CENTRAL FALLS, R.I. — When Natalia Molina began teaching her second grade students word problems earlier this school year, every lesson felt difficult. Most students were stymied by problems such as: “Sally went shopping. She spent $86 on groceries and $39 on clothing. How much more did Sally spend on groceries than on clothing?”
Both Molina, a first-year teacher, and her students had been trained to tackle word problems by zeroing in on key words like “and,” “more” and “total” — a simplistic approach that Molina said too often led her students astray. After recognizing the word “and,” for instance, they might mistakenly assume that they needed to add two nearby numbers together to arrive at an answer.
Some weaker readers, lost in a sea of text, couldn’t recognize any words at all.
“I saw how overwhelmed they would get,” said Molina, who teaches at Segue Institute for Learning, a predominantly Hispanic charter school in this small city just north of Providence.
Related: Kindergarten math is often too basic. Here’s why that’s a problem
So, with the help of a trainer doing work in Rhode Island through a state grant, Molina and some of her colleagues revamped their approach to teaching word problems this winter — an effort that they said is already paying off in terms of increased student confidence and ability. “It has been a game changer for them,” Molina said.
Perhaps no single educational task encompasses as many different skills as the word problem. Between reading, executive functioning, problem solving, computation and vocabulary, there are a lot of ways for students to go wrong. And for that reason, students perform significantly worse overall on word problems compared to questions more narrowly focused on computation or shapes (for example: “Solve 7 + _ = 22” or “What is 64 x 3?”).
If a student excels at word problems, it’s a good sign that they’re generally excelling at school. “Word-problem solving in lower grades is one of the better indicators of overall school success in K-12,” said Lynn Fuchs, a research professor at Vanderbilt University. In a large national survey, for instance, algebra teachers rated word-problem solving as the most important among 15 skills required to excel in the subject.
Teacher takeaways
- Don’t instruct students to focus mainly on “key words” in word problems such as “and” or “more”
- Mix question types in any lesson so that students don’t assume they just apply the same operation (addition, subtraction) again and again
- Teach students the underlying structure — or schema — of the word problem
Yet most experts and many educators agree that too many schools are doing it wrong, particularly in the elementary grades. And in a small but growing number of classrooms, teachers like Molina are working to change that. “With word problems, there are more struggling learners than non-struggling learners” because they are taught so poorly, said Nicole Bucka, who works with teachers throughout Rhode Island to provide strategies for struggling learners.
Too many teachers, particularly in the early grades, rely on key words to introduce math problems. Posters displaying the terms — sum, minus, fewer, etc. — tied to operations including addition and subtraction are a staple in elementary school classrooms across the country.
Key words can be a convenient crutch for both students and teachers, but they become virtually meaningless as the problems become harder, according to researchers. Key words can help first graders figure out whether to add or subtract more than half of the time, but the strategy rarely works for the multi-step problems students encounter starting in second and third grade. “With multi-step problems, key words don’t work 90 percent of the time,” said Sarah Powell, a professor at the University of Texas in Austin who studies word problems and whose research has highlighted the inefficacy of key words. “But the average kindergarten teacher is not thinking about that; they are teaching 5-year-olds, not 9-year-olds.”
Many teachers in the youngest grades hand out worksheets featuring the same type of word problem repeated over and over again. That’s what Molina’s colleague, Cassandra Santiago, did sometimes last year when leading a classroom on her own for the first time. “It was a mistake,” the first grade teacher said. “It’s really important to mix them up. It makes them think more critically about the parts they have to solve.”
Another flaw with word problem instruction is that the overwhelming majority of questions are divorced from the actual problem-solving a child might have to do outside the classroom in their daily life — or ever, really. “I’ve seen questions about two trains going on the same track,” said William Schmidt, a University Distinguished Professor at Michigan State University. “First, why would they be going on the same track and, second, who cares?”
Schmidt worked on an analysis of about 8,000 word problems used in 23 textbooks in 19 countries. He found that less than one percent had “real world applications” and involved “higher order math applications.”
“That is one of the reasons why children have problems with mathematics,” he said. “They don’t see the connection to the real world … We’re at this point in math right now where we are just teaching students how to manipulate numbers.”
Related: How to boost math skills in the early grades
He said a question, aimed at middle schoolers, that does have real world connections and involves more than manipulating numbers, might be: “Shopping at the new store in town includes a 43% discount on all items which are priced the same at $2. The state you live in has a 7% sales tax. You want to buy many things but only have a total of $52 to spend. Describe in words how many things you could buy.”
Schmidt added that relevancy of word problems is one area where few, if any, countries excel. “No one was a shining star leading the way,” he said.
***
In her brightly decorated classroom one Tuesday afternoon, Santiago, the first grade teacher, gave each student a set of animal-shaped objects and a sheet of blue paper (the water) and green (the grass). “We’re going to work on a number story,” she told them. “I want you to use your animals to tell me the story.”
“Once upon a time,” the story began. In this tale, three animals played in the water, and two animals played in the grass. Santiago allowed some time for the ducks, pigs and bears to frolic in the wilds of each student’s desk before she asked the children to write a number sentence that would tell them how many animals they have altogether.
Some of the students relied more on pictorial representations (three dots on one side of a line and two dots on the other) and others on the number sentence (3+2 = 5) but all of them eventually got to five. And Santiago made sure that her next question mixed up the order of operations (so students didn’t incorrectly assume that all they ever have to do is add): “Some more animals came and now there are seven. So how many more came?”
One approach to early elementary word problems that is taking off in some schools, including Segue Institute, has its origins in a special education intervention for struggling math students. Teachers avoid emphasizing key words and ask students instead to identify first the conceptual type of word problem (or schema, as many practitioners and researchers refer to it) they are dealing with: “Total problems,” for instance, involve combining two parts to find a new amount; “change problems” involve increasing or decreasing the amount of something. Total problems do not necessarily involve adding, however.
“The schemas that students learn in kindergarten will continue with them throughout their whole career,” said Powell, the word-problem researcher, who regularly works with districts across the country to help implement the approach.
In Olathe, Kansas — a district inspired by Powell’s work — teachers had struggled for years with word problems, said Kelly Ulmer, a math support specialist whose goal is to assist in closing academic gaps that resulted from lost instruction time during the pandemic. “We’ve all tried these traditional approaches that weren’t working,” she said. “Sometimes you get pushback on new initiatives from veteran teachers and one of the things that showed us how badly this was needed is that the veteran teachers were the most excited and engaged — they have tried so many things” that haven’t worked.
In Rhode Island, many elementary schools initially used the strategy with students who required extra help, including those in special education, but expanded this use to make it part of the core instruction for all, said Bucka. In some respects, it’s similar to the recent, well publicized evolution of reading instruction in which some special education interventions for struggling readers — most notably, a greater reliance on phonics in the early grades — have gone mainstream.
There is an extensive research base showing that focusing on the different conceptual types of word problems is an effective way of teaching math, although much of the research focuses specifically on students experiencing difficulties in the subject.
Molina has found asking students to identify word problems by type to be a useful tool with nearly all of her second graders; next school year she hopes to introduce the strategy much earlier.
One recent afternoon, a lesson on word problems started with everyone standing up and chanting in unison: “Part plus part equals total” (they brought two hands together). “Total minus part equals part” (they took one hand away).
It’s a way to help students remember different conceptual frameworks for word problems. And it’s especially effective for the students who learn well through listening and repeating. For visual learners, the different types of word problems were mapped out on individual dry erase mats.
The real work began when Molina passed out questions, and the students— organized into the Penguin, Flower Bloom, Red Panda and Marshmallow teams — had to figure out which framework they were dealing with on their own and then work toward an answer. A few months ago, many of them would have automatically shut down when they saw the text on the page, Molina said.
For the Red Pandas, the question under scrutiny was: “The clothing store had 47 shirts. They sold 21, how many do they have now?”
“It’s a total problem,” one student said.
“No, it’s not total,” responded another.
“I think it’s about change,” said a third.
None of the students seemed worried about their lack of consensus, however. And neither was Molina. A correct answer is always nice but those come more often now that most of the students have made a crucial leap. “I notice them thinking more and more,” she said, “about what the question is actually asking.”
This story about word problems was produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. Sign up for the Hechinger newsletter.
In the last example of the word problem involving the shirts would the teacher have been fine with either explanation. One thought would be that there is a total number of shirts (41) with one part being sold (21) leaving the other part on hand (26). Another way of thinking about the problem would be a change happening with the beginning number of (47) shirts being reduced or changed by the number being taken away (21), leaving the rest (26).
Do they have a list of the schemas for older grades? I like the ideas but as we head into 8th grade and Algebra 1 become more difficult. Would love to hear more.
I am a retired math teacher (high school, community college) and a book author, The Book for Math Empowerment (1997). I have a question: Why are “word” problems being taught to 2nd graders? They should be taught to read/spell well and they should be taught to memorize their number facts in second grade. Also – are they being taught math by teachers who themselves are math-phobic? What am I missing in this scenario?
When I read the first problem in your article, “Sally went shopping. She spent $86 on groceries and $39 on clothing. How much more did Sally spend on groceries than on clothing?” my thought was: Is such a language structure such as ‘How much more did Sally spend on groceries than on clothing?’ seen in the 2nd grade literacy program? When I was a K-12 Mathematics Supervisor, we had a similar problem in a grade 2 textbook which ended with, “How many more apples does Emily have than Harry?” I went to our Reading Supervisor and asked if such a language structure was in the reading program. The answer was, “No.” So the question is: Are textbooks reviewed by reading specialists especially in grades K to 2 to insure that the language structure of the sentences in the problems is in line with what is occurring in a literacy program? If not, then the children will obviously have difficulty with what they are supposed to do.
I have a PhD in Math Ed and MS in Psychology. Now retired, I write math materials/books for little kids (e.g., Very Early MATH: SET 1 – Numbers tell us “How Many”). This article sparked a long-held concern I have about word problems—clarity and conciseness.
Several years ago, I reviewed word problems from a state test and found many to be unclear and/or overly wordy. As a dyslexic who excelled in and loves math, I’ve been concerned about this since grade school. A significant minority of students (maybe over 10%) are dyslexic. Dyslexics live in a world in which words are time consuming to process and challenging to comprehend. Especially for dyslexic students, unnecessary words and ambiguity in math problems place a heavy tax on time and comprehension that’s unrelated to math ability. For example, my dyslexic brain is genuinely confused by several parts of the word problem featured in the article “…includes a 43% discount on all items which are priced the same at $2…you want to buy many things…describe in words how many things you could buy” (italics added). I could interpret this question in many ways depending on how I understand the italicized words. For example, does the question describe a store in which:
1. All items are normally $2.00, but today they’re all discounted 43%, or
2. Like most real stores, this store carries many items at many different prices. However, today all $2.00 items are discounted 43% (all other items are at full price).
Depending on how I interpret the italicized words, I could buy zero items because, although I want to buy many things, the ones I want each cost more than $52.00. Welcome to my dyslexic brain! As an adult, I’m better at guessing what’s intended. But my dyslexic middle-school brain would have twisted in knots. It’s discouraging, exhausting, and results in inaccurate math assessment.
Word problems help build and assess a student’s ability to model real-world scenarios mathematically. But, especially for dyslexic students, succinctness and clarity matter a lot. Without them, a math word problem becomes an exercise in reading comprehension and guessing intent. Clear and concise communication is harder than it appears, but worth the effort. If I’ve guessed the intended meaning of the above word problem correctly, it seems there’s little, if anything, lost and a lot gained by editing, shortening, and reformatting it. Perhaps something like …
At Carl’s Creamery, what’s the greatest number of ice cream cones you can buy today if:
o They normally cost $2.00.
o Today there’s a 43% discount.
o There’s a 7% sales tax.
o You have $52.00 to spend?
Just as improving reading instruction for struggling readers went mainstream because it also worked for other children, improving the clarity and conciseness of word problems will likely benefit all young math students.
Marty Epstein, PhD, MBA
Note: The edited word problem above is 35 words, the original that follows is 55 words: “Shopping at the new store in town includes a 43% discount on all items which are priced the same at $2. The state you live in has a 7% sales tax. You want to buy many things but only have a total of $52 to spend. Describe in words how many things you could buy.”