CENTRAL FALLS, RI — When Natalia Molina started teaching word problems to her second-grade students earlier this year, every lesson seemed daunting. Most of the students struggled with a problem like this: “Sally went shopping. She spent $86 on groceries and $39 on clothing. How much more did Sally spend on groceries than on clothing?”
Molina, a first-grade teacher, and her students were all trained to solve word problems by focusing on key words like “and,” “more,” and “total.” Molina said this simple approach often led students astray. For example, after recognizing the word “and,” students may mistakenly think that they need to add two close numbers to get the answer.
Some readers with poor reading skills were so lost in the sea of text that they could not make out any word.
“I saw how overwhelmed they were,” said Molina, who teaches at the Segue Institute for Learning, a predominantly Hispanic charter school in this small city just north of Providence.
So, with the help of an instructor working in Rhode Island through a state grant, Molina and some of her colleagues revamped their approach to teaching word problems this winter. ability. “This was a game changer for them,” Molina said.
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Perhaps no single instructional task encompasses as many skills as word problems. There are many ways for students to get it wrong: reading, executive functioning, problem solving, counting, and vocabulary. And for that reason, students tend to get it wrong on problems that focus more narrowly on counting or shapes (e.g., “Solve 7 + _”). = 22' or 'What is 64 x 3?').
If a student excels at word problems, it's a good sign that he or she is performing well in school overall. “Word problem solving in the early grades is one of the better indicators of overall school success in K-12,” said Lynn Fuchs, a research professor at Vanderbilt University. For example, in a large national survey, algebra teachers ranked word problem solving as the most important skill among 15 skills needed to excel in the subject.
lessons from teachers
- Avoid telling students to focus primarily on the “key words” in word problems, such as “and” or “more.”
- Mix up the question types in each lesson so that students don't feel like they're just repeating the same operations (addition, subtraction) over and over again.
- Teach students the basic structure, or schema, of word problems.
But most experts and educators agree that too many schools, especially elementary schools, are doing it wrong. And in a small but growing number of classrooms, teachers like Molina are working to change that. “When it comes to word problems, there are more struggling learners than non-struggling learners,” said Nicole Bucka, who works with teachers across Rhode Island to provide strategies for struggling learners.
Too many teachers, especially elementary school students, rely on keywords to introduce math problems. Posters that link terms like sum, subtraction, and addition to operations, including addition and subtraction, are a must in elementary school classrooms across the country.
According to the researchers, keywords can be a handy crutch for students and teachers alike, but they become virtually meaningless as problems become more difficult. Keywords may help first-grade students figure out whether to add or subtract more than half the time, but this strategy has little effect on the multistep problems students encounter starting in second and third grades. “Keywords are ineffective 90% of the time in multi-step problems,” said Sarah Powell, a professor at the University of Texas at Austin who studies word problems and emphasizes the ineffectiveness of keywords. “But the average kindergarten teacher doesn't think about that. They're teaching 5-year-olds, not 9-year-olds.”
Many teachers in the youngest grades hand out worksheets that feature the same type of word problem over and over again. That’s something Molina’s colleague, Cassandra Santiago, did sometimes when she first taught a class on her own last year. “It was a mistake,” the first-grade teacher said. “It’s really important to mix it up. It forces students to think more critically about what they’re trying to solve.”
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Another flaw in word problem maps is that the overwhelming majority of the questions are divorced from the real-world problems that children will have to solve in their everyday lives outside the classroom. Or, in fact, they may be. “I’ve seen questions about two trains running on the same track,” says William Schmidt, professor emeritus at Michigan State University. “First, why are they running on the same track? Second, who cares?”
Schmidt analyzed about 8,000 word problems used in 23 textbooks in 19 countries. He found that less than 1 percent had “real-world applications” and included “higher-order mathematical applications.”
“That’s one of the reasons kids have trouble with math,” he said. “They don’t see the connection to the real world… We’re at the point now where we’re teaching students in math how to manipulate numbers.”
He said questions aimed at middle school students that have real-world relevance and involve more than manipulating numbers might include: “Shop at the new store in town and you'll get a 43% discount on any item priced at $2. Your state has a 7% sales tax. You want to buy a lot, but you only have $52 total to spend. “Describe in words how many items you can buy.”
Schmidt added that word problems are one of the few areas where there are very few countries with a high level of relevance. “There was no one shining star leading the way,” he said.
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One Tuesday afternoon, in a brightly decorated classroom, Santiago, a first-grade teacher, gave each student a set of animal-shaped objects and a piece of paper, one blue (water) and one green (grass). “We’re going to tell a number story,” she told them. “I want you to tell me a story using your animal.”
“The story “Once upon a time” began. In this story, there were three animals playing in the water and two animals playing in the grass. After allowing the duck, pig, and bear to play at each student's desk, Santiago asked the children to write a number sentence telling him how many animals there were in total.
Some students relied more on the pictorial representation (three dots on one side of the line, two dots on the other), others on the number sentence (3+2=5), but in the end, they all came up with five. Then Santiago confused the order of operations with her next question (so that students did not mistakenly assume that they had to just add): “More animals came, so now there are seven. How many more?”
One approach to elementary word problems, which is being developed in some schools, including the Segue Institute, has its origins in special education interventions for struggling math students. Teachers avoid emphasizing key words and instead ask students to first identify the conceptual type (or schema, as many practitioners and researchers call it) of the word problem they are dealing with. For example, a “whole problem” involves combining two parts to find a new quantity. A “change problem” involves increasing or decreasing the quantity of something. However, a whole problem does not necessarily involve adding.
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“The schemas that students learn in kindergarten will continue for the rest of their lives,” said Powell, a word problem researcher who regularly works with districts across the country to implement the approach.
In Olathe, Kansas (a district inspired by Powell’s work), teachers have struggled with word problems for years, said Kelly Ulmer, a math support specialist who aims to fill the learning gap left by lost instructional time during the pandemic. “We’ve all tried these traditional approaches that haven’t worked,” she said. “Sometimes we get pushback from veteran teachers about new initiatives, but one of the things that shows how desperately this is needed is that veteran teachers are the most excited and engaged. They’ve tried a lot of things that haven’t worked.”
Many elementary schools in Rhode Island initially used the strategy for students with additional needs, including special education students, but have expanded its use to make it part of the core curriculum for all, Bucka said. In some ways, this is similar to recent, well-publicized developments in reading instruction, where some special education interventions for struggling readers—most notably, the increased reliance on phonics in the early grades—have become mainstream.
There is a large research base showing that focusing on problem solving across a variety of conceptual types is an effective way to teach mathematics, but most of the research focuses specifically on students who are struggling in the subject.
Molina has found that having students identify word problems by type is a useful tool for almost all of her second-graders, and she hopes to introduce this strategy much earlier next year.
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One recent afternoon, a class on word problems began with everyone standing up in unison and chanting, “Parts plus parts equals the whole” (they put their hands together). “The total minus part is equal to the part” (They took one hand away).
A way to help students remember the different conceptual frameworks for word problems. And it works especially well for students who learn well through listening and repetition. For visual learners, different types of word problems are mapped onto individual dry erase mats.
The real work began when Molina handed out the questions, and the students, who were part of the Penguin, Flower Bloom, Red Panda, and Marshmallow teams, had to figure out which framework they were working with and then work to find the answer. Just a few months ago, Molina said, many people would have just looked at the text on the page and quit automatically.
In the case of Red Panther, the question we looked at closely was, “We had 47 shirts in the store. We sold 21. How many do we have now?”
“This is a systemic problem,” one student said.
“No, not all of them,” answered another.
“I think it’s about change,” said a third.
But none of the students seemed concerned about the lack of consensus. And Molina was no exception. Correct answers are always good, but they tend to come up more often because most students have made significant leaps. “I could see that they were thinking more and more about what the question was actually asking,” she said.
This story about word problems was produced by: Hechinger Reportis a nonprofit, independent media outlet focused on inequality and innovation in education. Sign up Hechinger Newsletter.