This is a Think Math! feature or perspective
The joke is that to solve word problems, just look at the numbers. If there are more than two numbers, add. If there are two, and they're far apart, divide. Otherwise, multiply or subtract. How do we help children learn to read, understand, and solve word problems? A key approach is to arrange for them to be producers of the language that they will eventually need to be consumers of. One technique, described here, is to encounter partial word problems -- like the problem with the question-part left out -- recognize what part is missing, and generate the needed extra language and logic, focusing their attention on what that can say or ask, rather than wondering what they should say.
The solution: a developmental approach to word problems and problem-solving
To help children develop mathematical language and to prepare them to be creative problem solvers and good deductive thinkers, Think Math! regularly presents situations about which it is possible to ask many mathematical questions and make many true mathematical statements. This unique feature, the "Headline Stories," provides open-ended situations in which specific mathematical ideas are embedded. It has three main purposes:
- to help students learn to solve word problems by understanding how they are built,
- to develop students' skills at using both natural language and mathematical language to describe ideas drawn from mathematics or the physical world, and
- to help students learn to derive real-world meaning from mathematical statements and derive mathematical meaning from real-world situations.
Both creative thinking and deductive thinking require people to look at a situation and figure out what might follow from it: If I know this is true, then I can be sure that is also true. Like a newspaper headline, Headline Stories give clues about what might follow, but leave out the details. Only sometimes do they pose a specific problem. Envisioning a story and asking the right questions are left to the students. Part of the learning goal for students is to discover what mathematical questions are possible to ask, or answer, about a situation.
Depending on your class -- age, reading and writing ability, independence -- you may leave the problem on the board for them to think about, or even make journal entries about, on their own, but the essential element is a thoughtful mathematical discussion, students talking and thereby developing their ability to articulate and clarify mathematical ideas.
What is a Headline Story?
Headline Stories can take various forms, but can often be compared to word problems without the question.
Sometimes, a Headline Story is the setting for a problem without the actual problem -- a set of information presented in words or pictures or, occasionally, mathematical notation, challenging students to figure out what mathematical problems can reasonably be posed from the given information.
Example: There are about 235 million egg-laying hens in the United States. A laying hen produces roughly 250 to 300 eggs per year. In 2005, the population of the United States was estimated at about 295 million people. Pose some mathematical questions that could be answered using this information.
At other times Headline Stories may present a specific question, but too much (or not quite enough) information to answer that question, again requiring students to think about what can and can't be answered, and to pose sub-problems to help them solve the problem. For example, when information is missing, they might think about how to find or estimate the missing information or they might decide to give an answer, perhaps in table form, without nailing down that missing information.
Example: Emilio bought oranges, apples, and grapes for a school party. The oranges and grapes cost the same amount. If you were told the cost of the apples, and the total cost, how could you figure out the cost of oranges?
A well-crafted Headline Story embeds specific mathematical ideas into the setting, and yet provides the open-endedness of mathematical problem-posing that allows all students to participate in their way and at their level.
What do students learn from regular discussion of Headline Stories?
- Students practice translating among words, pictures, and mathematical symbols. (See also Translating among representations.)
- Students describe real world situations with these various languages.
- Students derive meaning from mathematical statements.
- Students pose mathematical problems.
- Students learn to solve word problems by creating word problems of various types.
- Students expand their repertoire of ways to look at things mathematically.
Teaching with a Headline Story
How often? How much time? When?
Every day! Brief but regular practice is the best way to develop skill at creating problems based on these headline situations, and to develop the language and discourse skills to hold a mathematical discussion.
Setting aside a special time of day—10 minutes at morning meeting, after snack, just after lunch, or some other regular time—for this every day exercise helps ensure students get the needed practice.Teaching idea: integrating Headline Stories with ELA. (Click on the notes to read the reasons for each recommendation.) Occasionally -- not too often -- hand each child a 3x5 card at the end of a Headline Story discussion, and give students an opportunity to write some part of the discussion, perhaps a story problem that they thought up based on the discussion. Post a small number of these in a special place, reserved for "Some excellent problems."
Presenting a Headline Story
To present a Headline Story, you might read it to the class or (if appropriate) write it on the board for students to read themselves. Ask the students to think about the situation it presents and to share their thoughts.
Liam earned $1.00 on Monday, $2.00 on Tuesday, $4.00 on Wednesday, $8.00 on Thursday,
The first challenge that students face in this example is that there doesn't seem to be a problem! This is just a statement; no question has been asked. At the beginning of each year, there will be children who have never encountered this kind of problem before. Even those who are familiar with Headline Stories from earlier grades may need a reminder of what to do.Anchor
With each headline story, the teacher guide presents statements like those that students might make that represent mathematical responses to the headline. To make these responses understandable in print, the language is often more complete than what students actually say. For the headline story given above, sample responses might include:
Each day he earned more than the day before.
By Thursday, Liam earned $15. 1 + 2 + 4 + 8 = 15.
I think he'll earn $16 on Friday.
If this pattern keeps up, Liam is going to be verrrry rich!
How much did he earn so far?
If the pattern keeps up, how much will he earn next Monday?
But maybe Liam doesn't work on weekends? I still don't think the pattern can continue.
Mathematical relevance vs. social relevance: Of course, students say other things, too. "I get an allowance" is a perfectly sensible social response in this situation, but it is not as mathematically relevant as the responses shown above. Young children do not automatically distinguish those two kinds of relevance. At first, you may need to accept relevant but not-very-mathematical ideas to help children feel free to express their thinking at all, but your goal is to sharpen, extend, and focus those ideas, helping your children develop (and recognize) mathematical ways of looking at the situations. It takes time.
Posing problems, solving problems, and clarifying problems: Asking "How much did he earn so far?" is like inventing a word problem. Stating that "by Thursday, Liam earned $15" is like answering a word problem. Both are important. The comments about whether the pattern can continue, and even about whether Liam works on the weekend, are also mathematically relevant, even though they involve no numbers and are pure opinion. These are attempts to make sense out of the numbers and pattern.
Different forms of responses: The sample responses for this story included a table. Learning to read and write tables is an important general skill, and the underlying skills—gaining information from the position of some item in two dimensions (horizontal and vertical position on a grid, or row and column in an array)—are often used in mathematics. Sometimes, to let children know that a table is an acceptable kind of response, a Headline Story might explicitly request one. More often, the form of the response is left to the student.
Managing the discussion
Try recording the most mathematically relevant responses so the class can see them (even if your students are not yet reading well). In doing so, you can call special attention to them while accepting a wide range of other responses.
When you think your students are ready, you may even directly ask questions such as:
- Can you think of a good problem to pose?
- Can you make a prediction?
- What can you figure out from this situation?
Criteria for good responses
Headline Stories vary greatly, so what a student can say in response will vary, too. In general, the best responses respond to the mathematical elements of the story.
Good mathematical responses might:
- describe a pattern implied by the information ("It looks like he doubles each day"),
- pose a question or new problem around the information given ("If this pattern continues, how much will he earn next Monday?" or "How long will it take before he has earned a total of $100"),
- make a prediction that seems relevant and likely ("I think he'll earn $16 on Friday"), or
- describe a result that can be derived logically ("By Thursday, he earned $15"),
- clarify or extend the situation (such as the table),
- question or clarify meaning in the problem, and attempt to make sense ("maybe Liam doesn't work on weekends").
Inventing appropriate mathematical questions is an especially important skill. (Click on the note to learn why.)
Though good responses focus on the mathematics, it can be appropriate, at times, for students to look at social and practical elements of the story as well. For example, with any pattern connected with the physical or social world, it is appropriate to ask:
- Can the pattern continue or must it eventually change or be limited in some way? (Liam's earnings must eventually limit, as it will not be too long before he has all the money in the world!),
- Does the pattern have a natural cause, or is it just an agreed-upon rule? (Liam's earnings are presumably an agreement; the rise and fall of day/night temperatures have a natural cause.)
How can I help students who don’t know where to start?
At the beginning of each year, you may need to help with one or two specific questions, depending on the nature of the Headline Story:
- What can you figure out from this information?
- What can you figure out about the situation?
- What questions can you ask?
- What problems can you pose?
- What additional information do you need?
- What information do you not need?
These questions are deliberately broad, not like "How much did Liam make altogether on Monday and Tuesday?" Their purpose is to get students thinking about what questions or ideas they can find in the problem situation. Over time, they will become more independent, and will not need prompts to get started.
How can I help students move beyond superficial answers or approaches?
One goal of Headline Stories is for students to learn to translate situations and information into problems that can be answered mathematically. The "sample responses" are deliberately broader than what a class is likely to produce in any session, just to suggest the kind of variety you might find. Over the course of the year, you want to help students learn to give more varied responses and, eventually, produce this kind of variety. With each Headline Story, you might lead to one novel response by asking one new question explicitly, perhaps based on a "sample response" that you feel your students are ready for. Use the sample responses to give you ideas about approaches to the problem. Some Headline Stories will give you special suggestions Here are four strategies you may use with Headline Stories, at your discretion, to expand children's thinking. You can use these for Think Math! problems or other open-ended problems that you might make up. (See Challenging and extending for ideas about making up good open-ended problems.)
1. Sometimes, if students are not sure where to start, it helps for them to review what they do know and what they don't know.
Debbie drew four coins out of a jar that contained only nickels and quarters. What can you figure out?
- They know that Debbie took coins from a jar; they know how many; and they know that the coins can only be nickels and/or quarters.
- They don't know what coins Debbie actually took. What's possible? What's impossible?
2. After figuring out what information they have, it can be useful to see if information is missing. Is there information we don't need? Can we fill in missing information with reasonable guesses? How can we figure out what is a "reasonable" guess?
Ra'anan is in charge of bringing pretzels, chips, and juice for the class party. The pretzels are $1.29, the chips are $2.49, and cookies are $3.19. He figured he'd need 3 quart-containers of orange juice. How much will he need to spend?
We can ignore the cookies. Pretzels and chips, together, cost 2¢ less than $3.80, but we don't know how much a quart of orange juice costs. A quart of juice probably costs more than a bag of pretzels, but probably less than $5 (or we can't afford it!). Ra'anan will probably need more than $7 and certainly less than $20.
3. Pose additional mathematical questions that relate to the context, or encourage students to pose those additional questions. Questions that look at the limits—what's the most/least—are especially useful questions for opening up discussion.
What might Debbie's four coins be? or What is the greatest (smallest) amount Debbie could get? or Can we find all the possibilities?
What is the most that Ra'anan is likely to need? What is the least?
Some questions are not mathematical: why might Debbie have wanted those coins? Why didn't Ra'anan buy the cookies, too? These may be important real-life questions in some situations, but they are not mathematical questions. Certain questions are especially mathematical. What makes a question sensible depends on the context.
4. Set limits—the most/least—that haven't been specified. Ask: How can I narrow down the problem?
The Ice Cream Counter has six flavors of ice cream today: chocolate, vanilla, mint chip, mocha, strawberry, and blueberry. They'll serve it in a cup, a sugar cone, or a cake cone. Sorry, no toppings today! What can you figure out?
Asking about most or least seems to make no sense in this context: it is not something one can "figure out" from the information. In fact it is hard to figure out anything without focusing the problem in some way: "suppose we bought just one scoop" or "suppose there is a limit of two scoops" or "let's start by assuming people buy only one flavor of ice-cream, but can get one, two, or three scoops." One reason that we present situations that require more constraints (like the number of scoops) and pose the general question "what can you figure out?" is that the real problem we want students to encounter is how to narrow, simplify, and focus a mathematical situation.
After setting some limit on the problem—like "one scoop per person, today"—here are some of the most common and important kinds of mathematical questions. They also make sense for the problem about Debbie and the coins.
- Is there any solution? What might a solution look like? (We can buy a mocha-chip cake-cone at the Ice Cream Counter. Debbie might have taken two quarters and two nickels.)
- Is this the only solution? (more than one possibility in the Ice Cream Counter; more than one possible collection of coins for Debbie)
- How many solutions are there? (How many different sets of four coins could Debbie take? How many different purchases at the Ice Cream Counter?)
- What is the maximum/minimum? (How much/little might Debbie have taken? This question makes no obvious sense in the ice cream example.)
- Could we solve this problem with less information? (This is best for puzzles with several clues. Are all the clues needed?)
- What if we change one number in the problem? (What if the Ice Cream Counter had only 5 flavors? What if Debbie had taken 5 coins? How would that change the problem?)
Over time, you will discover other ways to help students generate more focused mathematical questions and more varied and creative mathematical answers.
Challenging and extending
I’ve got 9¢ in my hand.
This Headline Story seems to provide too little information, but even such meager information can lead to good ideas and questions. Presenting the question concretely -- with your closed fist in the air holding some coins -- helps children think about the coins themselves, and not just their total value. That can lead to guesses about what coins might be in your hand, what coins must be in your hand, how many coins must be in your hand (at least, and at most), and so on.
Developing students' sense for word problems involves getting them to see the variety of questions that can be asked about a given situation. One creative strategy in situations like this is to add information (for example, "If I had one more penny…") or invent problems ("What coins might I have?" or "Can it be shared among two people?" or "If I bought gum for 5¢…").
What do we know for sure? What can logically be deduced from the information? To help children realize what they can deduce from so little information, you might introduce this story by concealing some coins in your hand, saying "In my hand I have 9¢," and asking "Can you be sure what coins I am holding?" (No; you could have 9 pennies or you could have a nickel and 4 pennies.)
You might then ask, "Can you say anything for sure about the coins I'm holding?" The answer to this second question is "yes," but if your children have not played this kind of game before, you might help them get started by asking some more leading questions, like these:
- Can you say for sure whether or not I have any dimes? (Yes. If you have 9¢, then you can't have any dimes.)
- Can you say for sure that I have at least 5 coins? (Yes. A nickel and four pennies is the least number of coins you could have.)
- Can you say for sure that I do not have more than 9 coins? (Yes. If you had more than 9 coins, then even if they were pennies you'd have more than 9¢.)
Note that this game of If… then… encourages precise expression! Children often have a hard time saying what they mean, and this is an opportunity for you to help them refine both their thinking and their verbal skills.
- Can you say for sure that I have at least one nickel? (No.)
- Can you say for sure that I have at least 5 coins? (Yes.)
Change the coins in your hand: "Now, the coins I'm holding are worth 13¢ altogether. What can you say for sure about these coins?" Some acceptable answers are:
- You have at least 3 pennies.
- You have at least 4 coins.
- You do not have more than 13 coins.
- If you have 5 coins, they are 2 nickels and 3 pennies.
- If you have more than 9 coins then you have 13 pennies.
- If you have 4 pennies then you have at least 8 pennies.
There is another kind of deduction children can make in a situation like this. Without telling the children what coins you actually have in your hand, change the question: "Tell me some combinations of coins I could have." After a few have been suggested, ask: "How many different combinations of coins are there that are worth 13¢?" Students can make a chart, and conclude that there are exactly four possible combinations.
How to invent open-ended problem solving experiences for special purposes
Any word problem can be transformed into a headline story. Here are five suggestions for modifying your favorite word problems. The suggestions are illustrated with a problem from first grade, but word problems from any grade can be modified in the same ways.
Shira had a bag with 18 candies in it. She took four candies out of her bag, and then decided to put two of them back. How many candies are in the bag now?
1. Present all the information, but drop out the question and ask kids what they can figure out from the given information.
Shira had a bag with 18 candies in it. She took four candies out of her bag, and then decided to put two of them back. What can you figure out from this information?
Teacher note: If children need help getting started, they might try acting this out with counters and a bag or box.
Sample responses: Two candies are still left out of the bag. Sixteen candies are in the bag. The bag has less than it had before. …
2. Drop out one piece of information and ask kids what else they need to know to solve the problem.
Shira had a bag of candies. She took four out, and then decided to put two back. How many candies are in the bag now?
Teacher note: If children don't spontaneously bring it up, encourage them to think about what other information they would need to have to solve this problem.
3. Present the problem with some extraneous information that looks like it might be relevant but is not, and ask what they do not need to know to solve the problem.
Shira had a bag with 18 candies, and Leah's bag had 6. Shira took four candies out of her bag, but then put two back. How many candies are in Shira's bag now?
Teacher note: Beginning problem solvers are sometimes distracted by extraneous information. If they try using all the information, for example, adding Leah's 6 candies, suggest that they review what the question is, and then say what information they need and what information they don't need, to answer that question. (See also "Right Answer, Wrong Question!")
4. Combining ideas 1 and 3 (harder): Drop the specific question and insert extra information that allows more than one thing to be figured out. Then ask kids what they can figure out from the given information.
Shira had a bag with 18 candies, and Leah's bag had 6. Shira took four candies out of her bag, but then put two back. What can you figure out from this information?
Teacher note: There are many questions that children can answer. Encourage them to voice the questions as well as the answers.
Sample responses: How many candies does Shira have in her bag now? Who has more candy? How many more candies does she have? How much candy do they have together? Can they share equally what Shira took out?
5. Variant on idea 2 (harder): Present the problem without one or more of the numbers and ask children to describe how they would solve the problem if they had the numbers.
Shira had a candy bag. She took 4 candies out, but then put 2 back. If you knew how many used to be in the bag, explain how to figure out how many are there now?
Teacher note: If children need help getting started, they might try acting this out with counters and a bag or box.
One sophisticated response: If she took 4 out and then put 2 back, the bag has two less now than it did before. So, if I knew how many it had to start with, I could subtract 2.
↑ Writing adds difficulty and time. So that Headline Stories remain an opportunity to enjoy being mathematically creative and to have a mathematical discussion, it is important that they don't become too "heavy" and get associated with long and difficult sessions.
↑ 3x5 cards are small, so they "contain" the task and keep it from feeling too burdensome. They are easy to collect and post, too. Occasionally, a child might request a second card -- perhaps to make room for a picture, or to rewrite a neater version, or just to give a second contribution, or (rarely) because they have a really good idea that just doesn't fit such small space -- but the idea is to limit the space.
↑ By not posting all cards, those students whose cards are posted feel special. It is, of course, valuable to share this honor (over time) with all children, but you can still keep it feeling special by not making it an automatic turn-taking, but by selecting contributions each time, that are "interesting" in some way.
↑ Children's strengths differ, and even the same child varies over time. To include all children, you might vary your criteria from time to time, or even from child to child within a single posting. An "excellent" problem might be mathematically interesting (a multi-step problem, or...). Or the story aspect or context might be creative or amusing. Or the writing might be particularly articulate. Or the handwriting or a picture that goes with the story might be especially nice. Or -- and this is important too -- to reward a reluctant or rare contributor for having taken the risk!
↑ A common error in solving word problems is to scan the numbers, and guess what to do based on the numbers without actually understanding the question. When children see that many different questions can be asked about the same data, they are less prone to this error. For a much simpler example, the headline story "Jeremy has 3 dimes and Jason has 7 dimes" can lead to many questions:
- Who has more?
- How many more dimes?
- How much more money?
- If they combine their money, how much do they have?
- Is it possible to share this amount evenly between the two of them?