Word Problems

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:

• The well-organized and flexible use of both conceptual and procedural knowledge
• Strategies and metacognition
• Positive affect and beliefs (De Corte, Greer, & Verschaffel, 1996; Schoenfeld, 1992)

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.