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## Old Problems with New Questions

by Mary Garner, Kennesaw State University (News Bulletin, May/June 2006)

Word problems have long been part of mathematics curricula. In fact, word problems can be found in the Greek Anthology,compiled around 500 A.D. And it is clear that word problems have endured because, when used properly, they can challenge students and reveal their mathematical understanding to educators.

There is a wonderful example of a typical word problem in the 1994 film Little Big League. In the movie, a young boy is given the following question for homework, “If Sam can paint a house in three hours and Joe can paint the same house in five hours, how many hours will it take for the two of them to paint the house together?” He takes his homework to a ballpark, where an entire baseball team struggles to find the solution, and their answers vary widely. They suggest adding 5 and 3 to get 8 hours, multiplying 5 and 3 to get 15 hours, finding the average of 5 and 3 to get 4 hours, taking the product of 5 and 3 and dividing it by the sum to get 15/8 hours—which is the correct answer—and finally, they suggest buying a house that is already painted.

The scene illustrates a typical strategy that students use to solve word problems—they combine the numbers in some way, often without understanding, to produce another number. Rather than requiring only numerical answers, we should place more emphasis on students representing their problem-solving and reasoning processes in order to increase the likelihood that students develop more profound understandings of mathematics.

I recall that when I first started teaching algebra, I used word problems without really grasping their value as tools for teaching and assessing. I spent many an evening contemplating how I could teach my students the easiest way to answer word problems. Our textbook provided algebraic procedures for solving the problems, and I condensed those procedures into quick and easy routines for my students to memorize. If my students encountered a question like the house-painting word problem, they would be able to identify key words that indicate the appropriate routine to use to find the correct answer. When it was time for me to design assessments, I had my own routine to follow. First, I made sure to use language that I knew students would recognize and second, I kept the test items straightforward (no “trick” questions). My main goal was to help students get the correct answer with as little thought or stress as possible. I see now that I was not engaging them in problem solving or mathematical thinking. And, even though their assessment results were excellent, they probably did not understand the mathematical concepts they should have been learning.

Today I know better. There was nothing wrong with the test items I devised in those early days, but there was much wrong with the questions I asked and the responses that I required. Even simple word problems can be wonderful tools for teaching and assessing, but you need to ask students to provide more than just the final answer if you are going to use them effectively. For instance, deeper mathematical understanding is required of students when you ask them to offer different ways to visualize or solve word problems, to discuss the limitations of the solution in the real world, or to answer “what if” questions (what if Joe starts ahead of Sam? what if some preparation time is added to the total? what if Sam’s rate varies from day to day? what if three or four people are working instead of just two?).

Consider the answers you would receive if you were to ask students to create a visualization of the house-painting problem.

One solution might be a graph with lines that show how much Joe completes in x hours, how much Sam completes in x hours, and how much they complete together in x hours. The slopes would represent their rates and the slope of the sum would represent their rate together.

Another possible solution might include boxes and fractions.

The unpainted house:

The house after 1 hour of painting:
 =Portion painted by Sam =Portion painted by Joe
After 1 hour, Sam and Joe could finish 8/15 of a house.

The house after 2 hours:

They would definitely finish the house in less than 2 hours, but how much less? Eight pieces of the house correspond to 1 hour. We need only 7/8 of the next hour to complete the house, so we need 1 and 7/8 hours to paint the house.

These examples show how asking for more than the final answer can turn a stale word problem into an item that can reveal students’ understanding of lines, graphs, fractions, and rates, as well as their skill in algebraic manipulation. Just because word problems have been a part of mathematics curricula for a long time doesn’t mean they are old, out of date, or lacking in depth. They can be challenging, can include rich mathematics, and can be used to learn a great deal about students’ mathematical skills and understanding—if only we pair them with new questions.