Word Problems

In mathematics, word problems are problems in which the background information needed for the solution is presented in text rather than merely in numbers or other mathematical notions. This type of problem is often more challenging to students, as it not only requires them to perform certain mathematical operations but also to think of a way to formulate the way to the solution itself.

Solving a word problem usually requires three steps: first students need to examine the word formulation, then they should find out the underlying mathematical notions and finally, come up with the symbolic equation. Before trying to solve the problem it is important to understand what exactly needs to be done and what they are looking for. Then they need to name things, by putting variables to stand for the unknowns. If it is a geometry problem, they might draw and label the parts of the figure. Once they have worked out the answer, they should try to translate it back into words to see if it makes sense with the problem.

The most common types of word problems are distance problems, age problems, work problems, mixture problems and coin problems. An example of a distance problem: "A car and a bus set out at 2 p.m. from the same point, headed in the same direction. The average speed of the car is 30mph slower than twice the speed of the bus. In two hours, the car is 20 miles ahead of the bus. Find the average speed of the car." This type of problem usually involves three notions: distance, rate and time, with one of the three unknown.

An age problem might take this form: "One-half of Margaret's age two years from now plus one-third of her age three years ago is 20 years. How old is she now?" A common formulation of a work word problem is: "One pipe can fill a pool 1.25 times faster than a second pipe. When both pipes are opened, they fill the pool in five hours. How long would it take to fill the pool if only the slower pipe is used?"

Mixture problems very often deal with percentages or quotas, such as: "How many ounces of pure water must be added to 50 ounces of a 15 percent saline solution to make a saline solution that is 10 percent salt? Coin problems are usually similar: "A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many coins of each kind are there?" In problems like this, students need to consider the value of the coins, their number and the total value of the coins.

A useful strategy in solving word problems is looking for key words that can help students to easily imagine what mathematical operation they need to perform. For example, words like increased by, total of, summed and combined mean that the problem requires addition. Subtraction would usually be signaled by words such as decreased by, minus, difference between, less than or fewer than. The presence of times, multiplied by, product of or increased by a factor of would mean that you need multiplication, whereas out of, ratio of, quotient of and percent would stand for division. Finally, words such as is or are, was or were, gives, yields and sold for mean equation.

The major difficulties for students arise from their inability to comprehend the problem. A very famous study by Hudson (1985) illustrates this. She showed students a picture of five birds and three worms and asked the children how many more birds than worms they could see. Only 17 percent of nursery school children and 64 percent of first grade children gave correct answers. When the children were asked how many of the birds would get no worm, the percentage of correct answers sharply rose to 83 percent and 100 percent respectively. This study clearly demonstrates that even at such a young age, children could grasp the part-whole relations but did not have the knowledge to reshape them into linguistic statements.

Word Problems: Selected full-text books and articles

Thinking through Math Word Problems: Strategies for Intermediate Elementary School Students By Arthur Whimbey; Jack Lochhead; Paula Potter Lawrence Erlbaum Associates, 1990
Assessing Higher Order Thinking in Mathematics By Gerald Kulm American Association for the Advancement of Science, 1990
Librarian’s tip: Chap. 10 "The Assessment of Schema Knowledge for Arithmetic Story Problems: A Cognitive Science Perspective"
Direct Instruction in Math Word Problems: Students with Learning Disabilities By Wilson, Cynthia L.; Sindelar, Paul T Exceptional Children, Vol. 57, No. 6, May 1991
Word Problem-Solving by Students with and without Mild Disabilities By Parmar, Rene S.; Cawley, John F.; Frazita, Richard R Exceptional Children, Vol. 62, No. 5, March-April 1996
Children's Numbers By Catherine Sophian Westview Press, 1996
Librarian’s tip: "Solving Addition and Subtraction Story Problems" begins on p.79
The Nature of Mathematical Thinking By Robert J. Sternberg; Talia Ben-Zeev Lawrence Erlbaum Associates, 1996
Librarian’s tip: Chap. 2 "The Process of Understanding Mathematical Problems"
Beyond Problem Solving and Comprehension: An Exploration of Quantitative Reasoning By Arthur Whimbey; Jack Lochhead Lawrence Erlbaum Associates, 1984
Librarian’s tip: Chap. 3 "Analyzing Word Problems" and Chap. 5 "Ye Olde English Word Problems"
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