One of the most common forms of problems in mathematics are word problems. They are essentially mathematical exercises presented in the form of a hypothetical situation that requires an equation to solve. They are usually expressed in a narrative form. I consider this a verbal (literary) form of expressing math (See the blog “Expressing Mathematically”). There are two insights I want to share with you about word problems:
First, when I solve word problems, I am actually executing at most five steps, not necessarily in the order given: (1) I translate the hypothetical situation into a visual expression, like a diagram using geometric concepts, or straight into an equation or expression. (2) I find the values that we are given in the word problem, and (3) I find the variable(s) we need to solve for (the “unknown”) to find the solution to the hypothetical situation. (4) I then decide which concepts and/or equations we need to apply to the hypothetical situation. (5) Finally, I apply the proper skills from our math toolbox to find the solution to the word problem.
Second, a traditional word problem explicitly supplies the students all the necessary information to do steps (1),(2),and (3). When I am presented with a problem in a real-world situation, I usually have to define the situation in a way that I can solve it mathematically and express it as a word problem, or a labeled diagram. Many times, I have to conduct measurements in order to establish the “given” values. Usually, the “unknown(s)” end up being those values I need to know that I cannot measure in a practical or direct way. It usually takes some work to properly define the situation that is usually given in a classroom-type word problem.
Word problems are one of many ways we can test the competency of a math student. They are also effectively used to help the math student achieve mastery in applying certain skills. I understand that it is impractical to consistently present students with real-world situations to solve. However, it is just as important for students to know how to properly generate a problem as it is to solve it.