The Problem with Story Problems

Here’s my story of a problem, and it began with Frank. Pressing the math picture book to his chest, Frank explained how delighted he was to be reunited with this fun, clever text from his childhood. Frank, the leader of the mathematics department for our school district, said he wished we could buy a copy of this sweet book for every teacher in the district. I was intrigued.

As soon as I started The Dot and the Line: A Romance in Lower Mathematics, I felt uneasy. The book—by Norton Juster—favored all the characteristics that Frank himself embodied. The main character was an intelligent male—white, English-speaking, heterosexual, and someone with power—just like Frank. The other two characters were a man of color and a woman—both thinly portrayed and framed as vapid, frivolous, inept, and marginal. The female character was described as physically attractive and her body measurements were presented as a form of mathematics humor. As a female math educator myself, I was disappointed and outraged at the portrayals in this book—and even more disappointed at how dearly Frank loved this text. What did this say to me and about me, and what did it say to and about all the K-12 students in our care?

This got me thinking about the values our math texts promote. Story problems are supposed to be the most humanizing part of math education. Although this is sometimes the case, too often the assumptions inherent in story problems perpetuate consumerism, reinforce racist and sexist stereotypes, and maintain classism and unsustainable approaches to the Earth.

Because I know my insights are limited by my life experiences, my curiosity drove me to start asking others—teachers and future teachers—to share examples of math problems that stood out as damaging or exploitative, or that put forward a worldview that privileged a certain way of thinking or kind of person. I also asked them how they have used these problematic problems to help their students think critically about textbooks and the world.

It turns out I wasn’t alone in my concern about the messages in word problems. We found poisonous examples all over the place, in materials from the elementary level right through calculus. Fortunately, I also learned about inspiring examples of how math teachers are working with students to recognize and subvert biased and negative messages hidden in supposedly neutral material.

Consumerism

One of the most common themes in story problems is consumerism. Countless problems focus on purchasing “stuff,” with the stated goal of acquiring the maximum quantity for the minimum cost. The problems promote the act of shopping, reinforce the ideals of capitalism, and frame readers/students as buyers. Here’s an example from the National Council of Teachers of Mathematics Illuminations website for grades 3-5. As the website explains: “Students participate in an activity in which they develop number sense in and around the shopping mall. They develop their skills in determining percents and estimating area”:

The shopping mall is about as American as baseball and apple pie. Did you know that the United States has more shopping centers than movie theaters? Enclosed malls number more than cities, four-year colleges, or television stations.

Students are then asked to calculate area for parking spaces at a mall, reinforcing the normalization of a car-dependent community engaged in the “all-American” act of consumption. The parking spaces described in the problem are all large enough to accommodate the oversized, fuel-inefficient automobiles common in the United States.

A teacher in Tigard, Oregon, challenged her 4th-grade students to change the values of this problem but keep the math concepts. They decided to calculate how much green space was lost in the creation of each of these parking spaces. This reframing shifted the emphasis from a glorification of parking spaces to a commentary on the exploitation and loss of nature.

Another example of consumerism appears in an extended activity called “Hawaiian Dream Vacation,” in Bridges in Mathematics 2, which includes the following squares on a game board for 2nd graders:

You call home: $8. You check into your condo: $155. You buy a camera: $35. You charter a plane back to Oahu: $78. You rent a beach umbrella for the day: $35.

Rather than have her 2nd-grade students grapple with the unfamiliar and classist concepts set forth in this problem, one teacher modified the problem to be about a “Hawaiian Dream Donation.” Her students decided which organizations they would suggest as recipients of monetary donations, using the same dollar amounts as the items in the original text. Her students enthusiastically generated a list of possible recipients, including food pantries, litter and pollution cleanup projects, and animal rescue organizations.

The relationship between consumerism and middle- or upper-class values is often conveyed through math problems about living spaces. These examples (typically focused on calculating area and/or perimeter) often center on recarpeting, retiling, or repainting rooms, walls, or other surfaces. A typical upper elementary problem is found in Glencoe Pre-Algebra:

Ashley is going to retile a part of a wall in her shower. . . . The area of the square section to be retiled is 36 square feet. If each square tile covers an area of .25 square feet, how many . . . tiles will she need?

Another example, this one for 5th graders, appears in Everyday Mathematics: The University of Chicago School Mathematics Project:

Regina wants to cover one wall of her room with wallpaper. The wall is 9 feet high and 15 feet wide. There is a doorway in the wall that is 3 feet wide and 7 feet tall. How many square feet of wallpaper will she need to buy?

Looking at the wallpapering example with his students, a teacher in Herndon, Virginia, asked his students: “What’s ‘normal’ in this problem?” and “Who has power?” His students identified a number of issues. They questioned whether the wallpaper requires glue that might be dangerous to humans and/or the planet; they asked whether the wallpaper itself exploited natural resources; and they noted that the wall Regina is papering is higher than a typical home, which means she may live in an expensive residence, underscoring her privilege.

What is troubling about problems like this is that “re-anything” (except reduce, reuse, or recycle) implies a rejection of an environmental orientation and promotes the idea that there are fashion trends in home decor. Those who rework parts of their homes are typically homeowners rather than renters, and have the disposable income to support what are often decorative projects. As one teacher explained, “These problems tell me that it’s ‘normal’ to be a homeowner, and I am expected to be constantly striving to ‘improve’ my space in ways that cost money, usually with a focus on someone else’s standard of beauty and not on functionality.” To explore the environmental toxicity involved in activities like hanging wallpaper and installing tile, students could research the ecological and human consequences of the manufacture and installation of these products.

One “Ism” After Another

Many teachers I queried noted that the textbooks are filled with examples that are alien to their students’ lives. The class bias is particularly troubling. For example, here’s a problem from Brooks/Cole’s Precalculus: Mathematics for Calculus, 5th edition:

Craig is saving to buy a vacation home. He inherits some money from a wealthy uncle, then combines this with the $22,000 he has already saved and doubles the total in a lucky investment. He ends up with $134,000, just enough to buy a cabin on the lake. How much did he inherit?

Other examples involve inheriting precious gems, multiple horses in your corral, and arranging parking for your yacht, each framed as normal experience. Some problems focus on ways to invest money to reap the greatest profit, but none explore where the profit actually comes from or at whose expense.

Perhaps most worrisome are the examples that emphasize getting “cheap labor” and calculating ways to pay “the help” as little as possible. Here’s one found by a teacher in Hillsboro, Oregon, at mathhelpforum.com:

An orange grower in California hires migrant workers to pick oranges during the season. He has 12 employees, and each can pick 400 oranges per hour. He has discovered that if he adds more workers, the production per worker decreases due to lack of supervision. When x new workers (above the 12) are hired, each worker picks 400 2×2 oranges per hour.

A problem like this encourages students to consider a grower’s profit as the objective, rather than exploring the dehumanizing conditions of migrant workers. It also reinforces stereotypes, describing migrant workers as being in need of close supervision to work effectively. To shift the context of this problem, the teacher reworked it to highlight the work of the Fair Food Program, which focuses on farm labor exploitation.

Unfortunately, it’s also easy to find problems that hint at race or racialized ways of knowing and being. One common example is problems that focus on meals, like this one from Holt McDougal’s Algebra 1:

ou want to plan a nutritious breakfast. It should supply at least 500 calories or more. Be sure your choices would provide a reasonable breakfast.

Breakfast food	Calories
Plain bagel	195
Cereal, 1 cup	102
Apple juice, 1 glass	123
Tomato juice, 1 glass	41
Egg	75
Milk, 1 cup	150

The instructions, using the words “nutritious” and “reasonable,” assume a collective baseline agreement on what these terms mean. Yet everything on the menu comes from an “all-American” view of breakfast. Common breakfast foods in Latin American, Asian, Middle Eastern, African, or other cultures are not included. In fact, the inclusion of milk ignores the fact that the majority of people on the planet (about 60 percent) are lactose intolerant, and that it is primarily white people (those of European descent) who are able to digest cow’s milk. In addition, what’s emphasized in this problem is not the nutritional content, but the calories associated with each food.

Heteronormativity (the assumption that everyone is heterosexual) is pervasive in word problems, in part due to the profound absence of same-sex relationships. One common example of both heteronormativity and sexism, identified by a middle school teacher in Baltimore, is the use of “As I Was Going to St. Ives,” an 18th-century English nursery rhyme found in multiple textbooks and websites:

As I was going to St. Ives,
I met a man with seven wives.
Every wife had seven sacks,
Every sack had seven cats,
Every cat had seven kits.
Kits, cats, sacks, wives,
How many were going to St. Ives?

To help her students look critically at this old puzzler, the teacher asked: “What stands out to you about this poem and problem?” The students were quick to identify key issues: The poem’s second line is one more example that assumes heterosexuality (and, in fact, polygamy). The protagonist is, once again, the man; the wives are described by their marital status and not their sex or gender. This problem goes a step further to include animal cruelty. A student in the class wondered: “Why is this problem even in our book? What are they trying to teach us?”

Leisure

Many word problems focus on leisure-time activities, often those that cost money and have a negative environmental impact. Common examples include problems like this one found by a teacher in New Mexico, from Big Ideas Math:

It costs $175 to rent a jet ski for 2 hours. It costs $300 to rent a jet ski for 4 hours. Write an equation that represents the cost y (in dollars) of renting a jet ski for x hours.

The teacher, who works in a public school in Albuquerque, explains: “Every single child in my school qualifies for free meals, and their families work harder than anyone else I know. It’s insulting to have a problem where the cost to rent a jet ski for four hours is the same amount someone working full time for minimum wage makes in a week. Before taxes!”

In order to encourage her students to think critically about this problem, she asked them to either calculate the profit the person renting the jet skis would make and discuss whether that is fair or just, or research what calculations would quantify the environmental impact of jet skis on a nearby lake, Elephant Butte.

Other recreational examples focus on bowling, golf, scuba, carriage rides, pilot lessons, music lessons, dance lessons, snowboard lessons, and martial arts lessons. For most children, these require both disposable income and the time to engage in them, and many have a detrimental and unquestioned impact on the environment.

Textbooks and mathematics tests also often include travel-related problems. These are sometimes framed as generic vacations and sometimes described more specifically: skiing in Switzerland, hiking in Ireland, staying at an underwater hotel. Here is a typical example from Harcourt Math (3rd grade):

Two art students are touring Paris. They each buy a one-day museum pass for $14. Each student also buys a ticket to the Eiffel Tower for $11 and a boat ticket for $3. How much do the two students spend altogether? Explain.

What is portrayed as an affordable day in Paris is, in fact, part of an outing in one of the most expensive cities in the world. There is no mention of the additional expenses involved in this problem (airfare, lodging, meals); the problem is presented as if art students touring Paris is a typical experience. The environmental damage caused by air travel is not mentioned, nor is the pollution caused by the boat ride.

Math Is Not Neutral

As mathematics educators, we’ve been encouraged to ignore many things. We’ve been taught that we are exempt from responsibility—that math is value-free.

But clearly story problems convey far more than the opportunity to hone math skills. So what does this mean? At root, we must know our students and create a classroom climate where challenging the status quo is accepted, normal, and encouraged. We can craft classroom communities, as so many educators have done, where it’s expected that students will say, “This problem reinforces exploitation” or “This example normalizes thoughtless destruction of natural resources.” Setting a tone that encourages students to think critically about their textbooks and the content of their courses gives them the power to challenge and reshape story problems, truly humanizing math. Our students know (or can learn) how to call out toxic thinking in our materials, and can generate new and more progressive content with the same rigorous mathematics content. We can draw on the human and Earth-oriented work of Rico Gutstein and Bob Peterson (Rethinking Mathematics: Teaching Social Justice by the Numbers) or Bob Moses (Radical Equations: Civil Rights from Mississippi to the Algebra Project) and consider how we can each extend this foundational work. I invite other educators to join me in thinking about how we can best frame and re-frame the story problems we choose with and for our students.

Endnotes

• Boswell, Laurie, and Ron Larson. 2010. Big Ideas Math. Big Ideas Learning.
• Center for Elementary Mathematics and Science Education: Study Link 9.4. 2015. Everyday Mathematics Resource and Information Center. Retrieved from everydaymath.uchicago.edu.
• Juster, Norton. 1963. The Dot and the Line: A Romance in Lower Mathematics. Random House.
• Larson, Ron. 2010. Algebra 1. Holt McDougal.
• Maletsky, Evan M., et al. 2002. Harcourt Math. Harcourt School Publishers.
• Malloy, Carol. 2003. Glencoe Pre-Algebra. Glencoe-McGraw-Hill.
• National Council of Teachers of Mathematics. Illuminations: Resources for Teaching Math, Grades 3-5. illuminations.nctm.org.
• Snider, Alynn, and Donna Burk. 1999. Blackline Masters for Volumes 1Ð3: Bridges in Mathematics 2. The Math Learning Center.
• Stewart, James, Lothar Redlin, and Saleem Watson. 2005. Precalculus: Mathematics for Calculus (5th Ed). Brooks/Cole.