The Problem with Word Problems

By Matt McLeod

Word problems can be confusing. Period. They can be set in an unfamiliar context. They can include superfluous information. They can contain ambiguous language. They can be written to trick or confuse. They can be overly complex. Here’s an example that meets a few of these criteria:

The pet store sells crickets for lizards. They charge $3.65 per two-dozen crickets, but right now they are offering a 15% discount on any purchase over $40. What will be the total cost for 276 crickets if there is a 6% sales tax?

How many of your students visit pet stores? How many buy crickets or even know what crickets are and that lizards eat them? If crickets are sold by the “two-dozen” why would you ask for 276 of them, especially since that’s an odd number of dozens? If you are allowed to buy 11.5 two-dozen crickets (aka 276), the total is $41.975. Is this number rounded to cents before or after you take the discount and add the sales tax? Why would you need so many crickets?  How many of your students would look at this problem and say, “I don’t want to read all of that” or “Woah! There’s too much going on in that problem”?

One of the constructs within the Neurodevelopmental Framework that we explore in Math for All is language. Language plays many roles in the mathematics classroom, not the least of which is reading and understanding word problems. For a student with language challenges, there are several elements of word problems that can cause frustration. Decoding the words, understanding words with several meanings, remembering vocabulary, and identifying necessary versus unnecessary information are just some of the components that can create barriers to a student understanding a problem. And all of these barriers come into play before a learner can even decide on which calculations are important and then do any of them.

Word problems are a great way to get students into and better at problem solving, which should be something we strive to do in the mathematics classroom. Providing them with experience and guidance in building their skill and confidence in tackling problems is perhaps the most critical thing we do. However, any word problem that we ask students to solve must be carefully vetted. One great way to do this (that Math for All strongly encourages) is to work through the problem yourself to see where students might bump into challenges. Not all word problems have as much room for improvement as the example above, but a vast majority need at least some adaptation to be a good fit for a classroom or a student. Here are some things to ask yourself when considering a word problem for your class:

  • What is the mathematical goal of the problem and does this align with your teaching goals?
  • Is the context familiar to students?
  • Is the language at students’ reading and comprehension levels?
  • Is there too much language?
  • Are the numbers accessible to the students?
  • Are the numbers reasonable in the context?
  • Is there too much going on in the problem?

Another way to help students with word problems and expand their problem-solving skill is to consider when they are presented in the class. Many curricular materials use word problems as a means of applying a particular skill or algorithm after that skill has been learned and practiced ad nauseam through numeric problems. What if the order is reversed? In an issue of the Alexandria City Public Schools newsletter, Terri Mazingo presents the idea of teaching “Mathematics Through Problem Solving vs. Mathematics for Problem Solving.” This approach suggests presenting the problem as the starting point instead of the wrap-up, giving students an opportunity to engage in productive struggle, which can be effective in developing positive mindsets and confidence when approaching a new problem or type of problem. In a 2010 article, Nancy Ward from the Clairbourn School synthesizes this idea by saying that “students must change their thinking from ‘What does the teacher want me to do?’ to ‘What can I do?’”

How word problems are presented also matters. Both articles offer suggestions for how to improve students’ confidence and agency in problem solving, including the idea of “headless” and “tailless” word problems. This idea is explained in more detail in a video featuring Jane Kang of Education Development Center, who describes how these types of problems are used in Transition to Algebra. Headless word problems simply ask a question—“How many steps have you taken in your life?”—while tailless problems offer a simple context—“I have 17 cents in my pocket.” Both types of problems change how students engage with the text. A traditional word problem positions students as math consumers simply using what someone else has provided. On the flip side, headless and tailless word problems allow students to be math producers, presenting them with opportunities to contribute (and subsequently answer) their own questions, drawing them into the problem in a different way.

As you carefully consider how to adapt the lesson for your student, it is critical to remember to keep the mathematics rigorous! It is not helpful to students to lower the standards or expectations. Although not a comprehensive list, here are some ways that might be helpful when adapting a problem to better meet the needs of your students.

  • Change the context of the problem. Consider using a context that students are exploring in another subject area, such as science, reading, social studies, etc.
  • Change the language in the problem (remove ambiguities, meet students’ comprehension levels, etc.).
  • Reduce the number of words in the problem.
  • Add images.
  • Offer manipulatives.
  • Offer vocabulary reminders.
  • Consider the option of eliminating the words completely.

Here are a few parting thoughts about building your students’ skills and confidence in word problems.

  • Be less helpful. Give students time and space for productive struggle, letting them be creative in the way they solve a problem.
  • Avoid promoting key words for operations—total means add, less means subtract, etc.—as they are not always accurate and can lead to confusion.
  • 3 Reads is one framework that offers students a way to deepen understanding of the problem before they begin to solve it.
  • Playfulness is an important part of learning (for students and teachers).

And finally, here are a couple more word problems. Try to answer each before you jump to the link for a discussion about it.

Problem 1: There are 49 dogs signed up to compete in a dog show. There are 36 more small dogs than large dogs. How many small dogs are signed up to compete? Here’s the discussion about this problem being presented to a second-grade class for homework.

Problem 2: Some birds were flying and met a bird on their way. The bird greeted them, “Hello, hundred!” They said, “We are not hundred. We need half of us plus you to make us hundred.”  Here’s what others did with it.

The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.

This work is licensed under CC BY-NC-SA 4.0  

Math for All is a professional development program that brings general and special education teachers together to enhance their skills in
planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.

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