Word problems are often dreaded by students. To try to help the grandmother who wrote about her frustration with helping her fourth-grade granddaughter with word problems, I replied with the following message:

*Instead of focusing on traditional word problems at this time, which seem to be dreaded by both of you, maybe it would be better to focus on math questions in a more playful way. Try posing problems that have more than one right answer, more in the spirit of math riddles. Here’s an example of a traditional word problem that a fourth grader would be expected to solve, followed by a problem that’s more like a riddle:*

** Traditional Word Problem:** Billy had $163 and spent some. He has $15 left. How much money did Billy spend?

** A Different Word Problem:** The answer to a subtraction problem is 15. What might the problem be?

*What interests me when I pose a problem such as the second one is the size of the numbers students choose. That can provide information about their level of numerical comfort. Also, then I can ask them to write a word-problem situation that would match the math problem they created. And, as always, I’m interested in how students solve problems. I think it’s very, very important to encourage students to reason mentally, without paper and pencil, and at all times to explain their thinking.*

Reflections on My Message

This one activity probably won’t be sufficient for helping the fourth grader. Students need a great deal of experience with interpreting situations, connecting them to the appropriate mathematical operations, and finding and justifying solutions. But it can shift the focus from being right or wrong to connecting situations to math problems. (See Classroom Suggestions below for instructional suggestions.)

** A caution**: Teachers often resort to helping children solve textbook problems by providing word cues. For example, teachers might tell their students that if a problem says “altogether,” they are supposed to add, or if a problem says “how many more,” they should subtract. Teachers provide these cues to help students be more successful with word problems. However, instruction of this type does little more than offer students tricks for figuring out answers. The cues focus on doing something with the numbers at hand, instead of making sense of the situation and modeling it mathematically.

Watch a Short Video

Here’s a typical word problem that’s useful for seeing if students can interpret a remainder and give an answer that correctly responds to the situation.

There are 295 students.

Each school bus holds 25 students.

How many buses are needed to fit all of the students?

For a revealing and vivid example of how word cues can interfere, watch this one-minute video clip of Mallika Scott asking Marisa, a fifth-grader, to solve the problem. inappropriately using word cues.

*Note*: This video is from the Math Reasoning Inventory (MRI), a free online assessment tool for learning about students’ numerical reasoning, appropriate for students in grade 5 and up. I’ve found MRI to provide insights I never had access before about students’ numerical understanding and reasoning ability. The video clip of Marisa is part of the MRI Video Library of almost 100 clips that you can search by students or by problem. For more information about MRI, see my earlier blog post: We Ask, We Listen, We Learn.

**Classroom Suggestions
**

Following are ideas about how to approach word problem instruction with students.

- Give students problems to solve that have more than one right answer, such as the one suggested above:
*The answer to a subtraction problem is 15. What might the problem be?*For other problems like this one, my two go-to resources are both titled*Good Questions for Math Teaching: Why Ask Them and What to Ask*. One book is for grades K–6, written by Peter Sullivan and Pat Lilburn, and the other is for grades 5–8, written

- Another way that I’ve had success is to present word problems in a different way. For example, using the traditional word problem I described earlier, I would write on the board:

*Billy had $____.*

*He spent $____.*

*Billy has $____ left. *

I’d read aloud what I’d written: “Billy had ‘blank’ dollars. He spent ‘blank’ dollars. Billy has ‘blank’ dollars left.” Then I would have the students read it aloud with me. Next I’d insert numbers in two of the blanks. For example, for the problem I presented above, I would write:

*Billy had $163.*

*He spent $____.*

*Billy has $15 left. *

Using the think-pair-share instructional strategy, first I would ask students to work individually to solve the problem, then I’d have them talk with their partners, and finally I’d lead a class discussion for them to report their results. As always, I would have students explain how they reasoned. To continue, I’d change the numbers, always filling in two blanks but changing which amount was unknown. And, for independent practice, I sometimes ask students to choose their own numbers. As with having them write subtraction problems that have a specified answer (like 15), this gives me insights into students’ numerical comfort.

This is great, thanks! I have found a lot of success in teaching students the structures of word problems to help them form an approach when solving problems. For example in grades 1 and 2, once they begin to focus on the difference between Result, Change, and Start unknown problems, they begin to pay closer attention to the question. They also become very capable at creating their own word problems of various types. The same has been true with the multiplication and word problem types for grades 3 and 4 and moving in to multistep problems. Your blog is a wonderful resource:).

Thanks! I really appreciate your suggestions on how to guide students to be successful problem solvers. I like the idea of Open Number Lines.

My students dislike word problems immensely, and I know that I wasn’t fond of them either back in my day. I am always looking for ideas to teach how to do them, so I can use all the help I can get. I am sympathetic but I still want them to “try.” They give up easily or see that it is a word problem and say they can’t get it. I used to have those blinders on myself. Keep throwing out ideas; we can use them! Thanks.

I love the way your phrase this! I have read multiple times how detrimental teaching key words can be to students yet it still seems to be a very common practice. Can’t wait to share this!

Also I am looking forward to your presentation at the NCTM conference in Boston next month!

Thank you – reading the blogs, I am having a wonderful time remembering how we learned to think.

I have tried presenting open-ended problems such as the one you stated and struggled with students defaulting to simplistic questions and operations. How do you encourage your students to challenge themselves rather than simply just trying to get it done as quickly as possible?

This is a really important question. In a way, it may call for changing the culture of math learning for students. Too often, most of students’ experiences have been where quick right answers are valued. With Four Strikes and You’re Out, I stop frequently as we play and ask, “What do you know for sure?” Then, the answer requires that they share their thinking, rather than make a beeline for the answer. It takes patience for this sort of change to be made, both from us as teachers and from students who are used to a different kind of feedback.

I agree completely with the detrimental result of teaching word cues as a way to solve a word problem. However, my belief in this has been questioned recently because I am teaching math to an 11 year old Chinese boy who is a beginner in English. What are your thoughts about how to break down a word problem to help him learn the “language” of math as well as mathematical reasoning?

I think your approach would be influenced by whether the boy is more advanced with understanding English or speaking English. Typcially, children learning a new language develop receptive skills before speaking skills. If that’s the case, then it would be important to tell him a story problem and talk with him about it. Does he understand? Can he act it out or model with blocks? Starting with an addition or subtraction problem, where the math won’t be the difficulty, will give you information about whether he can understand and “mathematize” the situation. E.g., 20 students were reading books and 6 students were playing a game. How many more (or less) students were reading than were playing? (Choose numbers that seem appropriate.) Can he give you the answer? Can he write a math problem to solve that gives the answer? After practice with a variety of problems, depending on the math he’s doing, then try giving him an equation: 12 x 9 = ___. Ask him to think of 12 x 9 as the title of a story or the headline of a newspaper article, and to write a short story to match. I’m not sure either idea will work, but they’ve worked for me.

I have a question:

In a multiple choice examination of 25 questions, 4 marks are given for each correct answer and 2 marks are deducted for each wrong answer. 1 mark is deducted for any question which is not attempted. A candidate attempts q questions and get c correct. Write down an expression for the candidate’s total mark in term in terms of q and c.

I was hoping if anyone could help with this question.