engage in deep mathematical thinking. But, they were also less likely to get the problems correct than their peers who did standard word problems.
Other research, with young children, has found that teaching students how to transfer their knowledge can help them work through novel problems.
When students are given only problems that are all structured the same way, even minor changes to that format can prevent them from recognizing problem schemas, said Fuchs.
鈥淲hat we found in our line of work is that if you change the way the word problem reads, in only very minor ways, they no longer recognize that, this is a 鈥榗hange鈥� problem, or a 鈥榙ifference鈥� problem,鈥� she said, referencing different problem schemas.
In the early 2000s, she and her colleagues to help students transfer their knowledge to more complex, at times open-ended problems. They found that when children were taught about the notion of transfer, shown examples of different forms of the same problem type, and encouraged to find examples in their own lives, they performed better on novel, multistep problems than their peers who had only received schema instruction.
The results are an example of how explicit instruction can lay the groundwork for students to be successful with more open-ended problem-solving, Fuchs said.
Exactly how to sequence this learning鈥攚hen to lean into structure and when to release students into challenge鈥攊s an open question, McNeil said.
鈥淲e need more researchers focused on what are the best structures? What order should things go in? What is the appropriate scope and sequence for word problems?鈥� she said. 鈥淲e don鈥檛 have that information yet.鈥�
