# Word Problems

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Word problems are described as "verbal descriptions of problem situations wherein one or more questions are raised the answer to which can be obtained by the application of mathematical operations to numerical data available in the problem statement" (Verschaffel, Greer, & De Corte, 2000). Solving word problems involves:

## Is Solving Word Problems the Same as Mathematical Modeling?

Solving word problems is not considered to be the same as mathematical modeling. Mathematical modeling tends to be a more complex process involving identifying questions to answer about the real world, making assumptions, identifying variables, translating a phenomenon into a mathematical model, assessing the solution, and iterating on the process to refine and extend the model (COMAP & SIAM, 2016). The process to solve a word problem isn't necessarily as complex, as the problem itself usually gives the reader the question to answer and the information necessary to answer it, and doesn’t require modeling's level of meaning-making and interpretation. These differences are relative, however, depending on the abilities of the student and the nature of the solution required to answer the problem.

## Understanding the Challenge

### What Makes Word Problems Difficult for Students?

Students' primary difficulty in solving word problems is attributed to their "suspension of sense-making" (Schoenfeld, 1991; Template:Verschaffel, Greer, & De Corte). Instead of thinking through the context of the word problem to understand it, many students simply seek a simple application of arithmetic needed to produce an answer, whether it makes sense or not. In a video, Kaplinsky (2013) reproduces a result of early 1980s research conducted at the Institut de Recherche sur l'Enseignement des Mathématiques in France.

Math teachers are often concerned about students' abilities to transfer classroom learning into the world beyond the classroom, but this "suspension of sense-making" shows that the reverse is also difficult – students struggle to apply their knowledge and understanding of the world back into a mathematics classroom. Having been conditioned with years of arithmetic, almost always involving obvious operations and the expectation that each problem has a correct answer, students develop a "compulsion to calculate" (Stacey & MacGregor, 1999) that can interfere with the development of the algebraic thinking that is usually needed to solve word problems. Some (but not all) research findings suggest that "compulsion to calculate" worsens as students age and develop beliefs that math is a collection of rules (Radatz, 1983; Stern, 1992, both as cited in Verschaffel, Greer, & De Corte, 2000, p. 5).

Students can also struggle with word problems because they have difficulty with academic vocabulary, mathematical vocabulary, or both. Due to these difficulties, English language learners and students of low socioeconomic status score lower on standardized assessment items than proficient speakers of English (Abedi & Lord, 2001).

### What Makes Word Problems Difficult for Teachers?

Some teachers ignore or struggle to apply their real-world knowledge when solving word problems, just like students (Verschaffel, De Corte, & Borghart, 1997). During instruction, teachers often try to help students "strip away the stuff we don’t really need" (Chapman, 2006, p. 219) and reduce the problem to the numbers and keywords or phrases that indicate operations or relations. This dismissal of the real-world aspects of word problems can contribute to students' suspension of sense-making and their compulsion to calculate.

Most teachers believe or assume that students will have more difficulty solving a word problem than solving an algebraic equation that represents the same mathematics without the words. Because of this, they believe in teaching word problems only after students master solving similar problems as equations. Traditional math textbooks reinforce this belief by placing word problems at the end of practice sets. This belief or assumption has been shown to be false, at least under some conditions. When tested, students have shown that they can be more successful with word or verbal problems than they are with equivalent problems that are purely symbolic (Nathan & Koedinger, 2000a, 2000b). Other research suggests that skill in algorithmic computation may not correspond to students' ability to conceptualize the relationship between numbers in word problems (Fuchs et al., 2006).

## Recommendations

### Use Word Problems to Teach Students Mathematics

Word problems are not just for applications of already-known mathematics. In fact, the most powerful way to use word problems in the classroom is as a means to help students learn math. By situating mathematics in contexts that are understandable for students, word problems encourage students to pursue solution strategies that make sense to them and lead more often to correct answers (Koedinger & Nathan, 2004). These strategies can then be made more formal and symbolic with additional instruction.

This is obvious for teachers of young children. In early mathematics, problems are almost always situated in realistic contexts that children can make sense of. There is no reason that this should end in early childhood. Students at all levels should engage in mathematics in a sensible context before it is made formal and symbolic.

### Engage Student Reasoning

Instead of dismissing the context of word problems, teachers should take time with students to make sense of word problems and their supporting context. Teachers should push back against students' compulsion to calculate by focusing on the relationship between the knowns and unknowns in word problems, and not rush to find an answer (Kieran, 2014). Some types of word problems might be particularly useful for promoting reasoning because they either lack an obvious strategy, don't have one right answer, or could be "tricky" for students who assume the problem is straightforward. Some examples:

1. Pete organized a birthday party for his tenth birthday. He invited 8 boy friends and 4 girl friends. How many friends did Pete invite for his birthday party?
2. Carl has 5 friends and Georges has 6 friends. Carl and Georges decide to give a party together. They invite all their friends. All friends are present. How many friends are there at the party?
3. Kathy, Ingrid, Hans and Tom got from their grandfather a box with 14 chocolate bars, which they shared equally amongst themselves. How many chocolate bars did each grandchild get?
4. Grandfather gives his 4 grandchildren a box containing 18 balloons, which they share equally. How many balloons does each grandchild get?
5. A shopkeeper has two containers for apples. The first container contains 60 apples and the other 90 apples. He puts all the apples into a new, bigger container. How many apples are there in that new container?
6. What will be the temperature of water in a container if you pour 1 jug of water at 80 degrees F and 1 jug of water at 40 degrees F into it? (Nesher, 1980)

Verschaffel, De Corte, and Lasure (1994) used these word problems to see if students would reason differently with the odd- and even-numbered items. Their research and subsequent studies have shown that the vast majority of students – sometimes more than 90 percent – will calculate and produce answers for the even-numbered items just as they do for the odd-numbered items, without any additional reasoning about real-world considerations. Giving students a general warning, such as "these problems are not as easy as they look," did not significantly help students. Instead, teachers can promote student reasoning by providing supports specific to each problem, such as encouraging students to explain their answer and why it makes sense, to draw a picture of their solution, or to consider a hypothetical but contrasting solution from another student. While these strategies can increase the number of students who reason with these problems correctly, in multiple studies they rarely produced correct answers for much more than 50 percent of students (see Chapter 3, Verschaffel, Greer, De Corte). In other words, these strategies are helpful but unlikely by themselves to ensure success for all students.

### Numberless Word Problems

Perhaps the most direct way of pushing back against students' compulsion to calculate is to give them word problems without numbers. Brian Bushart, an elementary teacher and mathematics curriculum coordinator from Texas, popularized the idea of "numberless word problems" after a colleague tried the approach with some third-grade students. Numberless word problems aren’t entirely new, as the book Problems Without Figures (Gillan, 1909) presented something vaguely similar in the early 20th century. Bushart's approach goes much further by focusing on the instructional moves and opportunities for student discourse that century-old approaches did not. Bushart's blog post (2014) and subsequent collection of resources (n.d.) describe both his process for numberless word problems and numerous examples for a range of content and grade levels.

#### Example 1: A Single-Step Word Problem

For an example of a numberless word problem, consider this item released from the Grade 4 PARCC test (PARCC, 2016a):

A pitcher contains 2 liters of juice. A glass is filled with 180 milliliters of juice from the pitcher. How many milliliters of juice are left in the pitcher after filling the glass?

Instead of giving students this problem as written, teachers can present a numberless version of this problem on a series of slides:

Slide 1: A pitcher contains some juice.

Slide 2: A pitcher contains some juice. A glass is filled with some of the juice from the pitcher.

Slide 3: A pitcher contains some juice. A glass is filled with some of the juice from the pitcher. How much juice is left in the pitcher after filling the glass?

Slide 4: A pitcher contains 2 liters of juice. A glass is filled with some of the juice from the pitcher. How much juice is left in the pitcher after filling the glass?

Slide 5: A pitcher contains 2 liters of juice. A glass is filled with 180 milliliters of juice from the pitcher. How much juice is left in the pitcher after filling the glass?

Slide 6: A pitcher contains 2 liters of juice. A glass is filled with 180 milliliters of juice from the pitcher. How many milliliters of juice are left in the pitcher after filling the glass?

Teachers can adjust the number of slides depending on student ability, the difficulty of the problem, and the amount of time the teacher wishes to dedicate to building and sharing student sense-making. In this example, a teacher might stop after Slide 1 to make sure students know what a pitcher is and to have students estimate the capacity of a pitcher. After Slide 2, the teacher can ask students to explain what will happen to the amount of juice in the pitcher, and the relationship between the juice in the pitcher and the juice in the glass. With each subsequent slide, the teacher can continue to probe students’ sense-making and understanding of the relationships between the quantities described by the problem.

#### Example 2: A Multi-Step Word Problem

Teachers can also use numberless word problems with multi-step word problems. Consider this multi-step item released from the Grade 5 PARCC test (PARCC, 2016b):

Dana is making bean soup. The recipe she has makes 10 servings and uses $\displaystyle{ \frac{3}{4} }$ of a pound of beans. How many total pounds of beans does she need to make 5 servings of soup? She has $\displaystyle{ \frac{1}{16} }$ of a pound of beans in one container and $\displaystyle{ \frac{1}{4} }$ of a pound of beans in another container. How many more pounds of beans does Dana need to make 5 servings of soup?

Just as with Example 1, a teacher could present a numberless version of this word problem on a series of slides.

Slide 1: Dana is making bean soup. The recipe she has makes a number of servings and uses an amount beans.

The teacher could use Slide 1 to make sure students understand the basic context. Students may observe that "If she wants to make more soup she'll need more beans," or have other insights that help establish the relationship between the amount of soup and the amount of beans.

Slide 2: Dana is making bean soup. The recipe she has makes a number of servings and uses an amount of beans. What amount of beans does she need to make a smaller number of servings of soup?

With Slide 2, students should observe that less soup should need less beans. Students might begin to conjecture with statements like, "If she wants half as much soup, she'll need half the beans."

Slide 3: Dana is making bean soup. The recipe she has makes a number of servings and uses an amount of beans. What amount of beans does she need to make a smaller number of servings of soup? She already has a small amount of beans. How much more beans does Dana need to make her soup?

After seeing Slide 3, students should now grapple with an even more complex relationship: Not only does Dana need less beans because she's making less soup than called for by the recipe, but she already has some of the beans that she'll need. Student observations and conjectures at this point should suggest two operations, such as a first step involving division to find the amount of beans needed in the reduced recipe, and a second step to subtract the amount of beans Dana already has. With this multi-step complexity, it would be useful for students to draw or otherwise illustrate their thinking.

Slide 4: Dana is making bean soup. The recipe she has makes 10 servings and uses an amount of beans. What amount of beans does she need to make 5 servings of soup? She already has a small amount of beans. How much more beans does Dana need to make 5 servings of soup?

Slide 4 specifies the numbers of servings, but not the beans. With this information, students can revise their observations to make clear that the recipe's amount of beans needs to be divided by 2, with an additional amount subtracted, to find how many more beans Dana needs.

Slide 5: Dana is making bean soup. The recipe she has makes 10 servings and uses $\displaystyle{ \frac{3}{4} }$ of a pound of beans. How many total pounds of beans does she need to make 5 servings of soup? She already has a small amount of beans. How many more pounds of beans does Dana need to make 5 servings of soup?

With Slide 5, students should have already decided that the amount of beans in the recipe needs to be divided by 2, and now they can focus on finding $\displaystyle{ \frac{3}{4} \div 2 }$. Students may also notice that the problem now makes clear that all measurements of beans is in pounds.

Slide 6: Dana is making bean soup. The recipe she has makes 10 servings and uses $\displaystyle{ \frac{3}{4} }$ of a pound of beans. How many total pounds of beans does she need to make 5 servings of soup? She has $\displaystyle{ \frac{1}{16} }$ of a pound of beans in one container and $\displaystyle{ \frac{1}{4} }$ of a pound of beans in another container. How many more pounds of beans does Dana need to make 5 servings of soup?

Slide 6 presents the word problem in its original form, and students may be surprised that the small amount of beans Dana already had is represented as two amounts that need to be considered together. Because of the complexity of this problem, there may not be an ideal time to introduce this information, but a 5th grader who has reasoned through the problem and gotten this far should be able to reason with the two small quantities of beans, either by adding them first and then subtracting the sum from $\displaystyle{ \frac{3}{8} }$ or by performing two subtractions.

Students have shown to be more successful with word problems when they are required to engage with the context in authentic ways. For example, DeFranco and Curcio (1997, as cited by Verschaffel, Greer, De Corte) gave a group of 20 sixth grade students the following word problem: “328 senior citizens are going on a trip. A bus can seat 40 people. How many buses are needed so that all the senior citizens can go on the trip?” Later, the researchers gave the students a similar problem, except they presented it in the form of a fact sheet with the numbers of people and the size of vans, and instructions for making a phone call that simulated placing an actual order for the number of required vehicles. In the first scenario, only 2 of the 20 students answered correctly and properly reasoned with the remainder left by the division. In the second scenario, using a more authentic setting, 16 of the 20 students answered correctly and reasoned appropriately with the remainder, either by rounding up to the next whole van or requesting "like a car or something" to transport the small number of remaining passengers.

Language complexity can be addressed either by reducing the complexity or providing students with additional support. For students who struggle with the language of word problems, it can be helpful to rewrite the problem using simpler, more familiar language, or a student's native language (Bernardo, 1999). These kinds of modifications tend to help English learners and low-SES students more than their English-proficient and higher-SES counterparts, meaning this strategy could help reduce achievement gaps (Abedi & Lord, 2001).

Teachers can also provide extra support. Recommendations for teaching English language learners include focusing on student reasoning and discourse, rather than correctness of language use, and using language learners' knowledge and experiences as resources (Moschkovich, 2012). The Understanding Language website (ell.stanford.edu) is a recommended resource for understanding how to support language learners in mathematics.

## References

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Bernardo, A. B. I. (1999). Overcoming obstacles to understanding and solving word problems in mathematics. Educational Psychology, 19(2), 149–163. https://doi.org/10.1080/0144341990190203

Bushart, B. (n.d.). Numberless word problems. Retrieved November 16, 2017, from https://bstockus.wordpress.com/numberless-word-problems/

Bushart, B. (2014, October 6). Numberless word problems [Blog post]. Retrieved November 16, 2017, from https://bstockus.wordpress.com/2014/10/06/numberless-word-problems/

Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62(2), 211–230. https://doi.org/10.1007/s10649-006-7834-1

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